Research ArticleDynamics in Braess Paradox with Nonimpulsive Commuters

Arianna Dal Forno1 Ugo Merlone2 and Viktor Avrutin34

1Department of Economics and Statistics ldquoCognetti de Martiisrdquo University of Torino 10153 Torino Italy2Department of Psychology University of Torino 10124 Torino Italy3DESP University of Urbino ldquoCarlo Bordquo 61026 Urbino Italy4IST University of Stuttgart 70550 Stuttgart Germany

Correspondence should be addressed to Ugo Merlone ugomerloneunitoit

Received 16 May 2014 Accepted 27 July 2014

Academic Editor Nikos I Karachalios

Copyright copy 2015 Arianna Dal Forno et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

In Braess paradox the addiction of an extra resource creates a social dilemma in which the individual rationality leads to collectiveirrationality In the literature the dynamics has been analyzedwhen considering impulsive commuters ie those who switch choiceregardless of the actual difference between costs We analyze a dynamical version of the paradox with nonimpulsive commuterswho change road proportionally to the cost difference When only two roads are available we provide a rigorous proof of theexistence of a unique fixed point showing that it is globally attracting even if locally unstable When a new road is added the systembecomes discontinuous and two-dimensionalWe prove that still a unique fixed point exists and its global attractivity is numericallyevidenced also when the fixed point is locally unstable Our analysis adds a new insight in the understanding of dynamics in socialdilemma

1 Introduction

Assume that two different points of a networkmdashan origin anda destinationmdashare connected by two possible roads onlyTheBraess paradox states that under specific conditions addinga third road to the network decreases the efficiency of thenetwork This phenomenon is known in the transportationfield andmore general scientific literature (see [1ndash10]) Braessrsquoparadox occurs because commuters try to minimize theirown travel time ignoring the effect of their decisions on othercommuters on the network As a result the total travel timemay increase following an expansion of the network in facteven if some commuters are better off using the new link theycontribute to increase the congestion for other commuters

The theoretical literature of the Braess paradox is partic-ularly productive especially in transportation communica-tion and computer science (far from being exhaustive seeeg [10 11]) Almost all the existing works have consideredthe basic network similar to the one presented in this paperwith the addition of a single link Notably [12] proposes

a broader class of Braess graphs A measure of the robustnessof the dynamic network considering the influence of theflow on other links when certain component (node or link)is removed can be found in [13] The empirical literatureprovides evidence in support of the paradox For examplein [14] examples are reported that occurred on a modelednetwork of the city ofWinnipeg while [15] focus on a portionof the Boston road network The experimental works areinterested in studying the occurrence of the paradox in acontrolled setting (see eg [16ndash20]) not only in basic butalso in augmented networksThis literature provides evidencein strong support of the paradox in some cases (see [18])while statistically significant but weaker support in someother cases (see [16 17 19 20]) A comparison of public versusprivate monitoring using the same participants is performedin [21] to investigate how the type of monitoring affects routechoice Interest in possible behaviors and composition ofthe population facing the basic network can be found in[22] which analyze the data gathered from the observationof an experiment with human participants codes artificial

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2015 Article ID 345795 12 pageshttpdxdoiorg1011552015345795

2 Discrete Dynamics in Nature and Society

behaviors emerged by mean of grounded theory and usesABM simulations For a review on ABM with special focuson spacial interactions and networks see [23]

The paradox has also been presented in a dynamic gameframework A discrete time dynamic population model withsocial externalities and two available choices is studied in [24]simulating an adaptive adjustment process and in particularwith impulsive commuters in [25]

Following the clinical psychology literature (see [26])impulsive commuters have been introduced in the analysis ofbinary choices with externalities in [25 27] when consideringcommuters whose switching rate only depends on the signof the difference between payoffs no matter how much theydiffer This approach has been used to consider choices bothin small and in large groups in [28] and also when consid-ering the introduction of a third choice which dramaticallychanges the dynamics in terms of complexity (as analyzed in[29])

Analyzing the Braess paradox in terms of a ternary choicegame shows new interesting dynamical characteristics thatare investigated in [30] with a particular interest in thecoexistence of several equilibria Nevertheless as illustratedin [22 31] considering only impulsive commuters is not suffi-cient to describe the dynamics observed in experiments withhuman participants Furthermore as impulsive commutersconsider only the sign and not the size of difference betweencosts the population dynamics does not depend on the costfunctions as long as the indifference point remains the sameIn this paper we consider a different behavior suggested byAmnonRapoport (We are grateful to him for this helpful sug-gestion) and which was used in [22] for artificial commutersAlthough the full analysis of a homogenousmdashyet differentmdashpopulation seems to provide a limited contribution it may bea step forward to analyze the aggregate decision behavior insocial dilemmas In fact in order to analyze heterogeneouspopulations it is important to well understand the dynamicsproperties of component behaviors In particular we considercommuters which are concerned not only about the signdifference in payoffs but also on the relative difference Thiskind of behavior is similar to the one considered in [32]where the propensity to switch choice is modulated by thedifference between payoffs Unlike the impulsive adjustmentprocess which makes the commuters change their choice assoon as a better choice occurs regardless of the differencewe consider a decision strategy which prescribes a changeto the best available choice proportionally to the reporteddifference Intuitively this strategy seems to be more robustThe main result of the present work is that it is in fact robustBy contrast to the strategy considered in [30] whichmay leadto a stable cycle of any period the strategy considered in thispaper leads to a globally attracting fixed point Dependingon the parameters this unique fixed point may be locallystable or unstable However the trajectories are convergentin a few steps to the fixed point This result is rigorouslyproved for the case of a binary choice problem For theternary choice problem an analogous result is evidenced aswell

The plan of the work is as follows In Section 2 theformalization of the dynamic model is reported Section 21

includes the detailed description of the case with only tworoads which is represented by a one-dimensional piecewisesmooth continuous map We prove the existence of a uniquefixed point which may be locally stable or unstable in onepartition We prove that in any case it is globally attractingthat is all the trajectories are ultimately converging to thefixed point The case extended to three roads is consideredin Section 3 and it is reduced to a two-dimensional piecewisesmooth discontinuousmapWe prove the existence of a uniquefixed point which may be locally attracting or not We haveonly numerical evidence that also in this more general casethe fixed point is globally attracting This can be rigorouslyproved for the particular cases in which the fixed pointbelongs to the boundaries of the domain of interest The lastsection is devoted to the conclusions

2 The Dynamic Model with Two Roads

The Braess paradox can be illustrated by Figure 1 as followsAssume there is a unitary mass of commuters from start (119878)to end (119864) and there are two roads one passing through left(119871) and the other through right (119877) The cost of each road isgiven by the sumof the time spent along each segment In thisnetwork segments 119878-119877 and 119871-119864 do not depend on traffic andthey cost 119886

119871and 119886119877 respectively On the contrary time spent

along segments 119878-119871 and 119877-119864 is proportional to the numberof travellers and therefore they cost 119887

119871(1 minus 119909) and 119887

119877119909 where

119909 isin [0 1] is the fraction of the population using segment119877-119864At the Nash equilibrium commuters are distributed in such away that both roads (119878-119871-119864 and 119878-119877-119864) have equal cos thatis a fraction (119886

119877+119887119877minus119886119871)(119887119871+119887119877) of commuters will choose

119878-119871-119864 and a fraction (119886119871+ 119887119871minus 119886119877)(119887119871+ 119887119877) will choose 119878-119877-

119864 For example assume as in [22] that 119886 = 119886119871= 119886119877= 27

and 119887 = 119887119871= 119887119877= 24 Then the symmetry of costs makes

the population to split exactly into two equal fractions at theNash equilibrium ((119886

119877+ 119887119877minus 119886119871)(119887119871+ 119887119877) = 12) so that

the travel time for both roads is 24(12) + 27 = 39 Nowwe assume that a very fast road is built connecting 119871 to 119877at cost 119889 = 3 As we will see in Section 3 at the new Nashequilibrium commuters are distributed in such a way that thethree roads (119878-119871-119864 119878-119871-119877-119864 and 119878-119877-119864) have equal cost thatis a fraction (119889+119887

119877minus119886119871)119887119877of commuters will choose 119878-119871-119864

a fraction (119889 + 119887119871minus119886119877)119887119871will choose 119878-119877-119864 and the rest will

choose 119878-119871-119877-119864 In the numerical example this means thatthe entire population will choose 119878-119871-119877-119864 (since in the othertwo roads the fraction of commuter is (119889 + 119887 minus 119886)119887 = 0) witha total travel cost of 119887

119871+119887119877+119889 = 51 According to [19] this is

considered paradoxical as it shows how adding extra capacityto a network can reduce overall performance More correctlyit is just counterintuitive and the mathematical reason isthat there is a distinction between Nash equilibria andoptima

We consider the original network first made of two roadsonly (ie without the link 119871-119877) as in Figure 1(a)

As in [24] this process can be modeled as a discrete timedynamical system considering commuters facing a binarychoice between paths 119878-119871-119864 and 119878-119877-119864 For the sake of brevitythese paths will be denoted as 119871 and 119877 respectively To this

Discrete Dynamics in Nature and Society 3

L R

S

E

aL

aRbLxL

bRxR

(a)

aR

aL

L R

S

E

d

bL(xL+ x

M)

bR(xM

+ xR)

(b)

Figure 1 Basic network with the addition of the link from 119871 to 119877 in (b) and without in (a)

purpose assume the set of commuters is normalized to theinterval [0 1] At each time period 119905 ge 0 119909119877

119905ge 0 indicates the

fraction of commuters choosing 119877 and 119909119871119905ge 0 the fraction of

those choosing 119871 given by 119909119871119905= 1 minus 119909

119877

119905 Therefore the travel

times are given by

119871 = 119886119871+ 119887119871119909119871

119877 = 119886119877+ 119887119877119909119877

(1)

where 119886119871 119886119877 119887119871 119887119877gt 0 and can be written as

119871 (119909119877) = 119886119871+ 119887119871(1 minus 119909

119877)

119877 (119909119877) = 119886119877+ 119887119877119909119877

(2)

This way the travel times on the two roads are equal (andthe commuters are indifferent) when 119871(119909119877) = 119877(119909119877) that iswhen 119909119877 = 119909lowast where

119909lowast=119886119871+ 119887119871minus 119886119877

119887119871+ 119887119877

(3)

We say that the threshold 119909lowast is feasible when 119909lowast isin [0 1]However 119909lowast = 0 (resp 119909lowast = 1) clearly corresponds to thecase in which the whole population chooses action 119871 (resp119877) and we shall see that this is indeed satisfied in the modeldescribed in the next subsection We obtain the followingconstraints on the parameters in order to have 119909lowast isin (0 1)which we assume henceforth

1198961= 119886119871+ 119887119871minus 119886119877gt 0

1198962= 119886119877+ 119887119877minus 119886119871gt 0

(4)

21 The One-Dimensional PWS Map and Its DynamicsCommuters are homogeneous and at each period decidetheir strategy considering the previous period travel costsassuming they will remain the same At time 119905+1 the fraction

119909119877

119905is common knowledge and each commuter observes the

respective travel costs either 119871(119909119877119905) or 119877(119909119877

119905) depending on

their choice The commuters decide their future action attime 119905 + 1 comparing the costs 119871(119909119877

119905) and 119877(119909119877

119905) We assume

the commuters are nonimpulsive and myopically minimizetheir costs That is they change strategy proportionally tothe difference between their chosen road and the best onewhere the best road is the fastest We introduce parameters120575119871isin (0 1) and 120575

119877isin (0 1) as switching rates that is

the propensity for moving either to 119871 or to 119877 respectivelyWith nonimpulsive commuters these rates are adjusted by thenormalized difference of travel costs The difference 119871(119909119877

119905) minus

119877(119909119877

119905) is normalized by the largest value of this difference

denoted by 1198961and obtained in correspondence of 119909119877

119905= 0

Therefore we have 1198961= 119871(0) minus 119877(0) = 119886

119871+ 119887119871minus 119886119877as

already introduced in (4) Thus commuters are switchingroad proportionally to the side of the payoff difference Sim-ilarly we can give a reason with respect to the difference119877(119909119877

119905) minus 119871(119909

119877

119905) to get a normalizing value Now the largest

value is obtained in correspondence of 119909119871119905= 0 that is 119909119877

119905= 1

andwe get 1198962= 119877(1)minus119871(1) = 119886

119877+119887119877minus119886119871as already introduced

in (4)The resulting dynamics is 119909119877

119905+1= 119865(119909

119877

119905) with 119909119877

119905isin [0 1]

and the map 119865 [0 1] rarr [0 1] is defined as

119865 (119909119877)

=

119891119871(119909119877)

= 119909119877+ 120575119877

119871 (119909119877) minus 119877 (119909

119877)

1198961

(1 minus 119909119877)

if 0 le 119909119877 le 119909lowast

119891119877(119909119877)

= 119909119877minus 120575119871

119877 (119909119877) minus 119871 (119909

119877)

1198962

119909119877 if 119909lowast le 119909119877 le 1

(5)

4 Discrete Dynamics in Nature and Society

Substituting from (2) and rearranging we obtain

119865 (119909119877) =

119891119871(119909119877) = 120575119877+ (1 minus 120575

119877minus 120575119877

119887119871+ 119887119877

1198961

)119909119877

+120575119877

119887119871+ 119887119877

1198961

(119909119877)2

if 0 le 119909119877 le 119909lowast

119891119877(119909119877) = (1 + 120575

119871

1198961

1198962

)119909119877

minus120575119871

119887119871+ 119887119877

1198962

(119909119877)2

if 119909lowast le 119909119877 le 1

(6)

From the definition of 119909lowast given in (3) it can also be writtenas

119909lowast=119886119871+ 119887119871minus 119886119877

119887119871+ 119887119877

=1198961

119887119871+ 119887119877

=1198961

1198961+ 1198962

(7)

at which we have 119871(119909lowast) = 119877(119909lowast) it is also 119891119871(119909lowast) = 119891119877(119909lowast)

and thus 119865 is continuous in 119909lowast and it is a fixed point as119865(119909lowast) = 119909lowast We have the following proposition

Proposition 1 Map 119865 is continuous in [0 1] and under theconditions given in (4) 119909lowast = 119896

1(119887119871+ 119887119877) isin (0 1) is the unique

fixed point

Proof Only the uniqueness is left to prove The equation119891119871(119909119877) = 119909

119877 has two solutions 119909119877 = 119909lowast and 119909119877 = 1 whichis larger than 119909lowast and thus a virtual fixed point The equation119891119877(119909119877) = 119909

119877 has two solutions 119909119877 = 119909lowast and 119909119877 = 0 whichis smaller than 119909lowast and thus a virtual fixed point Thus theunique solution is 119909119877 = 119909lowast

Clearly the fixed point may be locally stable or instabledepending on the slopes of the functions on the right and leftsides of 119909lowast However we shall see that in any case it is globallyattractingThat is any ic in the interval [0 1] has a trajectorywhich converges to 119909lowast

Proposition 2 The fixed point 119909lowast = 1198961(119887119871+ 119887119877) isin (0 1) is

globally attracting

Proof Even if 119865(119909119877) is not smooth in its fixed point (asindeed 119909lowast is a kink point of this map) the two components119891119871(119909119877) and 119891

119877(119909119877) are smooth in 119909lowast so that the left and right

side derivatives of 119865(119909119877) in 119909lowast are well defined 1198651015840minus(119909lowast) =

lim119909119877rarr119909lowastminus1198651015840(119909119877) = 1198911015840

119871(119909lowast) and1198651015840

+(119909lowast) = lim

119909119877rarr119909lowast+1198651015840(119909119877) =

1198911015840

119877(119909lowast)

We have

1198911015840

119871(119909119877) = (1 minus 120575

119877minus 120575119877

119887119871+ 119887119877

1198961

) + 2120575119877

119887119871+ 119887119877

1198961

119909119877 (8)

1198911015840

119871(119909lowast) = 1 minus 120575

119877

1198962

1198961

(9)

while

1198911015840

119877(119909119877) = (1 + 120575

119871

1198961

1198962

) minus 2120575119871

119887119871+ 119887119877

1198962

119909119877 (10)

1198911015840

119877(119909lowast) = 1 minus 120575

119871

1198961

1198962

(11)

From

11989110158401015840

119871(119909119877) = 2120575

119877

119887119871+ 119887119877

1198961

gt 0 11989110158401015840

119877(119909119877) = minus2120575

119871

119887119871+ 119887119877

1198962

lt 0

(12)

we have that on the left side of the fixed point the function isconvex while on the right side of the fixed point the functionis concave

Notice that 11989621198961only depends on 119886

119871 119887119871 119886119877 119887119877 and it is

positive under our assumptions Depending on the values ofthe ratio 119896

21198961we can classify the rightleft stability of the

fixed point 119909lowast From (9) we clearly have always 1198911015840119871(119909lowast) lt 1

thus the stabilityinstability on the left side of the fixed pointdepends on the condition 1198911015840

119871(119909lowast) gt minus1 which holds if and

only if

1198962

1198961

lt2

120575119877

(13)

Similarly from (11) we clearly have always 1198911015840119877(119909lowast) lt 1 and

the stabilityinstability on the right side of the fixed pointdepends on the condition 1198911015840

119877(119909lowast) gt minus1 which holds if and

only if

1198962

1198961

gt120575119871

2 (14)

And clearly we have the following sequence of inequalities (tobe used below)

0 lt120575119871

2lt 120575119871lt 1 lt

1

120575119877

lt2

120575119877

(15)

To investigate the local stability of the fixed point we considerthree intervals for 119896

21198961

(i) 0 lt 11989621198961le 1205751198712

(ii) 1205751198712 lt 119896

21198961lt 2120575

119877

(iii) 2120575119877le 11989621198961

In case (i) it is 1198911015840119877(119909lowast) le minus1 moreover it is easy to see

that for any 119909119877 isin [119909lowast 1] it is 1198911015840119877(119909119877) lt 119891

1015840

119877(119909lowast) and thus

the function on the right side of the fixed point is strictlydecreasing and expansive each point on the right side of thefixed point is mapped to the left side of the fixed point in oneiteration While on the left side of the fixed point the convexfunction 119891

119871(119909119877) has 1198911015840

119871(119909lowast) = 1 minus 120575

119877(11989621198961) gt 0 (as this

holds for 11989621198961lt 1120575

119877which is satisfied in this interval)This

implies that the point of local minimum on the left side say119909119877

119897119898 is smaller than the fixed point 119909119877

119897119898lt 119909lowast Then the fixed

point is unique and the map is a contraction in the interval119882119904

loc = [119909119877

119897119898 119909lowast] and an absorbing interval which is mapped

into 119882119904loc in one iteration is given by 119869 = [119909119877119897119898 119891minus1

119877(119909119877

119897119898)]

when 119891119877(1) = 1 minus 120575

119871lt 119909119877

119897119898 or 119869 = [119909119877

119897119898 1] when 119891

119877(1) =

1 minus 120575119871ge 119909119877

119897119898 Any initial condition belonging to [0 1] 119869

has a trajectory which is mapped into 119869 in a finite number ofiterationsThus119909lowast is globally attracting An example is shownin Figure 2(a)

Similarly we can reason when 11989621198961belongs to the third

interval In fact in case (iii) it is 1198911015840119871(119909lowast) le minus1 moreover it is

Discrete Dynamics in Nature and Society 5xR t+1

xR

t

00

025

025 05

05

075

1

1075

(a)xR t+1

xR

t

00

025

025 05

05

075

1

1075

(b)

Figure 2 Qualitative shapes of the map In (a) case (i) when 0 lt 11989621198961le 1205751198712 the fixed point is locally unstable on the right side In (b)

case (iii) when 2120575119877le 11989621198961 the fixed point is locally unstable on the left side

0

1

0 1

xR t

xL

t

(a)

0

1

0 1

xR t

xL

t

(b)

0

1

0 1

xR t

xL

t

(c)

Figure 3 Qualitative shapes of the map in case (ii) when 1205751198712 lt 119896

21198961le 2120575

119877and the fixed point is locally stable on both sides

easy to see that for any 119909119877 isin [0 119909lowast] it is 1198911015840119871(119909lowast) lt 1198911015840

119871(119909lowast) and

thus the function on the left side of the fixed point is strictlydecreasing and expansive each point on the left side of thefixed point is mapped to the right side of the fixed point inone iteration While on the right side of the fixed point theconcave function 119891

119877(119909119877) has 1198911015840

119877(119909lowast) = 1 minus 120575

119871(11989611198962) gt 0 (as

this holds for 11989621198961gt 120575119871which is satisfied in this interval)

This implies that the point of local maximum on the rightside say 119909119877

119903119872 is larger than the fixed point 119909119877

119903119872gt 119909lowast Then

the map is a contraction in the interval 119882119904loc = [119909lowast 119909119877

119903119872]

and an absorbing interval which is mapped into119882119904loc in oneiteration is given by 119869 = [119891minus1

119871(119909119877

119903119872) 119909119877

119903119872] when 119891

119871(0) =

120575119877gt 119909119877

119903119872 or 119869 = [0 119909119877

119903119872] when 119891

119871(0) = 120575

119877le 119909119877

119903119872 Any

initial condition belonging to [0 1] 119869 has a trajectory whichis mapped into 119869 in a finite number of iterations Thus 119909lowast isglobally attracting An example is shown in Figure 2(b)

When 11989621198961belongs to the interval in (ii) 120575

1198712 lt 119896

21198961lt

2120575119877 we have that the fixed point is locally stable on both

sides Moreover inside this interval we can distinguish threeregions as already showed in (15) that is

(a) 1205751198712 lt 119896

21198961le 120575119871with 0 lt 1198911015840

119871(119909lowast) lt 1 and

minus1 lt 1198911015840

119877(119909lowast) lt 0 as in Figure 3(a)

(b) 120575119871lt 11989621198961lt 1120575

119877with 0 lt 1198911015840

119871(119909lowast) lt 1 and

0 lt 1198911015840

119877(119909lowast) lt 1 as in Figure 3(b)

6 Discrete Dynamics in Nature and Society

(c) 1120575119877le 11989621198961lt 2120575

119877with minus1 lt 1198911015840

119871(119909lowast) lt 0 and

0 lt 1198911015840

119877(119909lowast) lt 1 as in Figure 3(c)

In case (a) we have that the point of local minimum onthe left side say 119909119877

119897119898 is smaller than the fixed point119909119877

119897119898lt 119909lowast

Then themap is a contraction in the interval119882119904loc = [119909119877

119897119898 119909lowast]

and on the right side the map is decreasing so that anabsorbing interval which is mapped into119882119904loc in one iterationis given by 119869 = [119909119877

119897119898 119891minus1

119877(119909119877

119897119898)]when119891

119877(1) = 1minus120575

119871lt 119909119877

119897119898 or

119869 = [119909119877

119897119898 1]when119891

119877(1) = 1minus120575

119871ge 119909119877

119897119898 Any initial condition

belonging to [0 1] 119869 has a trajectory which is mapped into 119869in a finite number of iterationsThus 119909lowast is globally attracting

In case (c) we can reason similarly we have that the pointof local maximum on the right side say 119909119877

119903119872 is larger than

the fixed point 119909119877119903119872gt 119909lowast Then the map is a contraction in

the interval 119882119904loc = [119909lowast 119909119877

119903119872] and on the left side the map

is decreasing so that an absorbing interval which is mappedinto 119882119904loc in one iteration is given by 119869 = [119891minus1

119871(119909119877

119903119872) 119909119877

119903119872]

when 119891119871(0) = 120575

119877gt 119909119877

119903119872 or 119869 = [0 119909119877

119903119872] when 119891

119871(0) =

120575119877le 119909119877

119903119872 Any initial condition belonging to [0 1] 119869 has

a trajectory which is mapped into 119869 in a finite number ofiterations Thus 119909lowast is globally attracting

In case (b) we have that the point of local minimum onthe left side say 119909119877

119897119898 is smaller than the fixed point 119909119877

119897119898lt

119909lowast and also that the point of local maximum on the right

side say 119909119877119903119872

is larger than the fixed point 119909119877119903119872gt 119909lowast

Then the map is a contraction in the interval 119882119904loc =

[119909119877

119897119898 119909119877

119903119872] which includes the fixed point and an absorbing

interval which is mapped into119882119904loc in one iteration is givenby 119869 = [119891minus1

119871(119909119877

119903119872) 119891minus1

119877(119909119877

119897119898)] or 119869 = [0 119891minus1

119877(119909119877

119897119898)] or 119869 =

[119891minus1

119871(119909119877

119903119872) 1] or [0 1] depending under obvious conditions

on the values of 120575119871and 120575

119877 When 119869 sub [0 1] then any initial

condition belonging to [0 1] 119869 has a trajectory which ismapped into 119869 in a finite number of iterations Thus 119909lowast isglobally attracting

We notice that in Proposition 2 we prove that when thefixed point 119909lowast is internal to the interval [0 1] then it isglobally attracting Let us turn to comment the two extremawhich occur when 119909lowast = 0 or 119909lowast = 1 These solutions clearlyexist and are also globally attracting In fact the case 119909lowast = 0corresponds to 119896

1= 119871(0)minus119877(0) = 119886

119871+119887119871minus119886119877= 0 and only the

function 119891119877(119909119877) is applied in the interval [0 1] and from the

concavity of the function we have that this fixed point is glob-ally attracting even if the eigenvalue is equal to 1 The othercase 119909lowast = 1 corresponds to 119896

2= 119877(1)minus119871(1) = 119886

119877+119887119877minus119886119871= 0

and only the function 119891119871(119909119877) is applied in the interval [0 1]

and from the convexity of the function we have that this fixedpoint is globally attracting even if the eigenvalue is equal to 1

3 The Dynamics in the ExpandedNetwork with Three Roads

As it is well known the Braess paradox introduces a new linkand shows how this opportunity worsens both the individualand collective payoffs This link introduces a new alternativenow the commuters are facing three choices where the new

one consists of path 119878-119871-119877-119864 (see Figure 1(b)) which will bedenoted as 119872 in the following Models with three choiceswere introduced in [29] as it concerns discrete time and [33]as it concerns continuous time In this paper we analyze thedynamics when the travel time 119889 for the new resulting linkvaries in a range that makes this link a dominant choice forsome values and a dominated choice for others It is worth toobserve that in particular when the travel time on link 119871-119877 iszero the paradoxical result still holds

Now assume that commuters switch roadwhenever traveltime has become larger that is the other road is less costlyand therefore more attractive We indicate by 119909119871 ge 0 thefraction of population choosing action 119871 by 119909119872 ge 0 thefraction choosing action 119872 and by 119909119877 ge 0 the fractionchoosing action 119877 with the constraint 119909119871 + 119909119877 + 119909119872 = 1The travel times are given respectively as

119871 = 119886119871+ 119887119871(119909119871+ 119909119872)

119872 = 119887119871(119909119871+ 119909119872) + 119887119877(119909119872+ 119909119877) + 119889

119877 = 119886119877+ 119887119877(119909119872+ 119909119877)

(16)

By using the constraint let us define the fraction of populationchoosing119872 as 119909119872 = 1 minus 119909119871 minus 119909119877 ge 0 so to consider onlytwo variables x = (119909119871 119909119877) isin R2

+ and the phase space is the

triangle

1198632= x = (119909119871 119909119877) isin R2

+ 0 le 119909

119871+ 119909119877le 1 (17)

After substituting the expression of 119909119872 in the travel times weget

119871 (x) = 119886119871 + 119887119871 (1 minus 119909119877)

119872 (x) = 119887119871 (1 minus 119909119877) + 119887119877(1 minus 119909

119871) + 119889

119877 (x) = 119886119877 + 119887119877 (1 minus 119909119871)

(18)

The vertices of the phase space119875lowast119871(1 0)119875lowast

119872(0 0) and119875lowast

119877(0 1)

represent states in which the whole population choosesrespectively actions 119871119872 and 119877

Since commuters are interested in minimizing traveltime we denote by 119877

119860the region in which the choice 119860

(where119860 is 119871119872 or 119877) is preferable These regions are calleddominance regions of 119860 We have the following definition ofthe dominance regions

119877119871= x isin 1198632 119871 (x) le 119872 (x) 119871 (x) le 119877 (x)

119877119872= x isin 1198632 119872 (x) le 119871 (x) 119872 (x) le 119877 (x)

119877119877= x isin 1198632 119877 (x) le 119871 (x) 119877 (x) le 119872 (x)

(19)

Discrete Dynamics in Nature and Society 7

1

100

xR

RL

RM

RR

xL

(a)

1

100

xR

RL

RM

RR

xL

(b)

1

100

xR

RL

RM

RR

xL

(c)

Figure 4 Dominance regions in the three choices network with different values of the cost parameter 119889 of the map F at 119886119871= 27 119887

119871= 30

119886119877= 18 119887

119877= 30 and (a) 119889 = 6 (b) 119889 = 3 (c) 119889 = 0

By using the expressions in (18) we get the following defini-tions for the regions

119877119871= x isin 1198632 119909119871 le 119909lowast119871

119886119871+ 119887119871(1 minus 119909

119877) le 119886119877+ 119887119877(1 minus 119909

119871)

119877119872= x isin 1198632 119909119871 ge 119909lowast119871 119909119877 ge 119909lowast119877

119877119877= x isin 1198632 119886

119877+ 119887119877(1 minus 119909

119871) le 119886119871+ 119887119871(1 minus 119909

119877)

119909119877le 119909lowast119877

(20)

where we have introduced

119909lowast119871= 1 minus

119886119871minus 119889

119887119877

=119889 minus 119886119871+ 119887119877

119887119877

119909lowast119877= 1 minus

119886119877minus 119889

119887119871

=119889 minus 119886119877+ 119887119871

119887119871

(21)

The boundaries of the three regions are given by segments ofstraight lines of equations

1198811 119909119871= 119909lowast119871

1198812 119909119877= 119909lowast119877

1198813 119886119871+ 119887119871(1 minus 119909

119877) = 119886119877+ 119887119877(1 minus 119909

119871)

(22)

and it is immediate to see that the point xlowast = (119909lowast119871 119909lowast119877)satisfies also the third equation that is it belongs to all thethree straight lines xlowast isin 119881

119894for 119894 = 1 2 3

Using the geometric representation in the triangle 1198632the dominance regions are illustrated in Figure 4 (via anexample) using different colors Three cases are shown fordifferent values of the parameter 119889 in order to illustrate howthe region119877

119872depends on the parameters of the third branch

The point xlowast = (119909lowast119871 119909lowast119877) is the unique point on theboundaries 119881

119894of all the three regions Thus for our model we

have to require 119909lowast119871 ge 0 119909lowast119877 ge 0 and 119909lowast119871 + 119909lowast119877 le 1 leadingto 119909lowast119872 = 1 minus 119909lowast119871 minus 119909lowast119877 In order to have three regions in1198632

it must be 119909lowast119871 gt 0 119909lowast119877 gt 0 and 119909lowast119871 + 119909lowast119877 lt 1 The equalityin one of these three conditions implies the disappearance ofone region For example when 119909lowast119871 = 0 then 0 le 119909lowast119877 le 1and 119909lowast119872 = 1 minus 119909lowast119877 leaving only the two regions 119877

119877and 119877

119872

and so onThe conditions 119909lowast119871 ge 0 119909lowast119877 ge 0 and 119909lowast119871 + 119909lowast119877 le 1 lead

to the constraints that will be assumed satisfied henceforth

0 le119886119871minus 119889

119887119877

le 1 0 le119886119877minus 119889

119887119871

le 1

119886119871minus 119889

119887119877

+119886119877minus 119889

119887119871

ge 1

(23)

We notice that from the third condition in (23) we have thevalues of 119889 for which the third region is really present in 1198632that is

0 le 119889 le119886119871119887119871+ 119886119877119887119877minus 119887119871119887119877

119887119877+ 119887119871

(24)

Commuters are homogeneous and minimize their nextperiod travel cost At time 119905+1 the vector x

119905becomes common

knowledge and each commuter can observe the travel costs119871(x119905)119872(x

119905) and 119877(x

119905) We assume that if at time 119905 a fraction

119909119871

119905chooses path 119871 a fraction 119909119877

119905chooses action 119877 and travel

costs are such that 119877(x119905) lt 119871(x

119905) and 119877(x

119905) lt 119872(x

119905) then

both a fraction of the 119909119871119905commuters who chose action 119871 and

a fraction of the (1 minus 119909119871119905minus 119909119877

119905) commuters who chose action

119872 will switch to path 119877 in the next time period 119905 + 1 Thefraction of switching commuters is given by the switchingpropensities 120575

119871 120575119872 120575119877modulated by the relative differences

in payoffs This happens whenever a path gives a smallertravel cost In other words at any time 119905 all the commutersdecide their future action at time 119905 + 1 comparing the costs119871(x119905)119872(x

119905) and 119877(x

119905) As in the one-dimensional case here

also the differences of travel costs are normalized dividing

8 Discrete Dynamics in Nature and Society

each difference by the constants giving the related maximumvalue given in the different cases as follows

1198961= max

x [119871 (x) minus 119877 (x)]

= [119871 (1 0) minus 119877 (1 0)] = 119886119871 + 119887119871 minus 119886119877

1198962= max

x [119877 (x) minus 119871 (x)]

= [119877 (0 1) minus 119871 (0 1)] = 119886119877 + 119887119877 minus 119886119871

1198963= max

x [119871 (x) minus 119872 (x)]

= [119871 (1 119909119877) minus119872(1 119909

119877)] = 119886

119871minus 119889 forall119909

119877isin [0 1]

1198964= max

x [119872 (x) minus 119871 (x)]

= [119872(0 119909119877) minus 119871 (0 119909

119877)] = 119889 + 119887

119877minus 119886119871 forall119909

119877isin [0 1]

1198965= max

x [119877 (x) minus 119872 (x)]

= [119877 (119909119871 1) minus 119872(119909

119871 1)] = 119886

119877minus 119889 forall119909

119871isin [0 1]

1198966= max

x [119872 (x) minus 119877 (x)]

= [119872(119909119871 0) minus 119877 (119909

119871 0)] = 119889 + 119887

119871minus 119886119877 forall119909

119871isin [0 1]

(25)

The resulting dynamics are described by a two-dimensionalmap x

119905+1= F(x

119905) defined as

F x119905+1=

F119871(x119905) if x

119905isin 119877119871

F119872(x119905) if x

119905isin 119877119872

F119877(x119905) if x

119905isin 119877119877

(26)

where

F119871

119909119871

119905+1= 119909119871

119905+ 120575119871(119877 (x119905) minus 119871 (x

119905)

119886119877+ 119887119877minus 119886119871

119909119877

119905

+119872(x119905) minus 119871 (x

119905)

119889 + 119887119877minus 119886119871

119909119872

119905)

119909119877

119905+1= (1 minus 120575

119871

119877 (x119905) minus 119871 (x

119905)

119886119877+ 119887119877minus 119886119871

)119909119877

119905

F119872

119909119871

119905+1= (1 minus 120575

119872

119871 (x119905) minus 119872(x

119905)

119886119871minus 119889

)119909119871

119905

119909119877

119905+1= (1 minus 120575

119872

119877 (x119905) minus 119872(x

119905)

119886119877minus 119889

)119909119877

119905

F119877

119909119871

119905+1= (1 minus 120575

119877

119871 (x119905) minus 119877 (x

119905)

119886119871+ 119887119871minus 119886119877

)119909119871

119905

119909119877

119905+1= 119909119877

119905+ 120575119877(119871 (x119905) minus 119877 (x

119905)

119886119871+ 119887119871minus 119886119877

119909119871

119905

+119872(x119905) minus 119877 (x

119905)

119889 + 119887119871minus 119886119877

119909119872

119905)

(27)

By construction we have that F maps 1198632 into itself In factto show that F 1198632 rarr 119863

2 it is enough to show that

given (119909119871119905 119909119877

119905) isin 119863

2 then (119909119871119905+1 119909119877

119905+1) = F(119909119871

119905 119909119877

119905) isin 119863

2From (119909119871

119905 119909119877

119905) isin 119863

2 we have that 0 le 119909119871119905+ 119909119877

119905le 1 and

119909119871

119905+ 119909119877

119905+ 119909119872

119905= 1 Then if (119909119871

119905+1 119909119877

119905+1) = F119871(119909119871

119905 119909119877

119905) we have

119909119871

119905+1+ 119909119877

119905+1= 119909119871

119905+ 119909119877

119905+ 120575119871

119872(x119905) minus 119871 (x

119905)

119889 + 119887119877minus 119886119871

119909119872

119905

le 119909119871

119905+ 119909119877

119905+ 119909119872

119905= 1

(28)

119909119871

119905+1ge 0 and 119909119877

119905+1ge 0 thus (119909119871

119905+1 119909119877

119905+1) isin 1198632 If (119909119871

119905+1 119909119877

119905+1) =

F119872(119909119871

119905 119909119877

119905) we have 0 le 119909119871

119905+1le 119909119871

119905and 0 le 119909119877

119905+1le 119909119877

119905 thus

(119909119871

119905+1 119909119877

119905+1) isin 1198632 If (119909119871

119905+1 119909119877

119905+1) = F119877(119909119871

119905 119909119877

119905) we have

119909119871

119905+1+ 119909119877

119905+1= 119909119871

119905+ 119909119877

119905+ 120575119877

119872(x119905) minus 119877 (x

119905)

119889 + 119887119871minus 119886119877

119909119872

119905

le 119909119871

119905+ 119909119877

119905+ 119909119872

119905= 1

(29)

119909119871

119905+1ge 0 and 119909119877

119905+1ge 0 thus (119909119871

119905+1 119909119877

119905+1) isin 1198632

Substituting from (18) we can rewrite the functions inexplicit form as

F119871

119909119871

119905+1= 120575119871+ (1 minus 120575

119871minus 120575119871

119887119877

1198964

)119909119871

119905

minus120575119871

119887119871

1198962

119909119877

119905+ 120575119871

119887119871

1198962

(119909119877

119905)2

+120575119871119887119877(1

1198964

minus1

1198962

)119909119871

119905119909119877

119905+ 120575119871

119887119877

1198964

(119909119871

119905)2

119909119877

119905+1= (1 minus 120575

119871minus 120575119871

119887119871

1198962

)119909119877

119905

+120575119871

119887119877

1198962

119909119871

119905119909119877

119905minus 120575119871

119887119871

1198962

(119909119877

119905)2

F119872

119909119871

119905+1= (1 + 120575

119872

1198964

1198963

)119909119871

119905minus 120575119872

119887119877

1198963

(119909119871

119905)2

119909119877

119905+1= (1 + 120575

119872

1198966

1198965

)119909119877

119905minus 120575119872

119887119871

1198965

(119909119877

119905)2

F119877

119909119871

119905+1= (1 minus 120575

119877minus 120575119877

119887119877

1198961

)119909119871

119905

+120575119877

119887119871

1198961

119909119871

119905119909119877

119905minus 120575119877

119887119877

1198961

(119909119871

119905)2

119909119877

119905+1= 120575119877+ (1 minus 120575

119877minus 120575119877

119887119871

1198966

)119909119877

119905

minus120575119877

119887119877

1198961

119909119871

119905+ 120575119877

119887119871

1198961

(119909119877

119905)2

+120575119877119887119871(1

1198966

minus1

1198961

)119909119871

119905119909119877

119905+ 120575119877

119887119877

1198966

(119909119871

119905)2

(30)

Differently from the one-dimensional map 119865 considered inthe previous section the two-dimensional map F is notcontinuous in 1198632 More precisely the functions F

119871 F119872

and F119877are continuous and thus F is continuous in the

interior of each region inwhich the different definitions applyHowever in general except for the point xlowast the map F is notcontinuous along the borders 119881

119894 119894 = 1 2 3 of the regions

119877119871 119877119872 and 119877

119877 In fact it is easy to see that considering

Discrete Dynamics in Nature and Society 9

xL

t+1

xL

t

xR

t

00

1

1

(a)

xL

t+1

xL

t

xR

t

00

1

1

(b)

Figure 5 Graphics of the three functions defining (a) 119909119871119905+1

and (b) 119909119877119905+1

with parameters 119886119871= 29 119887

119871= 16 119886

119877= 19 119887

119877= 24 119889 = 8 120575

119871= 05

120575119872= 08 and 120575

119877= 06

a point (119909119871 119909119877) belonging to one of the lines defined in (22)at the boundary of two regions only we have that the twofunctions defined on the two different regions lead to twodifferent points of1198632 An example of how the three functionsdefining 119909119871

119905+1behave is shown in Figure 5(a) and the three

functions defining 119909119877119905+1

are illustrated in Figure 5(b) showingthat F is continuous only in xlowast

The point xlowast in which the three regions are in contactand F is continuous is clearly particular In fact solving theequation 119871(x) = 119872(x) we have the unique solution 119909119871 = 119909lowast119871while solving the equation 119877(x) = 119872(x) we have the uniquesolution119909119877 = 119909lowast119877 It follows that at the point xlowast = (119909lowast119871 119909lowast119877)we have the same cost in each choice as 119871(xlowast) = 119872(xlowast) =119877(xlowast) so that at the point xlowast the commuters are indifferent toany available choice andnone of themwill switch choiceThusF(xlowast) = xlowast and xlowast is a stationary state Under the conditionsin (23) the point xlowast is feasible belonging to 1198632 We can nowprove that this is indeed the unique fixed point of map F

Proposition 3 The point xlowast = (119909lowast119871 119909lowast119877) given in (21) is theunique fixed point of the map F belonging to 1198632 when theconditions in (23) hold

Proof The proof of this proposition follows directly from theobservation that

F (x) = x if and only if 119871 (x) = 119872 (x) = 119877 (x) (31)One way is already proved as we have detected the fixed pointstarting from the solution of 119871(x) = 119872(x) and 119877(x) = 119872(x)showing that it is the unique point xlowast Vice versa let us lookfor the solutions of F(x) = x We have the following systems(where 119909119872 = 1 minus 119909119871 minus 119909119877)

F119871

119909119871= 119909119871+ 120575119871(119877 (x) minus 119871 (x)

1198962

119909119877

+119872(x) minus 119871 (x)

1198964

119909119872)

119909119877= (1 minus 120575

119871

119877 (x) minus 119871 (x)1198962

)119909119877

if x isin 119877119871

F119872

119909119871= (1 minus 120575

119872

119871 (x) minus 119872 (x)119886119871minus 119889

)119909119871

119909119877= (1 minus 120575

119872

119877 (x) minus 119872 (x)119886119877minus 119889

)119909119877

if x isin 119877119872

F119877

119909119871= (1 minus 120575

119877

119871 (x) minus 119877 (x)1198961

)119909119871

119909119877= 119909119877+ 120575119877(119871 (x) minus 119877 (x)

1198961

119909119871

+119872(x) minus 119877 (x)

1198966

119909119872)

if x isin 119877119877

(32)

or equivalently (we recall that 120575119871 120575119872 and 120575

119877are different

from zero)

F119871

119877 (x) minus 119871 (x)1198962

119909119877+119872(x) minus 119871 (x)

1198964

119909119872= 0

119877 (x) minus 119871 (x)1198962

119909119877= 0

if x isin 119877119871

F119872

119871 (x) minus 119872 (x)119886119871minus 119889

119909119871= 0

119877 (x) minus 119872 (x)119886119877minus 119889

119909119877= 0

if x isin 119877119872

F119877

119871 (x) minus 119877 (x)1198961

119909119871= 0

119871 (x) minus 119877 (x)1198961

119909119871+119872(x) minus 119877 (x)

1198966

119909119872= 0

if x isin 119877119877

(33)

Considering the equation F119871(x) = x we see that it is certainly

satisfied for 119909119877 = 0 and 119909119872 = 0 which leads to thecorner point with 119909119871 = 1 However if the fixed point is an

10 Discrete Dynamics in Nature and Society

interior point of 1198632 then the point (1 0) does not belong tothe proper region 119877

119871 that is it is a virtual fixed point Clearly

this equation is satisfied when 119871(x) = 119872(x) and 119871(x) = 119877(x)which occurs only at the point xlowast It is possible to have xlowast =(1 0) and this occurs for (119886

119871minus 119889)119887

119877= 0 and (119886

119877minus 119889)119887

119871= 1

Thus we can state that the only solution of F119871(x) = x in 119877

119871

is given by the fixed point xlowast Similarly we can reason in theother regions

It is worth to mention that particular cases are obtainedwhen the fixed point xlowast belongs to a boundary of the region1198632 For example when 119909lowast119877+119909lowast119871 = 1 and 119909lowast119872 = 0 then only

the two regions119877119871and 119877

119877are involvedThis occurs when the

parameters satisfy

119889 =119886119871119887119871+ 119886119877119887119877minus 119887119871119887119877

119887119877+ 119887119871

0 le119886119871minus 119889

119887119877

le 1

0 le119886119877minus 119889

119887119871

le 1

(34)

From (119886119877minus 119889)119887

119871= (119886119877+ 119887119877minus 119886119871)(119887119877+ 119887119871) we have 119909lowast119877 =

1minus(119886119877minus119889)119887

119871= (119886119871+119887119871minus119886119877)(119887119877+119887119871) = 1198961(119887119877+119887119871) = 119909lowast as

in the one-dimensional map studied in Section 2 and 119909lowast119871 =1 minus 119909lowast119877= 1 minus (119886

119871minus 119889)119887

119877= 1198962(119887119877+ 119887119871) = 1 minus 119896

1(119887119877+ 119887119871) =

1minus119909lowast From themapsF

119871andF119877in (27) and for the parameters

occurring in this case it can be seen that 119909119871119905+1= 1minus119909

119877

119905+1holds

in both so that the dynamics can be studied by using onlythe variable 119909119877

119905and the resulting one-dimensional map is as

follows

119865 (119909119877)

=

119891119871(119909119877)

= 119909119877+ 120575119877

119871 (119909119877)minus119877 (119909

119877)

1198961

(1 minus 119909119877)

if 0 le 119909119877 le 119909lowast

119891119877(119909119877)

= 119909119877minus 120575119871

119877 (119909119877) minus 119871 (119909

119877)

1198962

119909119877 if 119909lowast le 119909119877 le 1

(35)

which corresponds to the system already studied in Section 2As we have seen this fixed point is globally attracting even iflocally we can have the eigenvalue on some side larger than 1in modulus

The case with 119909lowast119871 = 0 so that 0 le 119909lowast119877 le 1 and 119909lowast119872 =1 minus 119909lowast119877 leaves only the two regions 119877

119877and 119877

119872 This occurs

when the parameters satisfy

119886119871minus 119889

119887119877

= 1 0 le119886119877minus 119889

119887119871

le 1 (36)

From the maps F119872

and F119877in (27) it can be seen that 119909119871

119905

converges to119909lowast119871 = 0 so that the resulting asymptotic dynamic

behavior can be studied via the one-dimensional map givenby 119909119877119905+1= 119879(119909

119877

119905) with 119909119877

119905isin [0 1] (and 119909119872

119905= 1 minus 119909

119877

119905) given by

119879 (119909119877)

=

119879119871(119909119877)

= 119909119877+ 120575119877

119872(119909119877) minus 119877 (119909

119877)

1198966

(1 minus 119909119877)

if 0 le 119909119877 le 119909lowast

119879119877(119909119877)

= 119909119877minus 120575119872

119877 (119909119877) minus119872(119909

119877)

119886119877minus 119889

119909119877 if 119909lowast le 119909119877 le 1

(37)

where119872(119909119877) minus 119877(119909119877) = 119886119871+ 119887119871(1 minus 119909

119877) minus 119886119877minus 119887119877

The case with 119909lowast119877 = 0 so that 0 le 119909lowast119871 le 1 and119909lowast119872= 1 minus 119909

lowast119871 leaves only the two regions 119877119871and 119877

119872 This

occurs when the parameters satisfy 0 le (119886119871minus 119889)119887

119877le 1

(119886119877minus 119889)119887

119871= 1 From the maps F

119872and F

119871in (27) it can

be seen that 119909119877119905converges to 119909lowast119877 = 0 so that the resulting

asymptotic dynamic behavior can be studied via the one-dimensional map given by 119909119871

119905+1= 119879(119909

119871

119905) similar to the one

obtained in (37) with obvious changesBy using arguments as in Section 2 it can be shown that

these one-dimensional systems have a unique fixed pointwhich is globally attracting (although it can be locally stableor unstable)

In order to study the local stability of the fixed point ofthe two-dimensional map F we can use the linearization ofthe different definitions in the fixed point which leads to thefollowing Jacobian matrices one for each region 119877

119871 119877119872 and

119877119877 that we denote respectively with 119869

119871 119869119872 and 119869

119877

119869119871(xlowast) = (

1 + 120575119871minus 120575119871

119887119877

119887119871

1198965

1198964

minus 120575119871

119887119877

119887119871

1198966

1198961

120575119871

1198966

1198961

120575119871

119887119877

119887119871

1198966

1198961

1 minus 120575119871

1198966

1198961

)

119869119872(xlowast) = (

1 minus 120575119872

1198964

1198963

0

0 1 minus 120575119872

1198966

1198965

)

119869119877(xlowast) = (

1 minus 120575119877

1198965

1198962

120575119877

119887119871

119887119877

1198965

1198962

120575119877

1198965

1198962

1 + 120575119877minus 120575119877

119887119871

119887119877

1198963

1198966

minus 120575119877

119887119871

119887119877

1198965

1198962

)

(38)

The two real eigenvalues of the Jacobian matrix in 119877119872are

in explicit form and both are smaller than +1 However theycan be larger than minus1 or not Also the eigenvalues of the otherjacobian matrices can be in modulus smaller or larger than1 However the global dynamics observed numerically givealways trajectories which are convergent to the fixed point xlowastat any set of allowed parameters values

Discrete Dynamics in Nature and Society 11

1

1

075

05

025

00 025 05 075

xR

xL

(a)

1

1

075

05

025

00 025 05 075

xR

xL

(b)

Figure 6 Examples of trajectories converging to the unique fixed point with parameters (a) 119886119871= 28 119887

119871= 24 119886

119877= 8 119887

119877= 24 119889 = 5 120575

119871= 04

120575119872= 03 120575

119877= 07 and ic 119909119871 = 08 119909119877 = 005 (b) 119886

119871= 29 119887

119871= 16 119886

119877= 15 119887

119877= 25 119889 = 7 120575

119871= 05 120575

119872= 08 120575

119877= 06 and ic 119909119871 = 005

119909119877= 09 Dots are linked by a continuous line for illustrative purpose only

This global attractivity in the two-dimensional map isdifficult to be proved rigorously however the results in theparticular cases commented above that is when the fixedpoint xlowast belongs to the boundary of 1198632 suggests that thesame result is true also when the fixed point is in the interiorof1198632

In Figure 6 we show some examples in Figure 6(a) wecan see a trajectory starting at 119877

119877 in Figure 6(b) a trajectory

whose initial point in 119877119871is mapped into 119877

119872 The shape of

the functions in the case of Figure 6(b) is similar to the oneshown in Figure 5

4 Conclusion

In this paper we have analyzed a dynamical version of theBraess paradox with nonimpulsive commuters who changeroad proportionally to the cost differenceWe were interestedto prove that with nonimpulsive commuters there exists aunique equilibrium both in the original network with twochoices and in the ternary onewhen a single link is addedWehave shown that both models are quite robust We were ableto provide an analytical proof of the global attractivity of theunique fixed point in the one-dimensional case when onlytwo roads are available andweusednumerical techniques andsimulations to give evidence in the two-dimensional case (iewhen a new road is added) as a rigorous proof can be doneonly for particular border cases

A limitation of our contribution is considering homo-geneous populations in spite of the fact that the evidencesupports heterogeneity Nevertheless heterogeneity of com-mutersrsquo behavior cannot be easily classified yet even when

conducting experiments common individual patterns arejust identified and classified On the other hand whenobserving the aggregate behavior inmany cases the literatureprovides evidence in support of the paradox outcome thatis the convergence to the Nash equilibrium The impulsiveagents in [30] react faster than the proportional agentspresented in this paper yet as a population they fail toachieve the Nash equilibria in certain cases while in thispaper the aggregate behavior does not We proposed thisaggregate behavior because it replies the convergence to theNash equilibria and at the same time it may be the resultof modeling some observed individual behaviors such asimitation and free riding

Taking into account the limitation of homogeneous pop-ulation analysis a natural extension of our contribution isconsidering heterogeneous populations with both impulsiveand nonimpulsive commuters We have reasons to believethat in such a case some properties of the different behaviorsare inherited and some are lost Therefore the results weprovide in this paper match those for impulsive populationsand are paramount to extend the analysis to more realisticcases In factmdashas the empirical results showmdashheterogeneouspopulations dynamics provide a better fit of what has beenobserved when considering human participants interaction

Furthermore it will be interesting to consider general-izations to networks with richer architectures Finally weobserve that since with the class of behavior we considerin this paper that dynamics depends on the shape of thecost functions it will be interesting to extend the analysis tononlinear cost functions

12 Discrete Dynamics in Nature and Society

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

V Avrutin is supported by the European Community withinthe scope of the project ldquoMultiple-discontinuity inducedbifurcations in theory and applicationsrdquo (Marie Curie Actionof the 7th Framework Programme Contract Agreement noPIEF-GA-2011-300281) U Merlone has worked under theauspices of COST Action IS1104 ldquoThe EU in the new complexgeography of economic systems models tools and policyevaluationrdquo

References

[1] R Arnott and K Small ldquoThe economics of traffic congestionrdquoAmerican Scientist vol 82 no 5 pp 446ndash455 1994

[2] T Bass ldquoRoad to ruinrdquo Discover pp 56ndash61 1992[3] B Calvert and G Keady ldquoBraessrsquos paradox and power-law non-

linearities in networksrdquoThe Journal of the AustralianMathemat-ical Society B Applied Mathematics vol 35 no 1 pp 1ndash22 1993

[4] J E Cohen andC Jeffries ldquoCongestion resulting from increasedcapacity in single-server queueing networksrdquo IEEEACMTrans-actions on Networking vol 5 no 2 pp 305ndash310 1997

[5] J E Cohen and F P Kelly ldquoA paradox of congestion in a queuingnetworkrdquo Journal of Applied Probability vol 27 no 3 pp 730ndash734 1990

[6] S Dafermos and A Nagurney ldquoOn some traffic equilibriumtheory paradoxesrdquo Transportation Research B Methodologicalvol 18 no 2 pp 101ndash110 1984

[7] C Fisk ldquoMore paradoxes in the equilibrium assignment prob-lemrdquo Transportation Research B Methodological vol 13 no 4pp 305ndash309 1979

[8] W H K Lam ldquoEffects of road pricing on system performancerdquoTraffic Engineering amp Control vol 29 no 12 pp 631ndash635 1988

[9] J D Murchland ldquoBraessrsquos paradox of traffic flowrdquo Transporta-tion Research vol 4 no 4 pp 391ndash394 1970

[10] R Steinberg and W I Zangwill ldquoThe prevalence o f BraessrsquoparadoxrdquoTransportation Science vol 17 no 3 pp 301ndash318 1983

[11] M Frank ldquoThe Braess paradoxrdquo Mathematical Programmingvol 20 no 1 pp 283ndash302 1981

[12] T Roughgarden Selfish Routing and the Price of Anarchy MITpress 2005

[13] B B Fu C X Zhao and S B Li ldquoThe analysis of Braessrsquo paradoxand robustness based on dynamic traffic assignment modelsrdquoDiscrete Dynamics in Nature and Society vol 2013 Article ID796842 8 pages 2013

[14] C Fisk and S Pallottino ldquoEmpirical evidence for equilibriumparadoxes with implications for optimal planning strategiesrdquoTransportation Research Part A General vol 15 no 3 pp 245ndash248 1981

[15] H Youn M T Gastner and H Jeong ldquoPrice of anarchy intransportation networks efficiency and optimality controlrdquoPhysical Review Letters vol 101 no 12 Article ID 128701 2008

[16] K Aoki Y Ohtsubo A Rapoport and T Saijo ldquoEffects of priorinvestment and personal responsibility in a simple network

gamerdquo Current Research in Social Psychology vol 13 no 2 pp10ndash21 2007

[17] J Morgan H Orzen and M Sefton ldquoNetwork architectureand traffic flows experiments on the Pigou-Knight-Downs andBraess paradoxesrdquoGames and Economic Behavior vol 66 no 1pp 348ndash372 2009

[18] A Rapoport V Mak and R Zwick ldquoNavigating congested net-works with variable demand experimental evidencerdquo Journal ofEconomic Psychology vol 27 no 5 pp 648ndash666 2006

[19] A Rapoport T Kugler S Dugar and E J Gisches ldquoBraessparadox in the laboratory an experimental study of route choicein traffic networks with asymmetric costsrdquo inDecisionModelingand Behavior in Uncertain and Complex Environment T KuglerJ C Smith T Connolly and Y J Son Eds pp 309ndash337Springer New York NY USA 2008

[20] A Rapoport T Kugler S Dugar and E J Gisches ldquoChoice ofroutes in congested traffic networks experimental tests of theBraess paradoxrdquo Games and Economic Behavior vol 65 no 2pp 538ndash571 2009

[21] E J Gisches and A Rapoport ldquoDegrading network capacitymay improve performance private versus public monitoring inthe Braess paradoxrdquoTheory and Decision vol 73 no 2 pp 267ndash293 2012

[22] A Dal Forno and U Merlone ldquoReplicating human interactionin Braess paradoxrdquo in Proceedings of the Simulation Conference(WSC rsquo13) pp 1754ndash1765 IEEE 2013

[23] M Ausloos H Dawid and U Merlone ldquoSpatial interactionsin agent-based modelingrdquo in Complexity and GeographicalEconomics Topics and Tools P Commendatore S S Kayamand I Kubin Eds Springer Berlin Germany 2014

[24] G-I Bischi and U Merlone ldquoGlobal dynamics in binary choicemodels with social influencerdquo The Journal of MathematicalSociology vol 33 no 4 pp 277ndash302 2009

[25] G I Bischi L Gardini and U Merlone ldquoImpulsivity in binarychoices and the emergence of periodicityrdquoDiscrete Dynamics inNature and Society vol 2009 Article ID 407913 22 pages 2009

[26] JH PattonM S Stanford andE S Barratt ldquoFactor structure ofthe Barratt Impulsiveness Scalerdquo Journal of Clinical Psychologyvol 51 no 6 pp 768ndash774 1995

[27] G I Bischi L Gardini and U Merlone ldquoPeriodic cycles andbifurcation curves for one-dimensional maps with two discon-tinuitiesrdquo Journal of Dynamical Systems GeometricTheories vol7 no 2 pp 101ndash123 2009

[28] G-I Bischi and U Merlone ldquoBinary choices in small and largegroups a unifiedmodelrdquo Physica A vol 389 no 4 pp 843ndash8532010

[29] A Dal Forno L Gardini and U Merlone ldquoTernary choicesin repeated games and border collision bifurcationsrdquo ChaosSolitons amp Fractals vol 45 no 3 pp 294ndash305 2012

[30] A Dal Forno and U Merlone ldquoBorder-collision bifurcationsin a model of Braess paradoxrdquo Mathematics and Computers inSimulation vol 87 pp 1ndash18 2013

[31] A Dal Forno V Giorgino and U Merlone ldquoAgent-based mod-eling with GTmethodology An examplerdquoWorking Paper 2014

[32] L Gardini U Merlone and F Tramontana ldquoInertia in binarychoices continuity breaking and big-bang bifurcation pointsrdquoJournal of Economic Behavior amp Organization vol 80 no 1 pp153ndash167 2011

[33] W H Sandholm Population Games and Evolutionary Dynam-ics The MIT Press Cambridge Mass USA 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Discrete Dynamics in Nature and Society

behaviors emerged by mean of grounded theory and usesABM simulations For a review on ABM with special focuson spacial interactions and networks see [23]

The paradox has also been presented in a dynamic gameframework A discrete time dynamic population model withsocial externalities and two available choices is studied in [24]simulating an adaptive adjustment process and in particularwith impulsive commuters in [25]

Following the clinical psychology literature (see [26])impulsive commuters have been introduced in the analysis ofbinary choices with externalities in [25 27] when consideringcommuters whose switching rate only depends on the signof the difference between payoffs no matter how much theydiffer This approach has been used to consider choices bothin small and in large groups in [28] and also when consid-ering the introduction of a third choice which dramaticallychanges the dynamics in terms of complexity (as analyzed in[29])

Analyzing the Braess paradox in terms of a ternary choicegame shows new interesting dynamical characteristics thatare investigated in [30] with a particular interest in thecoexistence of several equilibria Nevertheless as illustratedin [22 31] considering only impulsive commuters is not suffi-cient to describe the dynamics observed in experiments withhuman participants Furthermore as impulsive commutersconsider only the sign and not the size of difference betweencosts the population dynamics does not depend on the costfunctions as long as the indifference point remains the sameIn this paper we consider a different behavior suggested byAmnonRapoport (We are grateful to him for this helpful sug-gestion) and which was used in [22] for artificial commutersAlthough the full analysis of a homogenousmdashyet differentmdashpopulation seems to provide a limited contribution it may bea step forward to analyze the aggregate decision behavior insocial dilemmas In fact in order to analyze heterogeneouspopulations it is important to well understand the dynamicsproperties of component behaviors In particular we considercommuters which are concerned not only about the signdifference in payoffs but also on the relative difference Thiskind of behavior is similar to the one considered in [32]where the propensity to switch choice is modulated by thedifference between payoffs Unlike the impulsive adjustmentprocess which makes the commuters change their choice assoon as a better choice occurs regardless of the differencewe consider a decision strategy which prescribes a changeto the best available choice proportionally to the reporteddifference Intuitively this strategy seems to be more robustThe main result of the present work is that it is in fact robustBy contrast to the strategy considered in [30] whichmay leadto a stable cycle of any period the strategy considered in thispaper leads to a globally attracting fixed point Dependingon the parameters this unique fixed point may be locallystable or unstable However the trajectories are convergentin a few steps to the fixed point This result is rigorouslyproved for the case of a binary choice problem For theternary choice problem an analogous result is evidenced aswell

The plan of the work is as follows In Section 2 theformalization of the dynamic model is reported Section 21

includes the detailed description of the case with only tworoads which is represented by a one-dimensional piecewisesmooth continuous map We prove the existence of a uniquefixed point which may be locally stable or unstable in onepartition We prove that in any case it is globally attractingthat is all the trajectories are ultimately converging to thefixed point The case extended to three roads is consideredin Section 3 and it is reduced to a two-dimensional piecewisesmooth discontinuousmapWe prove the existence of a uniquefixed point which may be locally attracting or not We haveonly numerical evidence that also in this more general casethe fixed point is globally attracting This can be rigorouslyproved for the particular cases in which the fixed pointbelongs to the boundaries of the domain of interest The lastsection is devoted to the conclusions

2 The Dynamic Model with Two Roads

The Braess paradox can be illustrated by Figure 1 as followsAssume there is a unitary mass of commuters from start (119878)to end (119864) and there are two roads one passing through left(119871) and the other through right (119877) The cost of each road isgiven by the sumof the time spent along each segment In thisnetwork segments 119878-119877 and 119871-119864 do not depend on traffic andthey cost 119886

119871and 119886119877 respectively On the contrary time spent

along segments 119878-119871 and 119877-119864 is proportional to the numberof travellers and therefore they cost 119887

119871(1 minus 119909) and 119887

119877119909 where

119909 isin [0 1] is the fraction of the population using segment119877-119864At the Nash equilibrium commuters are distributed in such away that both roads (119878-119871-119864 and 119878-119877-119864) have equal cos thatis a fraction (119886

119877+119887119877minus119886119871)(119887119871+119887119877) of commuters will choose

119878-119871-119864 and a fraction (119886119871+ 119887119871minus 119886119877)(119887119871+ 119887119877) will choose 119878-119877-

119864 For example assume as in [22] that 119886 = 119886119871= 119886119877= 27

and 119887 = 119887119871= 119887119877= 24 Then the symmetry of costs makes

the population to split exactly into two equal fractions at theNash equilibrium ((119886

119877+ 119887119877minus 119886119871)(119887119871+ 119887119877) = 12) so that

the travel time for both roads is 24(12) + 27 = 39 Nowwe assume that a very fast road is built connecting 119871 to 119877at cost 119889 = 3 As we will see in Section 3 at the new Nashequilibrium commuters are distributed in such a way that thethree roads (119878-119871-119864 119878-119871-119877-119864 and 119878-119877-119864) have equal cost thatis a fraction (119889+119887

119877minus119886119871)119887119877of commuters will choose 119878-119871-119864

a fraction (119889 + 119887119871minus119886119877)119887119871will choose 119878-119877-119864 and the rest will

choose 119878-119871-119877-119864 In the numerical example this means thatthe entire population will choose 119878-119871-119877-119864 (since in the othertwo roads the fraction of commuter is (119889 + 119887 minus 119886)119887 = 0) witha total travel cost of 119887

119871+119887119877+119889 = 51 According to [19] this is

considered paradoxical as it shows how adding extra capacityto a network can reduce overall performance More correctlyit is just counterintuitive and the mathematical reason isthat there is a distinction between Nash equilibria andoptima

We consider the original network first made of two roadsonly (ie without the link 119871-119877) as in Figure 1(a)

As in [24] this process can be modeled as a discrete timedynamical system considering commuters facing a binarychoice between paths 119878-119871-119864 and 119878-119877-119864 For the sake of brevitythese paths will be denoted as 119871 and 119877 respectively To this

Discrete Dynamics in Nature and Society 3

L R

S

E

aL

aRbLxL

bRxR

(a)

aR

aL

L R

S

E

d

bL(xL+ x

M)

bR(xM

+ xR)

(b)

Figure 1 Basic network with the addition of the link from 119871 to 119877 in (b) and without in (a)

purpose assume the set of commuters is normalized to theinterval [0 1] At each time period 119905 ge 0 119909119877

119905ge 0 indicates the

fraction of commuters choosing 119877 and 119909119871119905ge 0 the fraction of

those choosing 119871 given by 119909119871119905= 1 minus 119909

119877

119905 Therefore the travel

times are given by

119871 = 119886119871+ 119887119871119909119871

119877 = 119886119877+ 119887119877119909119877

(1)

where 119886119871 119886119877 119887119871 119887119877gt 0 and can be written as

119871 (119909119877) = 119886119871+ 119887119871(1 minus 119909

119877)

119877 (119909119877) = 119886119877+ 119887119877119909119877

(2)

This way the travel times on the two roads are equal (andthe commuters are indifferent) when 119871(119909119877) = 119877(119909119877) that iswhen 119909119877 = 119909lowast where

119909lowast=119886119871+ 119887119871minus 119886119877

119887119871+ 119887119877

(3)

We say that the threshold 119909lowast is feasible when 119909lowast isin [0 1]However 119909lowast = 0 (resp 119909lowast = 1) clearly corresponds to thecase in which the whole population chooses action 119871 (resp119877) and we shall see that this is indeed satisfied in the modeldescribed in the next subsection We obtain the followingconstraints on the parameters in order to have 119909lowast isin (0 1)which we assume henceforth

1198961= 119886119871+ 119887119871minus 119886119877gt 0

1198962= 119886119877+ 119887119877minus 119886119871gt 0

(4)

21 The One-Dimensional PWS Map and Its DynamicsCommuters are homogeneous and at each period decidetheir strategy considering the previous period travel costsassuming they will remain the same At time 119905+1 the fraction

119909119877

119905is common knowledge and each commuter observes the

respective travel costs either 119871(119909119877119905) or 119877(119909119877

119905) depending on

their choice The commuters decide their future action attime 119905 + 1 comparing the costs 119871(119909119877

119905) and 119877(119909119877

119905) We assume

the commuters are nonimpulsive and myopically minimizetheir costs That is they change strategy proportionally tothe difference between their chosen road and the best onewhere the best road is the fastest We introduce parameters120575119871isin (0 1) and 120575

119877isin (0 1) as switching rates that is

the propensity for moving either to 119871 or to 119877 respectivelyWith nonimpulsive commuters these rates are adjusted by thenormalized difference of travel costs The difference 119871(119909119877

119905) minus

119877(119909119877

119905) is normalized by the largest value of this difference

denoted by 1198961and obtained in correspondence of 119909119877

119905= 0

Therefore we have 1198961= 119871(0) minus 119877(0) = 119886

119871+ 119887119871minus 119886119877as

already introduced in (4) Thus commuters are switchingroad proportionally to the side of the payoff difference Sim-ilarly we can give a reason with respect to the difference119877(119909119877

119905) minus 119871(119909

119877

119905) to get a normalizing value Now the largest

value is obtained in correspondence of 119909119871119905= 0 that is 119909119877

119905= 1

andwe get 1198962= 119877(1)minus119871(1) = 119886

119877+119887119877minus119886119871as already introduced

in (4)The resulting dynamics is 119909119877

119905+1= 119865(119909

119877

119905) with 119909119877

119905isin [0 1]

and the map 119865 [0 1] rarr [0 1] is defined as

119865 (119909119877)

=

119891119871(119909119877)

= 119909119877+ 120575119877

119871 (119909119877) minus 119877 (119909

119877)

1198961

(1 minus 119909119877)

if 0 le 119909119877 le 119909lowast

119891119877(119909119877)

= 119909119877minus 120575119871

119877 (119909119877) minus 119871 (119909

119877)

1198962

119909119877 if 119909lowast le 119909119877 le 1

(5)

4 Discrete Dynamics in Nature and Society

Substituting from (2) and rearranging we obtain

119865 (119909119877) =

119891119871(119909119877) = 120575119877+ (1 minus 120575

119877minus 120575119877

119887119871+ 119887119877

1198961

)119909119877

+120575119877

119887119871+ 119887119877

1198961

(119909119877)2

if 0 le 119909119877 le 119909lowast

119891119877(119909119877) = (1 + 120575

119871

1198961

1198962

)119909119877

minus120575119871

119887119871+ 119887119877

1198962

(119909119877)2

if 119909lowast le 119909119877 le 1

(6)

From the definition of 119909lowast given in (3) it can also be writtenas

119909lowast=119886119871+ 119887119871minus 119886119877

119887119871+ 119887119877

=1198961

119887119871+ 119887119877

=1198961

1198961+ 1198962

(7)

at which we have 119871(119909lowast) = 119877(119909lowast) it is also 119891119871(119909lowast) = 119891119877(119909lowast)

and thus 119865 is continuous in 119909lowast and it is a fixed point as119865(119909lowast) = 119909lowast We have the following proposition

Proposition 1 Map 119865 is continuous in [0 1] and under theconditions given in (4) 119909lowast = 119896

1(119887119871+ 119887119877) isin (0 1) is the unique

fixed point

Proof Only the uniqueness is left to prove The equation119891119871(119909119877) = 119909

119877 has two solutions 119909119877 = 119909lowast and 119909119877 = 1 whichis larger than 119909lowast and thus a virtual fixed point The equation119891119877(119909119877) = 119909

119877 has two solutions 119909119877 = 119909lowast and 119909119877 = 0 whichis smaller than 119909lowast and thus a virtual fixed point Thus theunique solution is 119909119877 = 119909lowast

Clearly the fixed point may be locally stable or instabledepending on the slopes of the functions on the right and leftsides of 119909lowast However we shall see that in any case it is globallyattractingThat is any ic in the interval [0 1] has a trajectorywhich converges to 119909lowast

Proposition 2 The fixed point 119909lowast = 1198961(119887119871+ 119887119877) isin (0 1) is

globally attracting

Proof Even if 119865(119909119877) is not smooth in its fixed point (asindeed 119909lowast is a kink point of this map) the two components119891119871(119909119877) and 119891

119877(119909119877) are smooth in 119909lowast so that the left and right

side derivatives of 119865(119909119877) in 119909lowast are well defined 1198651015840minus(119909lowast) =

lim119909119877rarr119909lowastminus1198651015840(119909119877) = 1198911015840

119871(119909lowast) and1198651015840

+(119909lowast) = lim

119909119877rarr119909lowast+1198651015840(119909119877) =

1198911015840

119877(119909lowast)

We have

1198911015840

119871(119909119877) = (1 minus 120575

119877minus 120575119877

119887119871+ 119887119877

1198961

) + 2120575119877

119887119871+ 119887119877

1198961

119909119877 (8)

1198911015840

119871(119909lowast) = 1 minus 120575

119877

1198962

1198961

(9)

while

1198911015840

119877(119909119877) = (1 + 120575

119871

1198961

1198962

) minus 2120575119871

119887119871+ 119887119877

1198962

119909119877 (10)

1198911015840

119877(119909lowast) = 1 minus 120575

119871

1198961

1198962

(11)

From

11989110158401015840

119871(119909119877) = 2120575

119877

119887119871+ 119887119877

1198961

gt 0 11989110158401015840

119877(119909119877) = minus2120575

119871

119887119871+ 119887119877

1198962

lt 0

(12)

we have that on the left side of the fixed point the function isconvex while on the right side of the fixed point the functionis concave

Notice that 11989621198961only depends on 119886

119871 119887119871 119886119877 119887119877 and it is

positive under our assumptions Depending on the values ofthe ratio 119896

21198961we can classify the rightleft stability of the

fixed point 119909lowast From (9) we clearly have always 1198911015840119871(119909lowast) lt 1

thus the stabilityinstability on the left side of the fixed pointdepends on the condition 1198911015840

119871(119909lowast) gt minus1 which holds if and

only if

1198962

1198961

lt2

120575119877

(13)

Similarly from (11) we clearly have always 1198911015840119877(119909lowast) lt 1 and

the stabilityinstability on the right side of the fixed pointdepends on the condition 1198911015840

119877(119909lowast) gt minus1 which holds if and

only if

1198962

1198961

gt120575119871

2 (14)

And clearly we have the following sequence of inequalities (tobe used below)

0 lt120575119871

2lt 120575119871lt 1 lt

1

120575119877

lt2

120575119877

(15)

To investigate the local stability of the fixed point we considerthree intervals for 119896

21198961

(i) 0 lt 11989621198961le 1205751198712

(ii) 1205751198712 lt 119896

21198961lt 2120575

119877

(iii) 2120575119877le 11989621198961

In case (i) it is 1198911015840119877(119909lowast) le minus1 moreover it is easy to see

that for any 119909119877 isin [119909lowast 1] it is 1198911015840119877(119909119877) lt 119891

1015840

119877(119909lowast) and thus

the function on the right side of the fixed point is strictlydecreasing and expansive each point on the right side of thefixed point is mapped to the left side of the fixed point in oneiteration While on the left side of the fixed point the convexfunction 119891

119871(119909119877) has 1198911015840

119871(119909lowast) = 1 minus 120575

119877(11989621198961) gt 0 (as this

holds for 11989621198961lt 1120575

119877which is satisfied in this interval)This

implies that the point of local minimum on the left side say119909119877

119897119898 is smaller than the fixed point 119909119877

119897119898lt 119909lowast Then the fixed

point is unique and the map is a contraction in the interval119882119904

loc = [119909119877

119897119898 119909lowast] and an absorbing interval which is mapped

into 119882119904loc in one iteration is given by 119869 = [119909119877119897119898 119891minus1

119877(119909119877

119897119898)]

when 119891119877(1) = 1 minus 120575

119871lt 119909119877

119897119898 or 119869 = [119909119877

119897119898 1] when 119891

119877(1) =

1 minus 120575119871ge 119909119877

119897119898 Any initial condition belonging to [0 1] 119869

has a trajectory which is mapped into 119869 in a finite number ofiterationsThus119909lowast is globally attracting An example is shownin Figure 2(a)

Similarly we can reason when 11989621198961belongs to the third

interval In fact in case (iii) it is 1198911015840119871(119909lowast) le minus1 moreover it is

Discrete Dynamics in Nature and Society 5xR t+1

xR

t

00

025

025 05

05

075

1

1075

(a)xR t+1

xR

t

00

025

025 05

05

075

1

1075

(b)

Figure 2 Qualitative shapes of the map In (a) case (i) when 0 lt 11989621198961le 1205751198712 the fixed point is locally unstable on the right side In (b)

case (iii) when 2120575119877le 11989621198961 the fixed point is locally unstable on the left side

0

1

0 1

xR t

xL

t

(a)

0

1

0 1

xR t

xL

t

(b)

0

1

0 1

xR t

xL

t

(c)

Figure 3 Qualitative shapes of the map in case (ii) when 1205751198712 lt 119896

21198961le 2120575

119877and the fixed point is locally stable on both sides

easy to see that for any 119909119877 isin [0 119909lowast] it is 1198911015840119871(119909lowast) lt 1198911015840

119871(119909lowast) and

thus the function on the left side of the fixed point is strictlydecreasing and expansive each point on the left side of thefixed point is mapped to the right side of the fixed point inone iteration While on the right side of the fixed point theconcave function 119891

119877(119909119877) has 1198911015840

119877(119909lowast) = 1 minus 120575

119871(11989611198962) gt 0 (as

this holds for 11989621198961gt 120575119871which is satisfied in this interval)

This implies that the point of local maximum on the rightside say 119909119877

119903119872 is larger than the fixed point 119909119877

119903119872gt 119909lowast Then

the map is a contraction in the interval 119882119904loc = [119909lowast 119909119877

119903119872]

and an absorbing interval which is mapped into119882119904loc in oneiteration is given by 119869 = [119891minus1

119871(119909119877

119903119872) 119909119877

119903119872] when 119891

119871(0) =

120575119877gt 119909119877

119903119872 or 119869 = [0 119909119877

119903119872] when 119891

119871(0) = 120575

119877le 119909119877

119903119872 Any

initial condition belonging to [0 1] 119869 has a trajectory whichis mapped into 119869 in a finite number of iterations Thus 119909lowast isglobally attracting An example is shown in Figure 2(b)

When 11989621198961belongs to the interval in (ii) 120575

1198712 lt 119896

21198961lt

2120575119877 we have that the fixed point is locally stable on both

sides Moreover inside this interval we can distinguish threeregions as already showed in (15) that is

(a) 1205751198712 lt 119896

21198961le 120575119871with 0 lt 1198911015840

119871(119909lowast) lt 1 and

minus1 lt 1198911015840

119877(119909lowast) lt 0 as in Figure 3(a)

(b) 120575119871lt 11989621198961lt 1120575

119877with 0 lt 1198911015840

119871(119909lowast) lt 1 and

0 lt 1198911015840

119877(119909lowast) lt 1 as in Figure 3(b)

6 Discrete Dynamics in Nature and Society

(c) 1120575119877le 11989621198961lt 2120575

119877with minus1 lt 1198911015840

119871(119909lowast) lt 0 and

0 lt 1198911015840

119877(119909lowast) lt 1 as in Figure 3(c)

In case (a) we have that the point of local minimum onthe left side say 119909119877

119897119898 is smaller than the fixed point119909119877

119897119898lt 119909lowast

Then themap is a contraction in the interval119882119904loc = [119909119877

119897119898 119909lowast]

and on the right side the map is decreasing so that anabsorbing interval which is mapped into119882119904loc in one iterationis given by 119869 = [119909119877

119897119898 119891minus1

119877(119909119877

119897119898)]when119891

119877(1) = 1minus120575

119871lt 119909119877

119897119898 or

119869 = [119909119877

119897119898 1]when119891

119877(1) = 1minus120575

119871ge 119909119877

119897119898 Any initial condition

belonging to [0 1] 119869 has a trajectory which is mapped into 119869in a finite number of iterationsThus 119909lowast is globally attracting

In case (c) we can reason similarly we have that the pointof local maximum on the right side say 119909119877

119903119872 is larger than

the fixed point 119909119877119903119872gt 119909lowast Then the map is a contraction in

the interval 119882119904loc = [119909lowast 119909119877

119903119872] and on the left side the map

is decreasing so that an absorbing interval which is mappedinto 119882119904loc in one iteration is given by 119869 = [119891minus1

119871(119909119877

119903119872) 119909119877

119903119872]

when 119891119871(0) = 120575

119877gt 119909119877

119903119872 or 119869 = [0 119909119877

119903119872] when 119891

119871(0) =

120575119877le 119909119877

119903119872 Any initial condition belonging to [0 1] 119869 has

a trajectory which is mapped into 119869 in a finite number ofiterations Thus 119909lowast is globally attracting

In case (b) we have that the point of local minimum onthe left side say 119909119877

119897119898 is smaller than the fixed point 119909119877

119897119898lt

119909lowast and also that the point of local maximum on the right

side say 119909119877119903119872

is larger than the fixed point 119909119877119903119872gt 119909lowast

Then the map is a contraction in the interval 119882119904loc =

[119909119877

119897119898 119909119877

119903119872] which includes the fixed point and an absorbing

interval which is mapped into119882119904loc in one iteration is givenby 119869 = [119891minus1

119871(119909119877

119903119872) 119891minus1

119877(119909119877

119897119898)] or 119869 = [0 119891minus1

119877(119909119877

119897119898)] or 119869 =

[119891minus1

119871(119909119877

119903119872) 1] or [0 1] depending under obvious conditions

on the values of 120575119871and 120575

119877 When 119869 sub [0 1] then any initial

condition belonging to [0 1] 119869 has a trajectory which ismapped into 119869 in a finite number of iterations Thus 119909lowast isglobally attracting

We notice that in Proposition 2 we prove that when thefixed point 119909lowast is internal to the interval [0 1] then it isglobally attracting Let us turn to comment the two extremawhich occur when 119909lowast = 0 or 119909lowast = 1 These solutions clearlyexist and are also globally attracting In fact the case 119909lowast = 0corresponds to 119896

1= 119871(0)minus119877(0) = 119886

119871+119887119871minus119886119877= 0 and only the

function 119891119877(119909119877) is applied in the interval [0 1] and from the

concavity of the function we have that this fixed point is glob-ally attracting even if the eigenvalue is equal to 1 The othercase 119909lowast = 1 corresponds to 119896

2= 119877(1)minus119871(1) = 119886

119877+119887119877minus119886119871= 0

and only the function 119891119871(119909119877) is applied in the interval [0 1]

and from the convexity of the function we have that this fixedpoint is globally attracting even if the eigenvalue is equal to 1

3 The Dynamics in the ExpandedNetwork with Three Roads

As it is well known the Braess paradox introduces a new linkand shows how this opportunity worsens both the individualand collective payoffs This link introduces a new alternativenow the commuters are facing three choices where the new

one consists of path 119878-119871-119877-119864 (see Figure 1(b)) which will bedenoted as 119872 in the following Models with three choiceswere introduced in [29] as it concerns discrete time and [33]as it concerns continuous time In this paper we analyze thedynamics when the travel time 119889 for the new resulting linkvaries in a range that makes this link a dominant choice forsome values and a dominated choice for others It is worth toobserve that in particular when the travel time on link 119871-119877 iszero the paradoxical result still holds

Now assume that commuters switch roadwhenever traveltime has become larger that is the other road is less costlyand therefore more attractive We indicate by 119909119871 ge 0 thefraction of population choosing action 119871 by 119909119872 ge 0 thefraction choosing action 119872 and by 119909119877 ge 0 the fractionchoosing action 119877 with the constraint 119909119871 + 119909119877 + 119909119872 = 1The travel times are given respectively as

119871 = 119886119871+ 119887119871(119909119871+ 119909119872)

119872 = 119887119871(119909119871+ 119909119872) + 119887119877(119909119872+ 119909119877) + 119889

119877 = 119886119877+ 119887119877(119909119872+ 119909119877)

(16)

By using the constraint let us define the fraction of populationchoosing119872 as 119909119872 = 1 minus 119909119871 minus 119909119877 ge 0 so to consider onlytwo variables x = (119909119871 119909119877) isin R2

+ and the phase space is the

triangle

1198632= x = (119909119871 119909119877) isin R2

+ 0 le 119909

119871+ 119909119877le 1 (17)

After substituting the expression of 119909119872 in the travel times weget

119871 (x) = 119886119871 + 119887119871 (1 minus 119909119877)

119872 (x) = 119887119871 (1 minus 119909119877) + 119887119877(1 minus 119909

119871) + 119889

119877 (x) = 119886119877 + 119887119877 (1 minus 119909119871)

(18)

The vertices of the phase space119875lowast119871(1 0)119875lowast

119872(0 0) and119875lowast

119877(0 1)

represent states in which the whole population choosesrespectively actions 119871119872 and 119877

Since commuters are interested in minimizing traveltime we denote by 119877

119860the region in which the choice 119860

(where119860 is 119871119872 or 119877) is preferable These regions are calleddominance regions of 119860 We have the following definition ofthe dominance regions

119877119871= x isin 1198632 119871 (x) le 119872 (x) 119871 (x) le 119877 (x)

119877119872= x isin 1198632 119872 (x) le 119871 (x) 119872 (x) le 119877 (x)

119877119877= x isin 1198632 119877 (x) le 119871 (x) 119877 (x) le 119872 (x)

(19)

Discrete Dynamics in Nature and Society 7

1

100

xR

RL

RM

RR

xL

(a)

1

100

xR

RL

RM

RR

xL

(b)

1

100

xR

RL

RM

RR

xL

(c)

Figure 4 Dominance regions in the three choices network with different values of the cost parameter 119889 of the map F at 119886119871= 27 119887

119871= 30

119886119877= 18 119887

119877= 30 and (a) 119889 = 6 (b) 119889 = 3 (c) 119889 = 0

By using the expressions in (18) we get the following defini-tions for the regions

119877119871= x isin 1198632 119909119871 le 119909lowast119871

119886119871+ 119887119871(1 minus 119909

119877) le 119886119877+ 119887119877(1 minus 119909

119871)

119877119872= x isin 1198632 119909119871 ge 119909lowast119871 119909119877 ge 119909lowast119877

119877119877= x isin 1198632 119886

119877+ 119887119877(1 minus 119909

119871) le 119886119871+ 119887119871(1 minus 119909

119877)

119909119877le 119909lowast119877

(20)

where we have introduced

119909lowast119871= 1 minus

119886119871minus 119889

119887119877

=119889 minus 119886119871+ 119887119877

119887119877

119909lowast119877= 1 minus

119886119877minus 119889

119887119871

=119889 minus 119886119877+ 119887119871

119887119871

(21)

The boundaries of the three regions are given by segments ofstraight lines of equations

1198811 119909119871= 119909lowast119871

1198812 119909119877= 119909lowast119877

1198813 119886119871+ 119887119871(1 minus 119909

119877) = 119886119877+ 119887119877(1 minus 119909

119871)

(22)

and it is immediate to see that the point xlowast = (119909lowast119871 119909lowast119877)satisfies also the third equation that is it belongs to all thethree straight lines xlowast isin 119881

119894for 119894 = 1 2 3

Using the geometric representation in the triangle 1198632the dominance regions are illustrated in Figure 4 (via anexample) using different colors Three cases are shown fordifferent values of the parameter 119889 in order to illustrate howthe region119877

119872depends on the parameters of the third branch

The point xlowast = (119909lowast119871 119909lowast119877) is the unique point on theboundaries 119881

119894of all the three regions Thus for our model we

have to require 119909lowast119871 ge 0 119909lowast119877 ge 0 and 119909lowast119871 + 119909lowast119877 le 1 leadingto 119909lowast119872 = 1 minus 119909lowast119871 minus 119909lowast119877 In order to have three regions in1198632

it must be 119909lowast119871 gt 0 119909lowast119877 gt 0 and 119909lowast119871 + 119909lowast119877 lt 1 The equalityin one of these three conditions implies the disappearance ofone region For example when 119909lowast119871 = 0 then 0 le 119909lowast119877 le 1and 119909lowast119872 = 1 minus 119909lowast119877 leaving only the two regions 119877

119877and 119877

119872

and so onThe conditions 119909lowast119871 ge 0 119909lowast119877 ge 0 and 119909lowast119871 + 119909lowast119877 le 1 lead

to the constraints that will be assumed satisfied henceforth

0 le119886119871minus 119889

119887119877

le 1 0 le119886119877minus 119889

119887119871

le 1

119886119871minus 119889

119887119877

+119886119877minus 119889

119887119871

ge 1

(23)

We notice that from the third condition in (23) we have thevalues of 119889 for which the third region is really present in 1198632that is

0 le 119889 le119886119871119887119871+ 119886119877119887119877minus 119887119871119887119877

119887119877+ 119887119871

(24)

Commuters are homogeneous and minimize their nextperiod travel cost At time 119905+1 the vector x

119905becomes common

knowledge and each commuter can observe the travel costs119871(x119905)119872(x

119905) and 119877(x

119905) We assume that if at time 119905 a fraction

119909119871

119905chooses path 119871 a fraction 119909119877

119905chooses action 119877 and travel

costs are such that 119877(x119905) lt 119871(x

119905) and 119877(x

119905) lt 119872(x

119905) then

both a fraction of the 119909119871119905commuters who chose action 119871 and

a fraction of the (1 minus 119909119871119905minus 119909119877

119905) commuters who chose action

119872 will switch to path 119877 in the next time period 119905 + 1 Thefraction of switching commuters is given by the switchingpropensities 120575

119871 120575119872 120575119877modulated by the relative differences

in payoffs This happens whenever a path gives a smallertravel cost In other words at any time 119905 all the commutersdecide their future action at time 119905 + 1 comparing the costs119871(x119905)119872(x

119905) and 119877(x

119905) As in the one-dimensional case here

also the differences of travel costs are normalized dividing

8 Discrete Dynamics in Nature and Society

each difference by the constants giving the related maximumvalue given in the different cases as follows

1198961= max

x [119871 (x) minus 119877 (x)]

= [119871 (1 0) minus 119877 (1 0)] = 119886119871 + 119887119871 minus 119886119877

1198962= max

x [119877 (x) minus 119871 (x)]

= [119877 (0 1) minus 119871 (0 1)] = 119886119877 + 119887119877 minus 119886119871

1198963= max

x [119871 (x) minus 119872 (x)]

= [119871 (1 119909119877) minus119872(1 119909

119877)] = 119886

119871minus 119889 forall119909

119877isin [0 1]

1198964= max

x [119872 (x) minus 119871 (x)]

= [119872(0 119909119877) minus 119871 (0 119909

119877)] = 119889 + 119887

119877minus 119886119871 forall119909

119877isin [0 1]

1198965= max

x [119877 (x) minus 119872 (x)]

= [119877 (119909119871 1) minus 119872(119909

119871 1)] = 119886

119877minus 119889 forall119909

119871isin [0 1]

1198966= max

x [119872 (x) minus 119877 (x)]

= [119872(119909119871 0) minus 119877 (119909

119871 0)] = 119889 + 119887

119871minus 119886119877 forall119909

119871isin [0 1]

(25)

The resulting dynamics are described by a two-dimensionalmap x

119905+1= F(x

119905) defined as

F x119905+1=

F119871(x119905) if x

119905isin 119877119871

F119872(x119905) if x

119905isin 119877119872

F119877(x119905) if x

119905isin 119877119877

(26)

where

F119871

119909119871

119905+1= 119909119871

119905+ 120575119871(119877 (x119905) minus 119871 (x

119905)

119886119877+ 119887119877minus 119886119871

119909119877

119905

+119872(x119905) minus 119871 (x

119905)

119889 + 119887119877minus 119886119871

119909119872

119905)

119909119877

119905+1= (1 minus 120575

119871

119877 (x119905) minus 119871 (x

119905)

119886119877+ 119887119877minus 119886119871

)119909119877

119905

F119872

119909119871

119905+1= (1 minus 120575

119872

119871 (x119905) minus 119872(x

119905)

119886119871minus 119889

)119909119871

119905

119909119877

119905+1= (1 minus 120575

119872

119877 (x119905) minus 119872(x

119905)

119886119877minus 119889

)119909119877

119905

F119877

119909119871

119905+1= (1 minus 120575

119877

119871 (x119905) minus 119877 (x

119905)

119886119871+ 119887119871minus 119886119877

)119909119871

119905

119909119877

119905+1= 119909119877

119905+ 120575119877(119871 (x119905) minus 119877 (x

119905)

119886119871+ 119887119871minus 119886119877

119909119871

119905

+119872(x119905) minus 119877 (x

119905)

119889 + 119887119871minus 119886119877

119909119872

119905)

(27)

By construction we have that F maps 1198632 into itself In factto show that F 1198632 rarr 119863

2 it is enough to show that

given (119909119871119905 119909119877

119905) isin 119863

2 then (119909119871119905+1 119909119877

119905+1) = F(119909119871

119905 119909119877

119905) isin 119863

2From (119909119871

119905 119909119877

119905) isin 119863

2 we have that 0 le 119909119871119905+ 119909119877

119905le 1 and

119909119871

119905+ 119909119877

119905+ 119909119872

119905= 1 Then if (119909119871

119905+1 119909119877

119905+1) = F119871(119909119871

119905 119909119877

119905) we have

119909119871

119905+1+ 119909119877

119905+1= 119909119871

119905+ 119909119877

119905+ 120575119871

119872(x119905) minus 119871 (x

119905)

119889 + 119887119877minus 119886119871

119909119872

119905

le 119909119871

119905+ 119909119877

119905+ 119909119872

119905= 1

(28)

119909119871

119905+1ge 0 and 119909119877

119905+1ge 0 thus (119909119871

119905+1 119909119877

119905+1) isin 1198632 If (119909119871

119905+1 119909119877

119905+1) =

F119872(119909119871

119905 119909119877

119905) we have 0 le 119909119871

119905+1le 119909119871

119905and 0 le 119909119877

119905+1le 119909119877

119905 thus

(119909119871

119905+1 119909119877

119905+1) isin 1198632 If (119909119871

119905+1 119909119877

119905+1) = F119877(119909119871

119905 119909119877

119905) we have

119909119871

119905+1+ 119909119877

119905+1= 119909119871

119905+ 119909119877

119905+ 120575119877

119872(x119905) minus 119877 (x

119905)

119889 + 119887119871minus 119886119877

119909119872

119905

le 119909119871

119905+ 119909119877

119905+ 119909119872

119905= 1

(29)

119909119871

119905+1ge 0 and 119909119877

119905+1ge 0 thus (119909119871

119905+1 119909119877

119905+1) isin 1198632

Substituting from (18) we can rewrite the functions inexplicit form as

F119871

119909119871

119905+1= 120575119871+ (1 minus 120575

119871minus 120575119871

119887119877

1198964

)119909119871

119905

minus120575119871

119887119871

1198962

119909119877

119905+ 120575119871

119887119871

1198962

(119909119877

119905)2

+120575119871119887119877(1

1198964

minus1

1198962

)119909119871

119905119909119877

119905+ 120575119871

119887119877

1198964

(119909119871

119905)2

119909119877

119905+1= (1 minus 120575

119871minus 120575119871

119887119871

1198962

)119909119877

119905

+120575119871

119887119877

1198962

119909119871

119905119909119877

119905minus 120575119871

119887119871

1198962

(119909119877

119905)2

F119872

119909119871

119905+1= (1 + 120575

119872

1198964

1198963

)119909119871

119905minus 120575119872

119887119877

1198963

(119909119871

119905)2

119909119877

119905+1= (1 + 120575

119872

1198966

1198965

)119909119877

119905minus 120575119872

119887119871

1198965

(119909119877

119905)2

F119877

119909119871

119905+1= (1 minus 120575

119877minus 120575119877

119887119877

1198961

)119909119871

119905

+120575119877

119887119871

1198961

119909119871

119905119909119877

119905minus 120575119877

119887119877

1198961

(119909119871

119905)2

119909119877

119905+1= 120575119877+ (1 minus 120575

119877minus 120575119877

119887119871

1198966

)119909119877

119905

minus120575119877

119887119877

1198961

119909119871

119905+ 120575119877

119887119871

1198961

(119909119877

119905)2

+120575119877119887119871(1

1198966

minus1

1198961

)119909119871

119905119909119877

119905+ 120575119877

119887119877

1198966

(119909119871

119905)2

(30)

Differently from the one-dimensional map 119865 considered inthe previous section the two-dimensional map F is notcontinuous in 1198632 More precisely the functions F

119871 F119872

and F119877are continuous and thus F is continuous in the

interior of each region inwhich the different definitions applyHowever in general except for the point xlowast the map F is notcontinuous along the borders 119881

119894 119894 = 1 2 3 of the regions

119877119871 119877119872 and 119877

119877 In fact it is easy to see that considering

Discrete Dynamics in Nature and Society 9

xL

t+1

xL

t

xR

t

00

1

1

(a)

xL

t+1

xL

t

xR

t

00

1

1

(b)

Figure 5 Graphics of the three functions defining (a) 119909119871119905+1

and (b) 119909119877119905+1

with parameters 119886119871= 29 119887

119871= 16 119886

119877= 19 119887

119877= 24 119889 = 8 120575

119871= 05

120575119872= 08 and 120575

119877= 06

a point (119909119871 119909119877) belonging to one of the lines defined in (22)at the boundary of two regions only we have that the twofunctions defined on the two different regions lead to twodifferent points of1198632 An example of how the three functionsdefining 119909119871

119905+1behave is shown in Figure 5(a) and the three

functions defining 119909119877119905+1

are illustrated in Figure 5(b) showingthat F is continuous only in xlowast

The point xlowast in which the three regions are in contactand F is continuous is clearly particular In fact solving theequation 119871(x) = 119872(x) we have the unique solution 119909119871 = 119909lowast119871while solving the equation 119877(x) = 119872(x) we have the uniquesolution119909119877 = 119909lowast119877 It follows that at the point xlowast = (119909lowast119871 119909lowast119877)we have the same cost in each choice as 119871(xlowast) = 119872(xlowast) =119877(xlowast) so that at the point xlowast the commuters are indifferent toany available choice andnone of themwill switch choiceThusF(xlowast) = xlowast and xlowast is a stationary state Under the conditionsin (23) the point xlowast is feasible belonging to 1198632 We can nowprove that this is indeed the unique fixed point of map F

Proposition 3 The point xlowast = (119909lowast119871 119909lowast119877) given in (21) is theunique fixed point of the map F belonging to 1198632 when theconditions in (23) hold

Proof The proof of this proposition follows directly from theobservation that

F (x) = x if and only if 119871 (x) = 119872 (x) = 119877 (x) (31)One way is already proved as we have detected the fixed pointstarting from the solution of 119871(x) = 119872(x) and 119877(x) = 119872(x)showing that it is the unique point xlowast Vice versa let us lookfor the solutions of F(x) = x We have the following systems(where 119909119872 = 1 minus 119909119871 minus 119909119877)

F119871

119909119871= 119909119871+ 120575119871(119877 (x) minus 119871 (x)

1198962

119909119877

+119872(x) minus 119871 (x)

1198964

119909119872)

119909119877= (1 minus 120575

119871

119877 (x) minus 119871 (x)1198962

)119909119877

if x isin 119877119871

F119872

119909119871= (1 minus 120575

119872

119871 (x) minus 119872 (x)119886119871minus 119889

)119909119871

119909119877= (1 minus 120575

119872

119877 (x) minus 119872 (x)119886119877minus 119889

)119909119877

if x isin 119877119872

F119877

119909119871= (1 minus 120575

119877

119871 (x) minus 119877 (x)1198961

)119909119871

119909119877= 119909119877+ 120575119877(119871 (x) minus 119877 (x)

1198961

119909119871

+119872(x) minus 119877 (x)

1198966

119909119872)

if x isin 119877119877

(32)

or equivalently (we recall that 120575119871 120575119872 and 120575

119877are different

from zero)

F119871

119877 (x) minus 119871 (x)1198962

119909119877+119872(x) minus 119871 (x)

1198964

119909119872= 0

119877 (x) minus 119871 (x)1198962

119909119877= 0

if x isin 119877119871

F119872

119871 (x) minus 119872 (x)119886119871minus 119889

119909119871= 0

119877 (x) minus 119872 (x)119886119877minus 119889

119909119877= 0

if x isin 119877119872

F119877

119871 (x) minus 119877 (x)1198961

119909119871= 0

119871 (x) minus 119877 (x)1198961

119909119871+119872(x) minus 119877 (x)

1198966

119909119872= 0

if x isin 119877119877

(33)

Considering the equation F119871(x) = x we see that it is certainly

satisfied for 119909119877 = 0 and 119909119872 = 0 which leads to thecorner point with 119909119871 = 1 However if the fixed point is an

10 Discrete Dynamics in Nature and Society

interior point of 1198632 then the point (1 0) does not belong tothe proper region 119877

119871 that is it is a virtual fixed point Clearly

this equation is satisfied when 119871(x) = 119872(x) and 119871(x) = 119877(x)which occurs only at the point xlowast It is possible to have xlowast =(1 0) and this occurs for (119886

119871minus 119889)119887

119877= 0 and (119886

119877minus 119889)119887

119871= 1

Thus we can state that the only solution of F119871(x) = x in 119877

119871

is given by the fixed point xlowast Similarly we can reason in theother regions

It is worth to mention that particular cases are obtainedwhen the fixed point xlowast belongs to a boundary of the region1198632 For example when 119909lowast119877+119909lowast119871 = 1 and 119909lowast119872 = 0 then only

the two regions119877119871and 119877

119877are involvedThis occurs when the

parameters satisfy

119889 =119886119871119887119871+ 119886119877119887119877minus 119887119871119887119877

119887119877+ 119887119871

0 le119886119871minus 119889

119887119877

le 1

0 le119886119877minus 119889

119887119871

le 1

(34)

From (119886119877minus 119889)119887

119871= (119886119877+ 119887119877minus 119886119871)(119887119877+ 119887119871) we have 119909lowast119877 =

1minus(119886119877minus119889)119887

119871= (119886119871+119887119871minus119886119877)(119887119877+119887119871) = 1198961(119887119877+119887119871) = 119909lowast as

in the one-dimensional map studied in Section 2 and 119909lowast119871 =1 minus 119909lowast119877= 1 minus (119886

119871minus 119889)119887

119877= 1198962(119887119877+ 119887119871) = 1 minus 119896

1(119887119877+ 119887119871) =

1minus119909lowast From themapsF

119871andF119877in (27) and for the parameters

occurring in this case it can be seen that 119909119871119905+1= 1minus119909

119877

119905+1holds

in both so that the dynamics can be studied by using onlythe variable 119909119877

119905and the resulting one-dimensional map is as

follows

119865 (119909119877)

=

119891119871(119909119877)

= 119909119877+ 120575119877

119871 (119909119877)minus119877 (119909

119877)

1198961

(1 minus 119909119877)

if 0 le 119909119877 le 119909lowast

119891119877(119909119877)

= 119909119877minus 120575119871

119877 (119909119877) minus 119871 (119909

119877)

1198962

119909119877 if 119909lowast le 119909119877 le 1

(35)

which corresponds to the system already studied in Section 2As we have seen this fixed point is globally attracting even iflocally we can have the eigenvalue on some side larger than 1in modulus

The case with 119909lowast119871 = 0 so that 0 le 119909lowast119877 le 1 and 119909lowast119872 =1 minus 119909lowast119877 leaves only the two regions 119877

119877and 119877

119872 This occurs

when the parameters satisfy

119886119871minus 119889

119887119877

= 1 0 le119886119877minus 119889

119887119871

le 1 (36)

From the maps F119872

and F119877in (27) it can be seen that 119909119871

119905

converges to119909lowast119871 = 0 so that the resulting asymptotic dynamic

behavior can be studied via the one-dimensional map givenby 119909119877119905+1= 119879(119909

119877

119905) with 119909119877

119905isin [0 1] (and 119909119872

119905= 1 minus 119909

119877

119905) given by

119879 (119909119877)

=

119879119871(119909119877)

= 119909119877+ 120575119877

119872(119909119877) minus 119877 (119909

119877)

1198966

(1 minus 119909119877)

if 0 le 119909119877 le 119909lowast

119879119877(119909119877)

= 119909119877minus 120575119872

119877 (119909119877) minus119872(119909

119877)

119886119877minus 119889

119909119877 if 119909lowast le 119909119877 le 1

(37)

where119872(119909119877) minus 119877(119909119877) = 119886119871+ 119887119871(1 minus 119909

119877) minus 119886119877minus 119887119877

The case with 119909lowast119877 = 0 so that 0 le 119909lowast119871 le 1 and119909lowast119872= 1 minus 119909

lowast119871 leaves only the two regions 119877119871and 119877

119872 This

occurs when the parameters satisfy 0 le (119886119871minus 119889)119887

119877le 1

(119886119877minus 119889)119887

119871= 1 From the maps F

119872and F

119871in (27) it can

be seen that 119909119877119905converges to 119909lowast119877 = 0 so that the resulting

asymptotic dynamic behavior can be studied via the one-dimensional map given by 119909119871

119905+1= 119879(119909

119871

119905) similar to the one

obtained in (37) with obvious changesBy using arguments as in Section 2 it can be shown that

these one-dimensional systems have a unique fixed pointwhich is globally attracting (although it can be locally stableor unstable)

In order to study the local stability of the fixed point ofthe two-dimensional map F we can use the linearization ofthe different definitions in the fixed point which leads to thefollowing Jacobian matrices one for each region 119877

119871 119877119872 and

119877119877 that we denote respectively with 119869

119871 119869119872 and 119869

119877

119869119871(xlowast) = (

1 + 120575119871minus 120575119871

119887119877

119887119871

1198965

1198964

minus 120575119871

119887119877

119887119871

1198966

1198961

120575119871

1198966

1198961

120575119871

119887119877

119887119871

1198966

1198961

1 minus 120575119871

1198966

1198961

)

119869119872(xlowast) = (

1 minus 120575119872

1198964

1198963

0

0 1 minus 120575119872

1198966

1198965

)

119869119877(xlowast) = (

1 minus 120575119877

1198965

1198962

120575119877

119887119871

119887119877

1198965

1198962

120575119877

1198965

1198962

1 + 120575119877minus 120575119877

119887119871

119887119877

1198963

1198966

minus 120575119877

119887119871

119887119877

1198965

1198962

)

(38)

The two real eigenvalues of the Jacobian matrix in 119877119872are

in explicit form and both are smaller than +1 However theycan be larger than minus1 or not Also the eigenvalues of the otherjacobian matrices can be in modulus smaller or larger than1 However the global dynamics observed numerically givealways trajectories which are convergent to the fixed point xlowastat any set of allowed parameters values

Discrete Dynamics in Nature and Society 11

1

1

075

05

025

00 025 05 075

xR

xL

(a)

1

1

075

05

025

00 025 05 075

xR

xL

(b)

Figure 6 Examples of trajectories converging to the unique fixed point with parameters (a) 119886119871= 28 119887

119871= 24 119886

119877= 8 119887

119877= 24 119889 = 5 120575

119871= 04

120575119872= 03 120575

119877= 07 and ic 119909119871 = 08 119909119877 = 005 (b) 119886

119871= 29 119887

119871= 16 119886

119877= 15 119887

119877= 25 119889 = 7 120575

119871= 05 120575

119872= 08 120575

119877= 06 and ic 119909119871 = 005

119909119877= 09 Dots are linked by a continuous line for illustrative purpose only

This global attractivity in the two-dimensional map isdifficult to be proved rigorously however the results in theparticular cases commented above that is when the fixedpoint xlowast belongs to the boundary of 1198632 suggests that thesame result is true also when the fixed point is in the interiorof1198632

In Figure 6 we show some examples in Figure 6(a) wecan see a trajectory starting at 119877

119877 in Figure 6(b) a trajectory

whose initial point in 119877119871is mapped into 119877

119872 The shape of

the functions in the case of Figure 6(b) is similar to the oneshown in Figure 5

4 Conclusion

In this paper we have analyzed a dynamical version of theBraess paradox with nonimpulsive commuters who changeroad proportionally to the cost differenceWe were interestedto prove that with nonimpulsive commuters there exists aunique equilibrium both in the original network with twochoices and in the ternary onewhen a single link is addedWehave shown that both models are quite robust We were ableto provide an analytical proof of the global attractivity of theunique fixed point in the one-dimensional case when onlytwo roads are available andweusednumerical techniques andsimulations to give evidence in the two-dimensional case (iewhen a new road is added) as a rigorous proof can be doneonly for particular border cases

A limitation of our contribution is considering homo-geneous populations in spite of the fact that the evidencesupports heterogeneity Nevertheless heterogeneity of com-mutersrsquo behavior cannot be easily classified yet even when

conducting experiments common individual patterns arejust identified and classified On the other hand whenobserving the aggregate behavior inmany cases the literatureprovides evidence in support of the paradox outcome thatis the convergence to the Nash equilibrium The impulsiveagents in [30] react faster than the proportional agentspresented in this paper yet as a population they fail toachieve the Nash equilibria in certain cases while in thispaper the aggregate behavior does not We proposed thisaggregate behavior because it replies the convergence to theNash equilibria and at the same time it may be the resultof modeling some observed individual behaviors such asimitation and free riding

Taking into account the limitation of homogeneous pop-ulation analysis a natural extension of our contribution isconsidering heterogeneous populations with both impulsiveand nonimpulsive commuters We have reasons to believethat in such a case some properties of the different behaviorsare inherited and some are lost Therefore the results weprovide in this paper match those for impulsive populationsand are paramount to extend the analysis to more realisticcases In factmdashas the empirical results showmdashheterogeneouspopulations dynamics provide a better fit of what has beenobserved when considering human participants interaction

Furthermore it will be interesting to consider general-izations to networks with richer architectures Finally weobserve that since with the class of behavior we considerin this paper that dynamics depends on the shape of thecost functions it will be interesting to extend the analysis tononlinear cost functions

12 Discrete Dynamics in Nature and Society

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

V Avrutin is supported by the European Community withinthe scope of the project ldquoMultiple-discontinuity inducedbifurcations in theory and applicationsrdquo (Marie Curie Actionof the 7th Framework Programme Contract Agreement noPIEF-GA-2011-300281) U Merlone has worked under theauspices of COST Action IS1104 ldquoThe EU in the new complexgeography of economic systems models tools and policyevaluationrdquo

References

[1] R Arnott and K Small ldquoThe economics of traffic congestionrdquoAmerican Scientist vol 82 no 5 pp 446ndash455 1994

[2] T Bass ldquoRoad to ruinrdquo Discover pp 56ndash61 1992[3] B Calvert and G Keady ldquoBraessrsquos paradox and power-law non-

linearities in networksrdquoThe Journal of the AustralianMathemat-ical Society B Applied Mathematics vol 35 no 1 pp 1ndash22 1993

[4] J E Cohen andC Jeffries ldquoCongestion resulting from increasedcapacity in single-server queueing networksrdquo IEEEACMTrans-actions on Networking vol 5 no 2 pp 305ndash310 1997

[5] J E Cohen and F P Kelly ldquoA paradox of congestion in a queuingnetworkrdquo Journal of Applied Probability vol 27 no 3 pp 730ndash734 1990

[6] S Dafermos and A Nagurney ldquoOn some traffic equilibriumtheory paradoxesrdquo Transportation Research B Methodologicalvol 18 no 2 pp 101ndash110 1984

[7] C Fisk ldquoMore paradoxes in the equilibrium assignment prob-lemrdquo Transportation Research B Methodological vol 13 no 4pp 305ndash309 1979

[8] W H K Lam ldquoEffects of road pricing on system performancerdquoTraffic Engineering amp Control vol 29 no 12 pp 631ndash635 1988

[9] J D Murchland ldquoBraessrsquos paradox of traffic flowrdquo Transporta-tion Research vol 4 no 4 pp 391ndash394 1970

[10] R Steinberg and W I Zangwill ldquoThe prevalence o f BraessrsquoparadoxrdquoTransportation Science vol 17 no 3 pp 301ndash318 1983

[11] M Frank ldquoThe Braess paradoxrdquo Mathematical Programmingvol 20 no 1 pp 283ndash302 1981

[12] T Roughgarden Selfish Routing and the Price of Anarchy MITpress 2005

[13] B B Fu C X Zhao and S B Li ldquoThe analysis of Braessrsquo paradoxand robustness based on dynamic traffic assignment modelsrdquoDiscrete Dynamics in Nature and Society vol 2013 Article ID796842 8 pages 2013

[14] C Fisk and S Pallottino ldquoEmpirical evidence for equilibriumparadoxes with implications for optimal planning strategiesrdquoTransportation Research Part A General vol 15 no 3 pp 245ndash248 1981

[15] H Youn M T Gastner and H Jeong ldquoPrice of anarchy intransportation networks efficiency and optimality controlrdquoPhysical Review Letters vol 101 no 12 Article ID 128701 2008

[16] K Aoki Y Ohtsubo A Rapoport and T Saijo ldquoEffects of priorinvestment and personal responsibility in a simple network

gamerdquo Current Research in Social Psychology vol 13 no 2 pp10ndash21 2007

[17] J Morgan H Orzen and M Sefton ldquoNetwork architectureand traffic flows experiments on the Pigou-Knight-Downs andBraess paradoxesrdquoGames and Economic Behavior vol 66 no 1pp 348ndash372 2009

[18] A Rapoport V Mak and R Zwick ldquoNavigating congested net-works with variable demand experimental evidencerdquo Journal ofEconomic Psychology vol 27 no 5 pp 648ndash666 2006

[19] A Rapoport T Kugler S Dugar and E J Gisches ldquoBraessparadox in the laboratory an experimental study of route choicein traffic networks with asymmetric costsrdquo inDecisionModelingand Behavior in Uncertain and Complex Environment T KuglerJ C Smith T Connolly and Y J Son Eds pp 309ndash337Springer New York NY USA 2008

[20] A Rapoport T Kugler S Dugar and E J Gisches ldquoChoice ofroutes in congested traffic networks experimental tests of theBraess paradoxrdquo Games and Economic Behavior vol 65 no 2pp 538ndash571 2009

[21] E J Gisches and A Rapoport ldquoDegrading network capacitymay improve performance private versus public monitoring inthe Braess paradoxrdquoTheory and Decision vol 73 no 2 pp 267ndash293 2012

[22] A Dal Forno and U Merlone ldquoReplicating human interactionin Braess paradoxrdquo in Proceedings of the Simulation Conference(WSC rsquo13) pp 1754ndash1765 IEEE 2013

[23] M Ausloos H Dawid and U Merlone ldquoSpatial interactionsin agent-based modelingrdquo in Complexity and GeographicalEconomics Topics and Tools P Commendatore S S Kayamand I Kubin Eds Springer Berlin Germany 2014

[24] G-I Bischi and U Merlone ldquoGlobal dynamics in binary choicemodels with social influencerdquo The Journal of MathematicalSociology vol 33 no 4 pp 277ndash302 2009

[25] G I Bischi L Gardini and U Merlone ldquoImpulsivity in binarychoices and the emergence of periodicityrdquoDiscrete Dynamics inNature and Society vol 2009 Article ID 407913 22 pages 2009

[26] JH PattonM S Stanford andE S Barratt ldquoFactor structure ofthe Barratt Impulsiveness Scalerdquo Journal of Clinical Psychologyvol 51 no 6 pp 768ndash774 1995

[27] G I Bischi L Gardini and U Merlone ldquoPeriodic cycles andbifurcation curves for one-dimensional maps with two discon-tinuitiesrdquo Journal of Dynamical Systems GeometricTheories vol7 no 2 pp 101ndash123 2009

[28] G-I Bischi and U Merlone ldquoBinary choices in small and largegroups a unifiedmodelrdquo Physica A vol 389 no 4 pp 843ndash8532010

[29] A Dal Forno L Gardini and U Merlone ldquoTernary choicesin repeated games and border collision bifurcationsrdquo ChaosSolitons amp Fractals vol 45 no 3 pp 294ndash305 2012

[30] A Dal Forno and U Merlone ldquoBorder-collision bifurcationsin a model of Braess paradoxrdquo Mathematics and Computers inSimulation vol 87 pp 1ndash18 2013

[31] A Dal Forno V Giorgino and U Merlone ldquoAgent-based mod-eling with GTmethodology An examplerdquoWorking Paper 2014

[32] L Gardini U Merlone and F Tramontana ldquoInertia in binarychoices continuity breaking and big-bang bifurcation pointsrdquoJournal of Economic Behavior amp Organization vol 80 no 1 pp153ndash167 2011

[33] W H Sandholm Population Games and Evolutionary Dynam-ics The MIT Press Cambridge Mass USA 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 3

L R

S

E

aL

aRbLxL

bRxR

(a)

aR

aL

L R

S

E

d

bL(xL+ x

M)

bR(xM

+ xR)

(b)

Figure 1 Basic network with the addition of the link from 119871 to 119877 in (b) and without in (a)

purpose assume the set of commuters is normalized to theinterval [0 1] At each time period 119905 ge 0 119909119877

119905ge 0 indicates the

fraction of commuters choosing 119877 and 119909119871119905ge 0 the fraction of

those choosing 119871 given by 119909119871119905= 1 minus 119909

119877

119905 Therefore the travel

times are given by

119871 = 119886119871+ 119887119871119909119871

119877 = 119886119877+ 119887119877119909119877

(1)

where 119886119871 119886119877 119887119871 119887119877gt 0 and can be written as

119871 (119909119877) = 119886119871+ 119887119871(1 minus 119909

119877)

119877 (119909119877) = 119886119877+ 119887119877119909119877

(2)

This way the travel times on the two roads are equal (andthe commuters are indifferent) when 119871(119909119877) = 119877(119909119877) that iswhen 119909119877 = 119909lowast where

119909lowast=119886119871+ 119887119871minus 119886119877

119887119871+ 119887119877

(3)

We say that the threshold 119909lowast is feasible when 119909lowast isin [0 1]However 119909lowast = 0 (resp 119909lowast = 1) clearly corresponds to thecase in which the whole population chooses action 119871 (resp119877) and we shall see that this is indeed satisfied in the modeldescribed in the next subsection We obtain the followingconstraints on the parameters in order to have 119909lowast isin (0 1)which we assume henceforth

1198961= 119886119871+ 119887119871minus 119886119877gt 0

1198962= 119886119877+ 119887119877minus 119886119871gt 0

(4)

21 The One-Dimensional PWS Map and Its DynamicsCommuters are homogeneous and at each period decidetheir strategy considering the previous period travel costsassuming they will remain the same At time 119905+1 the fraction

119909119877

119905is common knowledge and each commuter observes the

respective travel costs either 119871(119909119877119905) or 119877(119909119877

119905) depending on

their choice The commuters decide their future action attime 119905 + 1 comparing the costs 119871(119909119877

119905) and 119877(119909119877

119905) We assume

the commuters are nonimpulsive and myopically minimizetheir costs That is they change strategy proportionally tothe difference between their chosen road and the best onewhere the best road is the fastest We introduce parameters120575119871isin (0 1) and 120575

119877isin (0 1) as switching rates that is

the propensity for moving either to 119871 or to 119877 respectivelyWith nonimpulsive commuters these rates are adjusted by thenormalized difference of travel costs The difference 119871(119909119877

119905) minus

119877(119909119877

119905) is normalized by the largest value of this difference

denoted by 1198961and obtained in correspondence of 119909119877

119905= 0

Therefore we have 1198961= 119871(0) minus 119877(0) = 119886

119871+ 119887119871minus 119886119877as

already introduced in (4) Thus commuters are switchingroad proportionally to the side of the payoff difference Sim-ilarly we can give a reason with respect to the difference119877(119909119877

119905) minus 119871(119909

119877

119905) to get a normalizing value Now the largest

value is obtained in correspondence of 119909119871119905= 0 that is 119909119877

119905= 1

andwe get 1198962= 119877(1)minus119871(1) = 119886

119877+119887119877minus119886119871as already introduced

in (4)The resulting dynamics is 119909119877

119905+1= 119865(119909

119877

119905) with 119909119877

119905isin [0 1]

and the map 119865 [0 1] rarr [0 1] is defined as

119865 (119909119877)

=

119891119871(119909119877)

= 119909119877+ 120575119877

119871 (119909119877) minus 119877 (119909

119877)

1198961

(1 minus 119909119877)

if 0 le 119909119877 le 119909lowast

119891119877(119909119877)

= 119909119877minus 120575119871

119877 (119909119877) minus 119871 (119909

119877)

1198962

119909119877 if 119909lowast le 119909119877 le 1

(5)

4 Discrete Dynamics in Nature and Society

Substituting from (2) and rearranging we obtain

119865 (119909119877) =

119891119871(119909119877) = 120575119877+ (1 minus 120575

119877minus 120575119877

119887119871+ 119887119877

1198961

)119909119877

+120575119877

119887119871+ 119887119877

1198961

(119909119877)2

if 0 le 119909119877 le 119909lowast

119891119877(119909119877) = (1 + 120575

119871

1198961

1198962

)119909119877

minus120575119871

119887119871+ 119887119877

1198962

(119909119877)2

if 119909lowast le 119909119877 le 1

(6)

From the definition of 119909lowast given in (3) it can also be writtenas

119909lowast=119886119871+ 119887119871minus 119886119877

119887119871+ 119887119877

=1198961

119887119871+ 119887119877

=1198961

1198961+ 1198962

(7)

at which we have 119871(119909lowast) = 119877(119909lowast) it is also 119891119871(119909lowast) = 119891119877(119909lowast)

and thus 119865 is continuous in 119909lowast and it is a fixed point as119865(119909lowast) = 119909lowast We have the following proposition

Proposition 1 Map 119865 is continuous in [0 1] and under theconditions given in (4) 119909lowast = 119896

1(119887119871+ 119887119877) isin (0 1) is the unique

fixed point

Proof Only the uniqueness is left to prove The equation119891119871(119909119877) = 119909

119877 has two solutions 119909119877 = 119909lowast and 119909119877 = 1 whichis larger than 119909lowast and thus a virtual fixed point The equation119891119877(119909119877) = 119909

119877 has two solutions 119909119877 = 119909lowast and 119909119877 = 0 whichis smaller than 119909lowast and thus a virtual fixed point Thus theunique solution is 119909119877 = 119909lowast

Clearly the fixed point may be locally stable or instabledepending on the slopes of the functions on the right and leftsides of 119909lowast However we shall see that in any case it is globallyattractingThat is any ic in the interval [0 1] has a trajectorywhich converges to 119909lowast

Proposition 2 The fixed point 119909lowast = 1198961(119887119871+ 119887119877) isin (0 1) is

globally attracting

Proof Even if 119865(119909119877) is not smooth in its fixed point (asindeed 119909lowast is a kink point of this map) the two components119891119871(119909119877) and 119891

119877(119909119877) are smooth in 119909lowast so that the left and right

side derivatives of 119865(119909119877) in 119909lowast are well defined 1198651015840minus(119909lowast) =

lim119909119877rarr119909lowastminus1198651015840(119909119877) = 1198911015840

119871(119909lowast) and1198651015840

+(119909lowast) = lim

119909119877rarr119909lowast+1198651015840(119909119877) =

1198911015840

119877(119909lowast)

We have

1198911015840

119871(119909119877) = (1 minus 120575

119877minus 120575119877

119887119871+ 119887119877

1198961

) + 2120575119877

119887119871+ 119887119877

1198961

119909119877 (8)

1198911015840

119871(119909lowast) = 1 minus 120575

119877

1198962

1198961

(9)

while

1198911015840

119877(119909119877) = (1 + 120575

119871

1198961

1198962

) minus 2120575119871

119887119871+ 119887119877

1198962

119909119877 (10)

1198911015840

119877(119909lowast) = 1 minus 120575

119871

1198961

1198962

(11)

From

11989110158401015840

119871(119909119877) = 2120575

119877

119887119871+ 119887119877

1198961

gt 0 11989110158401015840

119877(119909119877) = minus2120575

119871

119887119871+ 119887119877

1198962

lt 0

(12)

we have that on the left side of the fixed point the function isconvex while on the right side of the fixed point the functionis concave

Notice that 11989621198961only depends on 119886

119871 119887119871 119886119877 119887119877 and it is

positive under our assumptions Depending on the values ofthe ratio 119896

21198961we can classify the rightleft stability of the

fixed point 119909lowast From (9) we clearly have always 1198911015840119871(119909lowast) lt 1

thus the stabilityinstability on the left side of the fixed pointdepends on the condition 1198911015840

119871(119909lowast) gt minus1 which holds if and

only if

1198962

1198961

lt2

120575119877

(13)

Similarly from (11) we clearly have always 1198911015840119877(119909lowast) lt 1 and

the stabilityinstability on the right side of the fixed pointdepends on the condition 1198911015840

119877(119909lowast) gt minus1 which holds if and

only if

1198962

1198961

gt120575119871

2 (14)

And clearly we have the following sequence of inequalities (tobe used below)

0 lt120575119871

2lt 120575119871lt 1 lt

1

120575119877

lt2

120575119877

(15)

To investigate the local stability of the fixed point we considerthree intervals for 119896

21198961

(i) 0 lt 11989621198961le 1205751198712

(ii) 1205751198712 lt 119896

21198961lt 2120575

119877

(iii) 2120575119877le 11989621198961

In case (i) it is 1198911015840119877(119909lowast) le minus1 moreover it is easy to see

that for any 119909119877 isin [119909lowast 1] it is 1198911015840119877(119909119877) lt 119891

1015840

119877(119909lowast) and thus

the function on the right side of the fixed point is strictlydecreasing and expansive each point on the right side of thefixed point is mapped to the left side of the fixed point in oneiteration While on the left side of the fixed point the convexfunction 119891

119871(119909119877) has 1198911015840

119871(119909lowast) = 1 minus 120575

119877(11989621198961) gt 0 (as this

holds for 11989621198961lt 1120575

119877which is satisfied in this interval)This

implies that the point of local minimum on the left side say119909119877

119897119898 is smaller than the fixed point 119909119877

119897119898lt 119909lowast Then the fixed

point is unique and the map is a contraction in the interval119882119904

loc = [119909119877

119897119898 119909lowast] and an absorbing interval which is mapped

into 119882119904loc in one iteration is given by 119869 = [119909119877119897119898 119891minus1

119877(119909119877

119897119898)]

when 119891119877(1) = 1 minus 120575

119871lt 119909119877

119897119898 or 119869 = [119909119877

119897119898 1] when 119891

119877(1) =

1 minus 120575119871ge 119909119877

119897119898 Any initial condition belonging to [0 1] 119869

has a trajectory which is mapped into 119869 in a finite number ofiterationsThus119909lowast is globally attracting An example is shownin Figure 2(a)

Similarly we can reason when 11989621198961belongs to the third

interval In fact in case (iii) it is 1198911015840119871(119909lowast) le minus1 moreover it is

Discrete Dynamics in Nature and Society 5xR t+1

xR

t

00

025

025 05

05

075

1

1075

(a)xR t+1

xR

t

00

025

025 05

05

075

1

1075

(b)

Figure 2 Qualitative shapes of the map In (a) case (i) when 0 lt 11989621198961le 1205751198712 the fixed point is locally unstable on the right side In (b)

case (iii) when 2120575119877le 11989621198961 the fixed point is locally unstable on the left side

0

1

0 1

xR t

xL

t

(a)

0

1

0 1

xR t

xL

t

(b)

0

1

0 1

xR t

xL

t

(c)

Figure 3 Qualitative shapes of the map in case (ii) when 1205751198712 lt 119896

21198961le 2120575

119877and the fixed point is locally stable on both sides

easy to see that for any 119909119877 isin [0 119909lowast] it is 1198911015840119871(119909lowast) lt 1198911015840

119871(119909lowast) and

thus the function on the left side of the fixed point is strictlydecreasing and expansive each point on the left side of thefixed point is mapped to the right side of the fixed point inone iteration While on the right side of the fixed point theconcave function 119891

119877(119909119877) has 1198911015840

119877(119909lowast) = 1 minus 120575

119871(11989611198962) gt 0 (as

this holds for 11989621198961gt 120575119871which is satisfied in this interval)

This implies that the point of local maximum on the rightside say 119909119877

119903119872 is larger than the fixed point 119909119877

119903119872gt 119909lowast Then

the map is a contraction in the interval 119882119904loc = [119909lowast 119909119877

119903119872]

and an absorbing interval which is mapped into119882119904loc in oneiteration is given by 119869 = [119891minus1

119871(119909119877

119903119872) 119909119877

119903119872] when 119891

119871(0) =

120575119877gt 119909119877

119903119872 or 119869 = [0 119909119877

119903119872] when 119891

119871(0) = 120575

119877le 119909119877

119903119872 Any

initial condition belonging to [0 1] 119869 has a trajectory whichis mapped into 119869 in a finite number of iterations Thus 119909lowast isglobally attracting An example is shown in Figure 2(b)

When 11989621198961belongs to the interval in (ii) 120575

1198712 lt 119896

21198961lt

2120575119877 we have that the fixed point is locally stable on both

sides Moreover inside this interval we can distinguish threeregions as already showed in (15) that is

(a) 1205751198712 lt 119896

21198961le 120575119871with 0 lt 1198911015840

119871(119909lowast) lt 1 and

minus1 lt 1198911015840

119877(119909lowast) lt 0 as in Figure 3(a)

(b) 120575119871lt 11989621198961lt 1120575

119877with 0 lt 1198911015840

119871(119909lowast) lt 1 and

0 lt 1198911015840

119877(119909lowast) lt 1 as in Figure 3(b)

6 Discrete Dynamics in Nature and Society

(c) 1120575119877le 11989621198961lt 2120575

119877with minus1 lt 1198911015840

119871(119909lowast) lt 0 and

0 lt 1198911015840

119877(119909lowast) lt 1 as in Figure 3(c)

In case (a) we have that the point of local minimum onthe left side say 119909119877

119897119898 is smaller than the fixed point119909119877

119897119898lt 119909lowast

Then themap is a contraction in the interval119882119904loc = [119909119877

119897119898 119909lowast]

and on the right side the map is decreasing so that anabsorbing interval which is mapped into119882119904loc in one iterationis given by 119869 = [119909119877

119897119898 119891minus1

119877(119909119877

119897119898)]when119891

119877(1) = 1minus120575

119871lt 119909119877

119897119898 or

119869 = [119909119877

119897119898 1]when119891

119877(1) = 1minus120575

119871ge 119909119877

119897119898 Any initial condition

belonging to [0 1] 119869 has a trajectory which is mapped into 119869in a finite number of iterationsThus 119909lowast is globally attracting

In case (c) we can reason similarly we have that the pointof local maximum on the right side say 119909119877

119903119872 is larger than

the fixed point 119909119877119903119872gt 119909lowast Then the map is a contraction in

the interval 119882119904loc = [119909lowast 119909119877

119903119872] and on the left side the map

is decreasing so that an absorbing interval which is mappedinto 119882119904loc in one iteration is given by 119869 = [119891minus1

119871(119909119877

119903119872) 119909119877

119903119872]

when 119891119871(0) = 120575

119877gt 119909119877

119903119872 or 119869 = [0 119909119877

119903119872] when 119891

119871(0) =

120575119877le 119909119877

119903119872 Any initial condition belonging to [0 1] 119869 has

a trajectory which is mapped into 119869 in a finite number ofiterations Thus 119909lowast is globally attracting

In case (b) we have that the point of local minimum onthe left side say 119909119877

119897119898 is smaller than the fixed point 119909119877

119897119898lt

119909lowast and also that the point of local maximum on the right

side say 119909119877119903119872

is larger than the fixed point 119909119877119903119872gt 119909lowast

Then the map is a contraction in the interval 119882119904loc =

[119909119877

119897119898 119909119877

119903119872] which includes the fixed point and an absorbing

interval which is mapped into119882119904loc in one iteration is givenby 119869 = [119891minus1

119871(119909119877

119903119872) 119891minus1

119877(119909119877

119897119898)] or 119869 = [0 119891minus1

119877(119909119877

119897119898)] or 119869 =

[119891minus1

119871(119909119877

119903119872) 1] or [0 1] depending under obvious conditions

on the values of 120575119871and 120575

119877 When 119869 sub [0 1] then any initial

condition belonging to [0 1] 119869 has a trajectory which ismapped into 119869 in a finite number of iterations Thus 119909lowast isglobally attracting

We notice that in Proposition 2 we prove that when thefixed point 119909lowast is internal to the interval [0 1] then it isglobally attracting Let us turn to comment the two extremawhich occur when 119909lowast = 0 or 119909lowast = 1 These solutions clearlyexist and are also globally attracting In fact the case 119909lowast = 0corresponds to 119896

1= 119871(0)minus119877(0) = 119886

119871+119887119871minus119886119877= 0 and only the

function 119891119877(119909119877) is applied in the interval [0 1] and from the

concavity of the function we have that this fixed point is glob-ally attracting even if the eigenvalue is equal to 1 The othercase 119909lowast = 1 corresponds to 119896

2= 119877(1)minus119871(1) = 119886

119877+119887119877minus119886119871= 0

and only the function 119891119871(119909119877) is applied in the interval [0 1]

and from the convexity of the function we have that this fixedpoint is globally attracting even if the eigenvalue is equal to 1

3 The Dynamics in the ExpandedNetwork with Three Roads

As it is well known the Braess paradox introduces a new linkand shows how this opportunity worsens both the individualand collective payoffs This link introduces a new alternativenow the commuters are facing three choices where the new

one consists of path 119878-119871-119877-119864 (see Figure 1(b)) which will bedenoted as 119872 in the following Models with three choiceswere introduced in [29] as it concerns discrete time and [33]as it concerns continuous time In this paper we analyze thedynamics when the travel time 119889 for the new resulting linkvaries in a range that makes this link a dominant choice forsome values and a dominated choice for others It is worth toobserve that in particular when the travel time on link 119871-119877 iszero the paradoxical result still holds

Now assume that commuters switch roadwhenever traveltime has become larger that is the other road is less costlyand therefore more attractive We indicate by 119909119871 ge 0 thefraction of population choosing action 119871 by 119909119872 ge 0 thefraction choosing action 119872 and by 119909119877 ge 0 the fractionchoosing action 119877 with the constraint 119909119871 + 119909119877 + 119909119872 = 1The travel times are given respectively as

119871 = 119886119871+ 119887119871(119909119871+ 119909119872)

119872 = 119887119871(119909119871+ 119909119872) + 119887119877(119909119872+ 119909119877) + 119889

119877 = 119886119877+ 119887119877(119909119872+ 119909119877)

(16)

By using the constraint let us define the fraction of populationchoosing119872 as 119909119872 = 1 minus 119909119871 minus 119909119877 ge 0 so to consider onlytwo variables x = (119909119871 119909119877) isin R2

+ and the phase space is the

triangle

1198632= x = (119909119871 119909119877) isin R2

+ 0 le 119909

119871+ 119909119877le 1 (17)

After substituting the expression of 119909119872 in the travel times weget

119871 (x) = 119886119871 + 119887119871 (1 minus 119909119877)

119872 (x) = 119887119871 (1 minus 119909119877) + 119887119877(1 minus 119909

119871) + 119889

119877 (x) = 119886119877 + 119887119877 (1 minus 119909119871)

(18)

The vertices of the phase space119875lowast119871(1 0)119875lowast

119872(0 0) and119875lowast

119877(0 1)

represent states in which the whole population choosesrespectively actions 119871119872 and 119877

Since commuters are interested in minimizing traveltime we denote by 119877

119860the region in which the choice 119860

(where119860 is 119871119872 or 119877) is preferable These regions are calleddominance regions of 119860 We have the following definition ofthe dominance regions

119877119871= x isin 1198632 119871 (x) le 119872 (x) 119871 (x) le 119877 (x)

119877119872= x isin 1198632 119872 (x) le 119871 (x) 119872 (x) le 119877 (x)

119877119877= x isin 1198632 119877 (x) le 119871 (x) 119877 (x) le 119872 (x)

(19)

Discrete Dynamics in Nature and Society 7

1

100

xR

RL

RM

RR

xL

(a)

1

100

xR

RL

RM

RR

xL

(b)

1

100

xR

RL

RM

RR

xL

(c)

Figure 4 Dominance regions in the three choices network with different values of the cost parameter 119889 of the map F at 119886119871= 27 119887

119871= 30

119886119877= 18 119887

119877= 30 and (a) 119889 = 6 (b) 119889 = 3 (c) 119889 = 0

By using the expressions in (18) we get the following defini-tions for the regions

119877119871= x isin 1198632 119909119871 le 119909lowast119871

119886119871+ 119887119871(1 minus 119909

119877) le 119886119877+ 119887119877(1 minus 119909

119871)

119877119872= x isin 1198632 119909119871 ge 119909lowast119871 119909119877 ge 119909lowast119877

119877119877= x isin 1198632 119886

119877+ 119887119877(1 minus 119909

119871) le 119886119871+ 119887119871(1 minus 119909

119877)

119909119877le 119909lowast119877

(20)

where we have introduced

119909lowast119871= 1 minus

119886119871minus 119889

119887119877

=119889 minus 119886119871+ 119887119877

119887119877

119909lowast119877= 1 minus

119886119877minus 119889

119887119871

=119889 minus 119886119877+ 119887119871

119887119871

(21)

The boundaries of the three regions are given by segments ofstraight lines of equations

1198811 119909119871= 119909lowast119871

1198812 119909119877= 119909lowast119877

1198813 119886119871+ 119887119871(1 minus 119909

119877) = 119886119877+ 119887119877(1 minus 119909

119871)

(22)

and it is immediate to see that the point xlowast = (119909lowast119871 119909lowast119877)satisfies also the third equation that is it belongs to all thethree straight lines xlowast isin 119881

119894for 119894 = 1 2 3

Using the geometric representation in the triangle 1198632the dominance regions are illustrated in Figure 4 (via anexample) using different colors Three cases are shown fordifferent values of the parameter 119889 in order to illustrate howthe region119877

119872depends on the parameters of the third branch

The point xlowast = (119909lowast119871 119909lowast119877) is the unique point on theboundaries 119881

119894of all the three regions Thus for our model we

have to require 119909lowast119871 ge 0 119909lowast119877 ge 0 and 119909lowast119871 + 119909lowast119877 le 1 leadingto 119909lowast119872 = 1 minus 119909lowast119871 minus 119909lowast119877 In order to have three regions in1198632

it must be 119909lowast119871 gt 0 119909lowast119877 gt 0 and 119909lowast119871 + 119909lowast119877 lt 1 The equalityin one of these three conditions implies the disappearance ofone region For example when 119909lowast119871 = 0 then 0 le 119909lowast119877 le 1and 119909lowast119872 = 1 minus 119909lowast119877 leaving only the two regions 119877

119877and 119877

119872

and so onThe conditions 119909lowast119871 ge 0 119909lowast119877 ge 0 and 119909lowast119871 + 119909lowast119877 le 1 lead

to the constraints that will be assumed satisfied henceforth

0 le119886119871minus 119889

119887119877

le 1 0 le119886119877minus 119889

119887119871

le 1

119886119871minus 119889

119887119877

+119886119877minus 119889

119887119871

ge 1

(23)

We notice that from the third condition in (23) we have thevalues of 119889 for which the third region is really present in 1198632that is

0 le 119889 le119886119871119887119871+ 119886119877119887119877minus 119887119871119887119877

119887119877+ 119887119871

(24)

Commuters are homogeneous and minimize their nextperiod travel cost At time 119905+1 the vector x

119905becomes common

knowledge and each commuter can observe the travel costs119871(x119905)119872(x

119905) and 119877(x

119905) We assume that if at time 119905 a fraction

119909119871

119905chooses path 119871 a fraction 119909119877

119905chooses action 119877 and travel

costs are such that 119877(x119905) lt 119871(x

119905) and 119877(x

119905) lt 119872(x

119905) then

both a fraction of the 119909119871119905commuters who chose action 119871 and

a fraction of the (1 minus 119909119871119905minus 119909119877

119905) commuters who chose action

119872 will switch to path 119877 in the next time period 119905 + 1 Thefraction of switching commuters is given by the switchingpropensities 120575

119871 120575119872 120575119877modulated by the relative differences

in payoffs This happens whenever a path gives a smallertravel cost In other words at any time 119905 all the commutersdecide their future action at time 119905 + 1 comparing the costs119871(x119905)119872(x

119905) and 119877(x

119905) As in the one-dimensional case here

also the differences of travel costs are normalized dividing

8 Discrete Dynamics in Nature and Society

each difference by the constants giving the related maximumvalue given in the different cases as follows

1198961= max

x [119871 (x) minus 119877 (x)]

= [119871 (1 0) minus 119877 (1 0)] = 119886119871 + 119887119871 minus 119886119877

1198962= max

x [119877 (x) minus 119871 (x)]

= [119877 (0 1) minus 119871 (0 1)] = 119886119877 + 119887119877 minus 119886119871

1198963= max

x [119871 (x) minus 119872 (x)]

= [119871 (1 119909119877) minus119872(1 119909

119877)] = 119886

119871minus 119889 forall119909

119877isin [0 1]

1198964= max

x [119872 (x) minus 119871 (x)]

= [119872(0 119909119877) minus 119871 (0 119909

119877)] = 119889 + 119887

119877minus 119886119871 forall119909

119877isin [0 1]

1198965= max

x [119877 (x) minus 119872 (x)]

= [119877 (119909119871 1) minus 119872(119909

119871 1)] = 119886

119877minus 119889 forall119909

119871isin [0 1]

1198966= max

x [119872 (x) minus 119877 (x)]

= [119872(119909119871 0) minus 119877 (119909

119871 0)] = 119889 + 119887

119871minus 119886119877 forall119909

119871isin [0 1]

(25)

The resulting dynamics are described by a two-dimensionalmap x

119905+1= F(x

119905) defined as

F x119905+1=

F119871(x119905) if x

119905isin 119877119871

F119872(x119905) if x

119905isin 119877119872

F119877(x119905) if x

119905isin 119877119877

(26)

where

F119871

119909119871

119905+1= 119909119871

119905+ 120575119871(119877 (x119905) minus 119871 (x

119905)

119886119877+ 119887119877minus 119886119871

119909119877

119905

+119872(x119905) minus 119871 (x

119905)

119889 + 119887119877minus 119886119871

119909119872

119905)

119909119877

119905+1= (1 minus 120575

119871

119877 (x119905) minus 119871 (x

119905)

119886119877+ 119887119877minus 119886119871

)119909119877

119905

F119872

119909119871

119905+1= (1 minus 120575

119872

119871 (x119905) minus 119872(x

119905)

119886119871minus 119889

)119909119871

119905

119909119877

119905+1= (1 minus 120575

119872

119877 (x119905) minus 119872(x

119905)

119886119877minus 119889

)119909119877

119905

F119877

119909119871

119905+1= (1 minus 120575

119877

119871 (x119905) minus 119877 (x

119905)

119886119871+ 119887119871minus 119886119877

)119909119871

119905

119909119877

119905+1= 119909119877

119905+ 120575119877(119871 (x119905) minus 119877 (x

119905)

119886119871+ 119887119871minus 119886119877

119909119871

119905

+119872(x119905) minus 119877 (x

119905)

119889 + 119887119871minus 119886119877

119909119872

119905)

(27)

By construction we have that F maps 1198632 into itself In factto show that F 1198632 rarr 119863

2 it is enough to show that

given (119909119871119905 119909119877

119905) isin 119863

2 then (119909119871119905+1 119909119877

119905+1) = F(119909119871

119905 119909119877

119905) isin 119863

2From (119909119871

119905 119909119877

119905) isin 119863

2 we have that 0 le 119909119871119905+ 119909119877

119905le 1 and

119909119871

119905+ 119909119877

119905+ 119909119872

119905= 1 Then if (119909119871

119905+1 119909119877

119905+1) = F119871(119909119871

119905 119909119877

119905) we have

119909119871

119905+1+ 119909119877

119905+1= 119909119871

119905+ 119909119877

119905+ 120575119871

119872(x119905) minus 119871 (x

119905)

119889 + 119887119877minus 119886119871

119909119872

119905

le 119909119871

119905+ 119909119877

119905+ 119909119872

119905= 1

(28)

119909119871

119905+1ge 0 and 119909119877

119905+1ge 0 thus (119909119871

119905+1 119909119877

119905+1) isin 1198632 If (119909119871

119905+1 119909119877

119905+1) =

F119872(119909119871

119905 119909119877

119905) we have 0 le 119909119871

119905+1le 119909119871

119905and 0 le 119909119877

119905+1le 119909119877

119905 thus

(119909119871

119905+1 119909119877

119905+1) isin 1198632 If (119909119871

119905+1 119909119877

119905+1) = F119877(119909119871

119905 119909119877

119905) we have

119909119871

119905+1+ 119909119877

119905+1= 119909119871

119905+ 119909119877

119905+ 120575119877

119872(x119905) minus 119877 (x

119905)

119889 + 119887119871minus 119886119877

119909119872

119905

le 119909119871

119905+ 119909119877

119905+ 119909119872

119905= 1

(29)

119909119871

119905+1ge 0 and 119909119877

119905+1ge 0 thus (119909119871

119905+1 119909119877

119905+1) isin 1198632

Substituting from (18) we can rewrite the functions inexplicit form as

F119871

119909119871

119905+1= 120575119871+ (1 minus 120575

119871minus 120575119871

119887119877

1198964

)119909119871

119905

minus120575119871

119887119871

1198962

119909119877

119905+ 120575119871

119887119871

1198962

(119909119877

119905)2

+120575119871119887119877(1

1198964

minus1

1198962

)119909119871

119905119909119877

119905+ 120575119871

119887119877

1198964

(119909119871

119905)2

119909119877

119905+1= (1 minus 120575

119871minus 120575119871

119887119871

1198962

)119909119877

119905

+120575119871

119887119877

1198962

119909119871

119905119909119877

119905minus 120575119871

119887119871

1198962

(119909119877

119905)2

F119872

119909119871

119905+1= (1 + 120575

119872

1198964

1198963

)119909119871

119905minus 120575119872

119887119877

1198963

(119909119871

119905)2

119909119877

119905+1= (1 + 120575

119872

1198966

1198965

)119909119877

119905minus 120575119872

119887119871

1198965

(119909119877

119905)2

F119877

119909119871

119905+1= (1 minus 120575

119877minus 120575119877

119887119877

1198961

)119909119871

119905

+120575119877

119887119871

1198961

119909119871

119905119909119877

119905minus 120575119877

119887119877

1198961

(119909119871

119905)2

119909119877

119905+1= 120575119877+ (1 minus 120575

119877minus 120575119877

119887119871

1198966

)119909119877

119905

minus120575119877

119887119877

1198961

119909119871

119905+ 120575119877

119887119871

1198961

(119909119877

119905)2

+120575119877119887119871(1

1198966

minus1

1198961

)119909119871

119905119909119877

119905+ 120575119877

119887119877

1198966

(119909119871

119905)2

(30)

Differently from the one-dimensional map 119865 considered inthe previous section the two-dimensional map F is notcontinuous in 1198632 More precisely the functions F

119871 F119872

and F119877are continuous and thus F is continuous in the

interior of each region inwhich the different definitions applyHowever in general except for the point xlowast the map F is notcontinuous along the borders 119881

119894 119894 = 1 2 3 of the regions

119877119871 119877119872 and 119877

119877 In fact it is easy to see that considering

Discrete Dynamics in Nature and Society 9

xL

t+1

xL

t

xR

t

00

1

1

(a)

xL

t+1

xL

t

xR

t

00

1

1

(b)

Figure 5 Graphics of the three functions defining (a) 119909119871119905+1

and (b) 119909119877119905+1

with parameters 119886119871= 29 119887

119871= 16 119886

119877= 19 119887

119877= 24 119889 = 8 120575

119871= 05

120575119872= 08 and 120575

119877= 06

a point (119909119871 119909119877) belonging to one of the lines defined in (22)at the boundary of two regions only we have that the twofunctions defined on the two different regions lead to twodifferent points of1198632 An example of how the three functionsdefining 119909119871

119905+1behave is shown in Figure 5(a) and the three

functions defining 119909119877119905+1

are illustrated in Figure 5(b) showingthat F is continuous only in xlowast

The point xlowast in which the three regions are in contactand F is continuous is clearly particular In fact solving theequation 119871(x) = 119872(x) we have the unique solution 119909119871 = 119909lowast119871while solving the equation 119877(x) = 119872(x) we have the uniquesolution119909119877 = 119909lowast119877 It follows that at the point xlowast = (119909lowast119871 119909lowast119877)we have the same cost in each choice as 119871(xlowast) = 119872(xlowast) =119877(xlowast) so that at the point xlowast the commuters are indifferent toany available choice andnone of themwill switch choiceThusF(xlowast) = xlowast and xlowast is a stationary state Under the conditionsin (23) the point xlowast is feasible belonging to 1198632 We can nowprove that this is indeed the unique fixed point of map F

Proposition 3 The point xlowast = (119909lowast119871 119909lowast119877) given in (21) is theunique fixed point of the map F belonging to 1198632 when theconditions in (23) hold

Proof The proof of this proposition follows directly from theobservation that

F (x) = x if and only if 119871 (x) = 119872 (x) = 119877 (x) (31)One way is already proved as we have detected the fixed pointstarting from the solution of 119871(x) = 119872(x) and 119877(x) = 119872(x)showing that it is the unique point xlowast Vice versa let us lookfor the solutions of F(x) = x We have the following systems(where 119909119872 = 1 minus 119909119871 minus 119909119877)

F119871

119909119871= 119909119871+ 120575119871(119877 (x) minus 119871 (x)

1198962

119909119877

+119872(x) minus 119871 (x)

1198964

119909119872)

119909119877= (1 minus 120575

119871

119877 (x) minus 119871 (x)1198962

)119909119877

if x isin 119877119871

F119872

119909119871= (1 minus 120575

119872

119871 (x) minus 119872 (x)119886119871minus 119889

)119909119871

119909119877= (1 minus 120575

119872

119877 (x) minus 119872 (x)119886119877minus 119889

)119909119877

if x isin 119877119872

F119877

119909119871= (1 minus 120575

119877

119871 (x) minus 119877 (x)1198961

)119909119871

119909119877= 119909119877+ 120575119877(119871 (x) minus 119877 (x)

1198961

119909119871

+119872(x) minus 119877 (x)

1198966

119909119872)

if x isin 119877119877

(32)

or equivalently (we recall that 120575119871 120575119872 and 120575

119877are different

from zero)

F119871

119877 (x) minus 119871 (x)1198962

119909119877+119872(x) minus 119871 (x)

1198964

119909119872= 0

119877 (x) minus 119871 (x)1198962

119909119877= 0

if x isin 119877119871

F119872

119871 (x) minus 119872 (x)119886119871minus 119889

119909119871= 0

119877 (x) minus 119872 (x)119886119877minus 119889

119909119877= 0

if x isin 119877119872

F119877

119871 (x) minus 119877 (x)1198961

119909119871= 0

119871 (x) minus 119877 (x)1198961

119909119871+119872(x) minus 119877 (x)

1198966

119909119872= 0

if x isin 119877119877

(33)

Considering the equation F119871(x) = x we see that it is certainly

satisfied for 119909119877 = 0 and 119909119872 = 0 which leads to thecorner point with 119909119871 = 1 However if the fixed point is an

10 Discrete Dynamics in Nature and Society

interior point of 1198632 then the point (1 0) does not belong tothe proper region 119877

119871 that is it is a virtual fixed point Clearly

this equation is satisfied when 119871(x) = 119872(x) and 119871(x) = 119877(x)which occurs only at the point xlowast It is possible to have xlowast =(1 0) and this occurs for (119886

119871minus 119889)119887

119877= 0 and (119886

119877minus 119889)119887

119871= 1

Thus we can state that the only solution of F119871(x) = x in 119877

119871

is given by the fixed point xlowast Similarly we can reason in theother regions

It is worth to mention that particular cases are obtainedwhen the fixed point xlowast belongs to a boundary of the region1198632 For example when 119909lowast119877+119909lowast119871 = 1 and 119909lowast119872 = 0 then only

the two regions119877119871and 119877

119877are involvedThis occurs when the

parameters satisfy

119889 =119886119871119887119871+ 119886119877119887119877minus 119887119871119887119877

119887119877+ 119887119871

0 le119886119871minus 119889

119887119877

le 1

0 le119886119877minus 119889

119887119871

le 1

(34)

From (119886119877minus 119889)119887

119871= (119886119877+ 119887119877minus 119886119871)(119887119877+ 119887119871) we have 119909lowast119877 =

1minus(119886119877minus119889)119887

119871= (119886119871+119887119871minus119886119877)(119887119877+119887119871) = 1198961(119887119877+119887119871) = 119909lowast as

in the one-dimensional map studied in Section 2 and 119909lowast119871 =1 minus 119909lowast119877= 1 minus (119886

119871minus 119889)119887

119877= 1198962(119887119877+ 119887119871) = 1 minus 119896

1(119887119877+ 119887119871) =

1minus119909lowast From themapsF

119871andF119877in (27) and for the parameters

occurring in this case it can be seen that 119909119871119905+1= 1minus119909

119877

119905+1holds

in both so that the dynamics can be studied by using onlythe variable 119909119877

119905and the resulting one-dimensional map is as

follows

119865 (119909119877)

=

119891119871(119909119877)

= 119909119877+ 120575119877

119871 (119909119877)minus119877 (119909

119877)

1198961

(1 minus 119909119877)

if 0 le 119909119877 le 119909lowast

119891119877(119909119877)

= 119909119877minus 120575119871

119877 (119909119877) minus 119871 (119909

119877)

1198962

119909119877 if 119909lowast le 119909119877 le 1

(35)

which corresponds to the system already studied in Section 2As we have seen this fixed point is globally attracting even iflocally we can have the eigenvalue on some side larger than 1in modulus

The case with 119909lowast119871 = 0 so that 0 le 119909lowast119877 le 1 and 119909lowast119872 =1 minus 119909lowast119877 leaves only the two regions 119877

119877and 119877

119872 This occurs

when the parameters satisfy

119886119871minus 119889

119887119877

= 1 0 le119886119877minus 119889

119887119871

le 1 (36)

From the maps F119872

and F119877in (27) it can be seen that 119909119871

119905

converges to119909lowast119871 = 0 so that the resulting asymptotic dynamic

behavior can be studied via the one-dimensional map givenby 119909119877119905+1= 119879(119909

119877

119905) with 119909119877

119905isin [0 1] (and 119909119872

119905= 1 minus 119909

119877

119905) given by

119879 (119909119877)

=

119879119871(119909119877)

= 119909119877+ 120575119877

119872(119909119877) minus 119877 (119909

119877)

1198966

(1 minus 119909119877)

if 0 le 119909119877 le 119909lowast

119879119877(119909119877)

= 119909119877minus 120575119872

119877 (119909119877) minus119872(119909

119877)

119886119877minus 119889

119909119877 if 119909lowast le 119909119877 le 1

(37)

where119872(119909119877) minus 119877(119909119877) = 119886119871+ 119887119871(1 minus 119909

119877) minus 119886119877minus 119887119877

The case with 119909lowast119877 = 0 so that 0 le 119909lowast119871 le 1 and119909lowast119872= 1 minus 119909

lowast119871 leaves only the two regions 119877119871and 119877

119872 This

occurs when the parameters satisfy 0 le (119886119871minus 119889)119887

119877le 1

(119886119877minus 119889)119887

119871= 1 From the maps F

119872and F

119871in (27) it can

be seen that 119909119877119905converges to 119909lowast119877 = 0 so that the resulting

asymptotic dynamic behavior can be studied via the one-dimensional map given by 119909119871

119905+1= 119879(119909

119871

119905) similar to the one

obtained in (37) with obvious changesBy using arguments as in Section 2 it can be shown that

these one-dimensional systems have a unique fixed pointwhich is globally attracting (although it can be locally stableor unstable)

In order to study the local stability of the fixed point ofthe two-dimensional map F we can use the linearization ofthe different definitions in the fixed point which leads to thefollowing Jacobian matrices one for each region 119877

119871 119877119872 and

119877119877 that we denote respectively with 119869

119871 119869119872 and 119869

119877

119869119871(xlowast) = (

1 + 120575119871minus 120575119871

119887119877

119887119871

1198965

1198964

minus 120575119871

119887119877

119887119871

1198966

1198961

120575119871

1198966

1198961

120575119871

119887119877

119887119871

1198966

1198961

1 minus 120575119871

1198966

1198961

)

119869119872(xlowast) = (

1 minus 120575119872

1198964

1198963

0

0 1 minus 120575119872

1198966

1198965

)

119869119877(xlowast) = (

1 minus 120575119877

1198965

1198962

120575119877

119887119871

119887119877

1198965

1198962

120575119877

1198965

1198962

1 + 120575119877minus 120575119877

119887119871

119887119877

1198963

1198966

minus 120575119877

119887119871

119887119877

1198965

1198962

)

(38)

The two real eigenvalues of the Jacobian matrix in 119877119872are

in explicit form and both are smaller than +1 However theycan be larger than minus1 or not Also the eigenvalues of the otherjacobian matrices can be in modulus smaller or larger than1 However the global dynamics observed numerically givealways trajectories which are convergent to the fixed point xlowastat any set of allowed parameters values

Discrete Dynamics in Nature and Society 11

1

1

075

05

025

00 025 05 075

xR

xL

(a)

1

1

075

05

025

00 025 05 075

xR

xL

(b)

Figure 6 Examples of trajectories converging to the unique fixed point with parameters (a) 119886119871= 28 119887

119871= 24 119886

119877= 8 119887

119877= 24 119889 = 5 120575

119871= 04

120575119872= 03 120575

119877= 07 and ic 119909119871 = 08 119909119877 = 005 (b) 119886

119871= 29 119887

119871= 16 119886

119877= 15 119887

119877= 25 119889 = 7 120575

119871= 05 120575

119872= 08 120575

119877= 06 and ic 119909119871 = 005

119909119877= 09 Dots are linked by a continuous line for illustrative purpose only

This global attractivity in the two-dimensional map isdifficult to be proved rigorously however the results in theparticular cases commented above that is when the fixedpoint xlowast belongs to the boundary of 1198632 suggests that thesame result is true also when the fixed point is in the interiorof1198632

In Figure 6 we show some examples in Figure 6(a) wecan see a trajectory starting at 119877

119877 in Figure 6(b) a trajectory

whose initial point in 119877119871is mapped into 119877

119872 The shape of

the functions in the case of Figure 6(b) is similar to the oneshown in Figure 5

4 Conclusion

In this paper we have analyzed a dynamical version of theBraess paradox with nonimpulsive commuters who changeroad proportionally to the cost differenceWe were interestedto prove that with nonimpulsive commuters there exists aunique equilibrium both in the original network with twochoices and in the ternary onewhen a single link is addedWehave shown that both models are quite robust We were ableto provide an analytical proof of the global attractivity of theunique fixed point in the one-dimensional case when onlytwo roads are available andweusednumerical techniques andsimulations to give evidence in the two-dimensional case (iewhen a new road is added) as a rigorous proof can be doneonly for particular border cases

A limitation of our contribution is considering homo-geneous populations in spite of the fact that the evidencesupports heterogeneity Nevertheless heterogeneity of com-mutersrsquo behavior cannot be easily classified yet even when

conducting experiments common individual patterns arejust identified and classified On the other hand whenobserving the aggregate behavior inmany cases the literatureprovides evidence in support of the paradox outcome thatis the convergence to the Nash equilibrium The impulsiveagents in [30] react faster than the proportional agentspresented in this paper yet as a population they fail toachieve the Nash equilibria in certain cases while in thispaper the aggregate behavior does not We proposed thisaggregate behavior because it replies the convergence to theNash equilibria and at the same time it may be the resultof modeling some observed individual behaviors such asimitation and free riding

Taking into account the limitation of homogeneous pop-ulation analysis a natural extension of our contribution isconsidering heterogeneous populations with both impulsiveand nonimpulsive commuters We have reasons to believethat in such a case some properties of the different behaviorsare inherited and some are lost Therefore the results weprovide in this paper match those for impulsive populationsand are paramount to extend the analysis to more realisticcases In factmdashas the empirical results showmdashheterogeneouspopulations dynamics provide a better fit of what has beenobserved when considering human participants interaction

Furthermore it will be interesting to consider general-izations to networks with richer architectures Finally weobserve that since with the class of behavior we considerin this paper that dynamics depends on the shape of thecost functions it will be interesting to extend the analysis tononlinear cost functions

12 Discrete Dynamics in Nature and Society

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

V Avrutin is supported by the European Community withinthe scope of the project ldquoMultiple-discontinuity inducedbifurcations in theory and applicationsrdquo (Marie Curie Actionof the 7th Framework Programme Contract Agreement noPIEF-GA-2011-300281) U Merlone has worked under theauspices of COST Action IS1104 ldquoThe EU in the new complexgeography of economic systems models tools and policyevaluationrdquo

References

[1] R Arnott and K Small ldquoThe economics of traffic congestionrdquoAmerican Scientist vol 82 no 5 pp 446ndash455 1994

[2] T Bass ldquoRoad to ruinrdquo Discover pp 56ndash61 1992[3] B Calvert and G Keady ldquoBraessrsquos paradox and power-law non-

linearities in networksrdquoThe Journal of the AustralianMathemat-ical Society B Applied Mathematics vol 35 no 1 pp 1ndash22 1993

[4] J E Cohen andC Jeffries ldquoCongestion resulting from increasedcapacity in single-server queueing networksrdquo IEEEACMTrans-actions on Networking vol 5 no 2 pp 305ndash310 1997

[5] J E Cohen and F P Kelly ldquoA paradox of congestion in a queuingnetworkrdquo Journal of Applied Probability vol 27 no 3 pp 730ndash734 1990

[6] S Dafermos and A Nagurney ldquoOn some traffic equilibriumtheory paradoxesrdquo Transportation Research B Methodologicalvol 18 no 2 pp 101ndash110 1984

[7] C Fisk ldquoMore paradoxes in the equilibrium assignment prob-lemrdquo Transportation Research B Methodological vol 13 no 4pp 305ndash309 1979

[8] W H K Lam ldquoEffects of road pricing on system performancerdquoTraffic Engineering amp Control vol 29 no 12 pp 631ndash635 1988

[9] J D Murchland ldquoBraessrsquos paradox of traffic flowrdquo Transporta-tion Research vol 4 no 4 pp 391ndash394 1970

[10] R Steinberg and W I Zangwill ldquoThe prevalence o f BraessrsquoparadoxrdquoTransportation Science vol 17 no 3 pp 301ndash318 1983

[11] M Frank ldquoThe Braess paradoxrdquo Mathematical Programmingvol 20 no 1 pp 283ndash302 1981

[12] T Roughgarden Selfish Routing and the Price of Anarchy MITpress 2005

[13] B B Fu C X Zhao and S B Li ldquoThe analysis of Braessrsquo paradoxand robustness based on dynamic traffic assignment modelsrdquoDiscrete Dynamics in Nature and Society vol 2013 Article ID796842 8 pages 2013

[14] C Fisk and S Pallottino ldquoEmpirical evidence for equilibriumparadoxes with implications for optimal planning strategiesrdquoTransportation Research Part A General vol 15 no 3 pp 245ndash248 1981

[15] H Youn M T Gastner and H Jeong ldquoPrice of anarchy intransportation networks efficiency and optimality controlrdquoPhysical Review Letters vol 101 no 12 Article ID 128701 2008

[16] K Aoki Y Ohtsubo A Rapoport and T Saijo ldquoEffects of priorinvestment and personal responsibility in a simple network

gamerdquo Current Research in Social Psychology vol 13 no 2 pp10ndash21 2007

[17] J Morgan H Orzen and M Sefton ldquoNetwork architectureand traffic flows experiments on the Pigou-Knight-Downs andBraess paradoxesrdquoGames and Economic Behavior vol 66 no 1pp 348ndash372 2009

[18] A Rapoport V Mak and R Zwick ldquoNavigating congested net-works with variable demand experimental evidencerdquo Journal ofEconomic Psychology vol 27 no 5 pp 648ndash666 2006

[19] A Rapoport T Kugler S Dugar and E J Gisches ldquoBraessparadox in the laboratory an experimental study of route choicein traffic networks with asymmetric costsrdquo inDecisionModelingand Behavior in Uncertain and Complex Environment T KuglerJ C Smith T Connolly and Y J Son Eds pp 309ndash337Springer New York NY USA 2008

[20] A Rapoport T Kugler S Dugar and E J Gisches ldquoChoice ofroutes in congested traffic networks experimental tests of theBraess paradoxrdquo Games and Economic Behavior vol 65 no 2pp 538ndash571 2009

[21] E J Gisches and A Rapoport ldquoDegrading network capacitymay improve performance private versus public monitoring inthe Braess paradoxrdquoTheory and Decision vol 73 no 2 pp 267ndash293 2012

[22] A Dal Forno and U Merlone ldquoReplicating human interactionin Braess paradoxrdquo in Proceedings of the Simulation Conference(WSC rsquo13) pp 1754ndash1765 IEEE 2013

[23] M Ausloos H Dawid and U Merlone ldquoSpatial interactionsin agent-based modelingrdquo in Complexity and GeographicalEconomics Topics and Tools P Commendatore S S Kayamand I Kubin Eds Springer Berlin Germany 2014

[24] G-I Bischi and U Merlone ldquoGlobal dynamics in binary choicemodels with social influencerdquo The Journal of MathematicalSociology vol 33 no 4 pp 277ndash302 2009

[25] G I Bischi L Gardini and U Merlone ldquoImpulsivity in binarychoices and the emergence of periodicityrdquoDiscrete Dynamics inNature and Society vol 2009 Article ID 407913 22 pages 2009

[26] JH PattonM S Stanford andE S Barratt ldquoFactor structure ofthe Barratt Impulsiveness Scalerdquo Journal of Clinical Psychologyvol 51 no 6 pp 768ndash774 1995

[27] G I Bischi L Gardini and U Merlone ldquoPeriodic cycles andbifurcation curves for one-dimensional maps with two discon-tinuitiesrdquo Journal of Dynamical Systems GeometricTheories vol7 no 2 pp 101ndash123 2009

[28] G-I Bischi and U Merlone ldquoBinary choices in small and largegroups a unifiedmodelrdquo Physica A vol 389 no 4 pp 843ndash8532010

[29] A Dal Forno L Gardini and U Merlone ldquoTernary choicesin repeated games and border collision bifurcationsrdquo ChaosSolitons amp Fractals vol 45 no 3 pp 294ndash305 2012

[30] A Dal Forno and U Merlone ldquoBorder-collision bifurcationsin a model of Braess paradoxrdquo Mathematics and Computers inSimulation vol 87 pp 1ndash18 2013

[31] A Dal Forno V Giorgino and U Merlone ldquoAgent-based mod-eling with GTmethodology An examplerdquoWorking Paper 2014

[32] L Gardini U Merlone and F Tramontana ldquoInertia in binarychoices continuity breaking and big-bang bifurcation pointsrdquoJournal of Economic Behavior amp Organization vol 80 no 1 pp153ndash167 2011

[33] W H Sandholm Population Games and Evolutionary Dynam-ics The MIT Press Cambridge Mass USA 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Discrete Dynamics in Nature and Society

Substituting from (2) and rearranging we obtain

119865 (119909119877) =

119891119871(119909119877) = 120575119877+ (1 minus 120575

119877minus 120575119877

119887119871+ 119887119877

1198961

)119909119877

+120575119877

119887119871+ 119887119877

1198961

(119909119877)2

if 0 le 119909119877 le 119909lowast

119891119877(119909119877) = (1 + 120575

119871

1198961

1198962

)119909119877

minus120575119871

119887119871+ 119887119877

1198962

(119909119877)2

if 119909lowast le 119909119877 le 1

(6)

From the definition of 119909lowast given in (3) it can also be writtenas

119909lowast=119886119871+ 119887119871minus 119886119877

119887119871+ 119887119877

=1198961

119887119871+ 119887119877

=1198961

1198961+ 1198962

(7)

at which we have 119871(119909lowast) = 119877(119909lowast) it is also 119891119871(119909lowast) = 119891119877(119909lowast)

and thus 119865 is continuous in 119909lowast and it is a fixed point as119865(119909lowast) = 119909lowast We have the following proposition

Proposition 1 Map 119865 is continuous in [0 1] and under theconditions given in (4) 119909lowast = 119896

1(119887119871+ 119887119877) isin (0 1) is the unique

fixed point

Proof Only the uniqueness is left to prove The equation119891119871(119909119877) = 119909

119877 has two solutions 119909119877 = 119909lowast and 119909119877 = 1 whichis larger than 119909lowast and thus a virtual fixed point The equation119891119877(119909119877) = 119909

119877 has two solutions 119909119877 = 119909lowast and 119909119877 = 0 whichis smaller than 119909lowast and thus a virtual fixed point Thus theunique solution is 119909119877 = 119909lowast

Clearly the fixed point may be locally stable or instabledepending on the slopes of the functions on the right and leftsides of 119909lowast However we shall see that in any case it is globallyattractingThat is any ic in the interval [0 1] has a trajectorywhich converges to 119909lowast

Proposition 2 The fixed point 119909lowast = 1198961(119887119871+ 119887119877) isin (0 1) is

globally attracting

Proof Even if 119865(119909119877) is not smooth in its fixed point (asindeed 119909lowast is a kink point of this map) the two components119891119871(119909119877) and 119891

119877(119909119877) are smooth in 119909lowast so that the left and right

side derivatives of 119865(119909119877) in 119909lowast are well defined 1198651015840minus(119909lowast) =

lim119909119877rarr119909lowastminus1198651015840(119909119877) = 1198911015840

119871(119909lowast) and1198651015840

+(119909lowast) = lim

119909119877rarr119909lowast+1198651015840(119909119877) =

1198911015840

119877(119909lowast)

We have

1198911015840

119871(119909119877) = (1 minus 120575

119877minus 120575119877

119887119871+ 119887119877

1198961

) + 2120575119877

119887119871+ 119887119877

1198961

119909119877 (8)

1198911015840

119871(119909lowast) = 1 minus 120575

119877

1198962

1198961

(9)

while

1198911015840

119877(119909119877) = (1 + 120575

119871

1198961

1198962

) minus 2120575119871

119887119871+ 119887119877

1198962

119909119877 (10)

1198911015840

119877(119909lowast) = 1 minus 120575

119871

1198961

1198962

(11)

From

11989110158401015840

119871(119909119877) = 2120575

119877

119887119871+ 119887119877

1198961

gt 0 11989110158401015840

119877(119909119877) = minus2120575

119871

119887119871+ 119887119877

1198962

lt 0

(12)

we have that on the left side of the fixed point the function isconvex while on the right side of the fixed point the functionis concave

Notice that 11989621198961only depends on 119886

119871 119887119871 119886119877 119887119877 and it is

positive under our assumptions Depending on the values ofthe ratio 119896

21198961we can classify the rightleft stability of the

fixed point 119909lowast From (9) we clearly have always 1198911015840119871(119909lowast) lt 1

thus the stabilityinstability on the left side of the fixed pointdepends on the condition 1198911015840

119871(119909lowast) gt minus1 which holds if and

only if

1198962

1198961

lt2

120575119877

(13)

Similarly from (11) we clearly have always 1198911015840119877(119909lowast) lt 1 and

the stabilityinstability on the right side of the fixed pointdepends on the condition 1198911015840

119877(119909lowast) gt minus1 which holds if and

only if

1198962

1198961

gt120575119871

2 (14)

And clearly we have the following sequence of inequalities (tobe used below)

0 lt120575119871

2lt 120575119871lt 1 lt

1

120575119877

lt2

120575119877

(15)

To investigate the local stability of the fixed point we considerthree intervals for 119896

21198961

(i) 0 lt 11989621198961le 1205751198712

(ii) 1205751198712 lt 119896

21198961lt 2120575

119877

(iii) 2120575119877le 11989621198961

In case (i) it is 1198911015840119877(119909lowast) le minus1 moreover it is easy to see

that for any 119909119877 isin [119909lowast 1] it is 1198911015840119877(119909119877) lt 119891

1015840

119877(119909lowast) and thus

the function on the right side of the fixed point is strictlydecreasing and expansive each point on the right side of thefixed point is mapped to the left side of the fixed point in oneiteration While on the left side of the fixed point the convexfunction 119891

119871(119909119877) has 1198911015840

119871(119909lowast) = 1 minus 120575

119877(11989621198961) gt 0 (as this

holds for 11989621198961lt 1120575

119877which is satisfied in this interval)This

implies that the point of local minimum on the left side say119909119877

119897119898 is smaller than the fixed point 119909119877

119897119898lt 119909lowast Then the fixed

point is unique and the map is a contraction in the interval119882119904

loc = [119909119877

119897119898 119909lowast] and an absorbing interval which is mapped

into 119882119904loc in one iteration is given by 119869 = [119909119877119897119898 119891minus1

119877(119909119877

119897119898)]

when 119891119877(1) = 1 minus 120575

119871lt 119909119877

119897119898 or 119869 = [119909119877

119897119898 1] when 119891

119877(1) =

1 minus 120575119871ge 119909119877

119897119898 Any initial condition belonging to [0 1] 119869

has a trajectory which is mapped into 119869 in a finite number ofiterationsThus119909lowast is globally attracting An example is shownin Figure 2(a)

Similarly we can reason when 11989621198961belongs to the third

interval In fact in case (iii) it is 1198911015840119871(119909lowast) le minus1 moreover it is

Discrete Dynamics in Nature and Society 5xR t+1

xR

t

00

025

025 05

05

075

1

1075

(a)xR t+1

xR

t

00

025

025 05

05

075

1

1075

(b)

Figure 2 Qualitative shapes of the map In (a) case (i) when 0 lt 11989621198961le 1205751198712 the fixed point is locally unstable on the right side In (b)

case (iii) when 2120575119877le 11989621198961 the fixed point is locally unstable on the left side

0

1

0 1

xR t

xL

t

(a)

0

1

0 1

xR t

xL

t

(b)

0

1

0 1

xR t

xL

t

(c)

Figure 3 Qualitative shapes of the map in case (ii) when 1205751198712 lt 119896

21198961le 2120575

119877and the fixed point is locally stable on both sides

easy to see that for any 119909119877 isin [0 119909lowast] it is 1198911015840119871(119909lowast) lt 1198911015840

119871(119909lowast) and

thus the function on the left side of the fixed point is strictlydecreasing and expansive each point on the left side of thefixed point is mapped to the right side of the fixed point inone iteration While on the right side of the fixed point theconcave function 119891

119877(119909119877) has 1198911015840

119877(119909lowast) = 1 minus 120575

119871(11989611198962) gt 0 (as

this holds for 11989621198961gt 120575119871which is satisfied in this interval)

This implies that the point of local maximum on the rightside say 119909119877

119903119872 is larger than the fixed point 119909119877

119903119872gt 119909lowast Then

the map is a contraction in the interval 119882119904loc = [119909lowast 119909119877

119903119872]

and an absorbing interval which is mapped into119882119904loc in oneiteration is given by 119869 = [119891minus1

119871(119909119877

119903119872) 119909119877

119903119872] when 119891

119871(0) =

120575119877gt 119909119877

119903119872 or 119869 = [0 119909119877

119903119872] when 119891

119871(0) = 120575

119877le 119909119877

119903119872 Any

initial condition belonging to [0 1] 119869 has a trajectory whichis mapped into 119869 in a finite number of iterations Thus 119909lowast isglobally attracting An example is shown in Figure 2(b)

When 11989621198961belongs to the interval in (ii) 120575

1198712 lt 119896

21198961lt

2120575119877 we have that the fixed point is locally stable on both

sides Moreover inside this interval we can distinguish threeregions as already showed in (15) that is

(a) 1205751198712 lt 119896

21198961le 120575119871with 0 lt 1198911015840

119871(119909lowast) lt 1 and

minus1 lt 1198911015840

119877(119909lowast) lt 0 as in Figure 3(a)

(b) 120575119871lt 11989621198961lt 1120575

119877with 0 lt 1198911015840

119871(119909lowast) lt 1 and

0 lt 1198911015840

119877(119909lowast) lt 1 as in Figure 3(b)

6 Discrete Dynamics in Nature and Society

(c) 1120575119877le 11989621198961lt 2120575

119877with minus1 lt 1198911015840

119871(119909lowast) lt 0 and

0 lt 1198911015840

119877(119909lowast) lt 1 as in Figure 3(c)

In case (a) we have that the point of local minimum onthe left side say 119909119877

119897119898 is smaller than the fixed point119909119877

119897119898lt 119909lowast

Then themap is a contraction in the interval119882119904loc = [119909119877

119897119898 119909lowast]

and on the right side the map is decreasing so that anabsorbing interval which is mapped into119882119904loc in one iterationis given by 119869 = [119909119877

119897119898 119891minus1

119877(119909119877

119897119898)]when119891

119877(1) = 1minus120575

119871lt 119909119877

119897119898 or

119869 = [119909119877

119897119898 1]when119891

119877(1) = 1minus120575

119871ge 119909119877

119897119898 Any initial condition

belonging to [0 1] 119869 has a trajectory which is mapped into 119869in a finite number of iterationsThus 119909lowast is globally attracting

In case (c) we can reason similarly we have that the pointof local maximum on the right side say 119909119877

119903119872 is larger than

the fixed point 119909119877119903119872gt 119909lowast Then the map is a contraction in

the interval 119882119904loc = [119909lowast 119909119877

119903119872] and on the left side the map

is decreasing so that an absorbing interval which is mappedinto 119882119904loc in one iteration is given by 119869 = [119891minus1

119871(119909119877

119903119872) 119909119877

119903119872]

when 119891119871(0) = 120575

119877gt 119909119877

119903119872 or 119869 = [0 119909119877

119903119872] when 119891

119871(0) =

120575119877le 119909119877

119903119872 Any initial condition belonging to [0 1] 119869 has

a trajectory which is mapped into 119869 in a finite number ofiterations Thus 119909lowast is globally attracting

In case (b) we have that the point of local minimum onthe left side say 119909119877

119897119898 is smaller than the fixed point 119909119877

119897119898lt

119909lowast and also that the point of local maximum on the right

side say 119909119877119903119872

is larger than the fixed point 119909119877119903119872gt 119909lowast

Then the map is a contraction in the interval 119882119904loc =

[119909119877

119897119898 119909119877

119903119872] which includes the fixed point and an absorbing

interval which is mapped into119882119904loc in one iteration is givenby 119869 = [119891minus1

119871(119909119877

119903119872) 119891minus1

119877(119909119877

119897119898)] or 119869 = [0 119891minus1

119877(119909119877

119897119898)] or 119869 =

[119891minus1

119871(119909119877

119903119872) 1] or [0 1] depending under obvious conditions

on the values of 120575119871and 120575

119877 When 119869 sub [0 1] then any initial

condition belonging to [0 1] 119869 has a trajectory which ismapped into 119869 in a finite number of iterations Thus 119909lowast isglobally attracting

We notice that in Proposition 2 we prove that when thefixed point 119909lowast is internal to the interval [0 1] then it isglobally attracting Let us turn to comment the two extremawhich occur when 119909lowast = 0 or 119909lowast = 1 These solutions clearlyexist and are also globally attracting In fact the case 119909lowast = 0corresponds to 119896

1= 119871(0)minus119877(0) = 119886

119871+119887119871minus119886119877= 0 and only the

function 119891119877(119909119877) is applied in the interval [0 1] and from the

concavity of the function we have that this fixed point is glob-ally attracting even if the eigenvalue is equal to 1 The othercase 119909lowast = 1 corresponds to 119896

2= 119877(1)minus119871(1) = 119886

119877+119887119877minus119886119871= 0

and only the function 119891119871(119909119877) is applied in the interval [0 1]

and from the convexity of the function we have that this fixedpoint is globally attracting even if the eigenvalue is equal to 1

3 The Dynamics in the ExpandedNetwork with Three Roads

As it is well known the Braess paradox introduces a new linkand shows how this opportunity worsens both the individualand collective payoffs This link introduces a new alternativenow the commuters are facing three choices where the new

one consists of path 119878-119871-119877-119864 (see Figure 1(b)) which will bedenoted as 119872 in the following Models with three choiceswere introduced in [29] as it concerns discrete time and [33]as it concerns continuous time In this paper we analyze thedynamics when the travel time 119889 for the new resulting linkvaries in a range that makes this link a dominant choice forsome values and a dominated choice for others It is worth toobserve that in particular when the travel time on link 119871-119877 iszero the paradoxical result still holds

Now assume that commuters switch roadwhenever traveltime has become larger that is the other road is less costlyand therefore more attractive We indicate by 119909119871 ge 0 thefraction of population choosing action 119871 by 119909119872 ge 0 thefraction choosing action 119872 and by 119909119877 ge 0 the fractionchoosing action 119877 with the constraint 119909119871 + 119909119877 + 119909119872 = 1The travel times are given respectively as

119871 = 119886119871+ 119887119871(119909119871+ 119909119872)

119872 = 119887119871(119909119871+ 119909119872) + 119887119877(119909119872+ 119909119877) + 119889

119877 = 119886119877+ 119887119877(119909119872+ 119909119877)

(16)

By using the constraint let us define the fraction of populationchoosing119872 as 119909119872 = 1 minus 119909119871 minus 119909119877 ge 0 so to consider onlytwo variables x = (119909119871 119909119877) isin R2

+ and the phase space is the

triangle

1198632= x = (119909119871 119909119877) isin R2

+ 0 le 119909

119871+ 119909119877le 1 (17)

After substituting the expression of 119909119872 in the travel times weget

119871 (x) = 119886119871 + 119887119871 (1 minus 119909119877)

119872 (x) = 119887119871 (1 minus 119909119877) + 119887119877(1 minus 119909

119871) + 119889

119877 (x) = 119886119877 + 119887119877 (1 minus 119909119871)

(18)

The vertices of the phase space119875lowast119871(1 0)119875lowast

119872(0 0) and119875lowast

119877(0 1)

represent states in which the whole population choosesrespectively actions 119871119872 and 119877

Since commuters are interested in minimizing traveltime we denote by 119877

119860the region in which the choice 119860

(where119860 is 119871119872 or 119877) is preferable These regions are calleddominance regions of 119860 We have the following definition ofthe dominance regions

119877119871= x isin 1198632 119871 (x) le 119872 (x) 119871 (x) le 119877 (x)

119877119872= x isin 1198632 119872 (x) le 119871 (x) 119872 (x) le 119877 (x)

119877119877= x isin 1198632 119877 (x) le 119871 (x) 119877 (x) le 119872 (x)

(19)

Discrete Dynamics in Nature and Society 7

1

100

xR

RL

RM

RR

xL

(a)

1

100

xR

RL

RM

RR

xL

(b)

1

100

xR

RL

RM

RR

xL

(c)

Figure 4 Dominance regions in the three choices network with different values of the cost parameter 119889 of the map F at 119886119871= 27 119887

119871= 30

119886119877= 18 119887

119877= 30 and (a) 119889 = 6 (b) 119889 = 3 (c) 119889 = 0

By using the expressions in (18) we get the following defini-tions for the regions

119877119871= x isin 1198632 119909119871 le 119909lowast119871

119886119871+ 119887119871(1 minus 119909

119877) le 119886119877+ 119887119877(1 minus 119909

119871)

119877119872= x isin 1198632 119909119871 ge 119909lowast119871 119909119877 ge 119909lowast119877

119877119877= x isin 1198632 119886

119877+ 119887119877(1 minus 119909

119871) le 119886119871+ 119887119871(1 minus 119909

119877)

119909119877le 119909lowast119877

(20)

where we have introduced

119909lowast119871= 1 minus

119886119871minus 119889

119887119877

=119889 minus 119886119871+ 119887119877

119887119877

119909lowast119877= 1 minus

119886119877minus 119889

119887119871

=119889 minus 119886119877+ 119887119871

119887119871

(21)

The boundaries of the three regions are given by segments ofstraight lines of equations

1198811 119909119871= 119909lowast119871

1198812 119909119877= 119909lowast119877

1198813 119886119871+ 119887119871(1 minus 119909

119877) = 119886119877+ 119887119877(1 minus 119909

119871)

(22)

and it is immediate to see that the point xlowast = (119909lowast119871 119909lowast119877)satisfies also the third equation that is it belongs to all thethree straight lines xlowast isin 119881

119894for 119894 = 1 2 3

Using the geometric representation in the triangle 1198632the dominance regions are illustrated in Figure 4 (via anexample) using different colors Three cases are shown fordifferent values of the parameter 119889 in order to illustrate howthe region119877

119872depends on the parameters of the third branch

The point xlowast = (119909lowast119871 119909lowast119877) is the unique point on theboundaries 119881

119894of all the three regions Thus for our model we

have to require 119909lowast119871 ge 0 119909lowast119877 ge 0 and 119909lowast119871 + 119909lowast119877 le 1 leadingto 119909lowast119872 = 1 minus 119909lowast119871 minus 119909lowast119877 In order to have three regions in1198632

it must be 119909lowast119871 gt 0 119909lowast119877 gt 0 and 119909lowast119871 + 119909lowast119877 lt 1 The equalityin one of these three conditions implies the disappearance ofone region For example when 119909lowast119871 = 0 then 0 le 119909lowast119877 le 1and 119909lowast119872 = 1 minus 119909lowast119877 leaving only the two regions 119877

119877and 119877

119872

and so onThe conditions 119909lowast119871 ge 0 119909lowast119877 ge 0 and 119909lowast119871 + 119909lowast119877 le 1 lead

to the constraints that will be assumed satisfied henceforth

0 le119886119871minus 119889

119887119877

le 1 0 le119886119877minus 119889

119887119871

le 1

119886119871minus 119889

119887119877

+119886119877minus 119889

119887119871

ge 1

(23)

We notice that from the third condition in (23) we have thevalues of 119889 for which the third region is really present in 1198632that is

0 le 119889 le119886119871119887119871+ 119886119877119887119877minus 119887119871119887119877

119887119877+ 119887119871

(24)

Commuters are homogeneous and minimize their nextperiod travel cost At time 119905+1 the vector x

119905becomes common

knowledge and each commuter can observe the travel costs119871(x119905)119872(x

119905) and 119877(x

119905) We assume that if at time 119905 a fraction

119909119871

119905chooses path 119871 a fraction 119909119877

119905chooses action 119877 and travel

costs are such that 119877(x119905) lt 119871(x

119905) and 119877(x

119905) lt 119872(x

119905) then

both a fraction of the 119909119871119905commuters who chose action 119871 and

a fraction of the (1 minus 119909119871119905minus 119909119877

119905) commuters who chose action

119872 will switch to path 119877 in the next time period 119905 + 1 Thefraction of switching commuters is given by the switchingpropensities 120575

119871 120575119872 120575119877modulated by the relative differences

in payoffs This happens whenever a path gives a smallertravel cost In other words at any time 119905 all the commutersdecide their future action at time 119905 + 1 comparing the costs119871(x119905)119872(x

119905) and 119877(x

119905) As in the one-dimensional case here

also the differences of travel costs are normalized dividing

8 Discrete Dynamics in Nature and Society

each difference by the constants giving the related maximumvalue given in the different cases as follows

1198961= max

x [119871 (x) minus 119877 (x)]

= [119871 (1 0) minus 119877 (1 0)] = 119886119871 + 119887119871 minus 119886119877

1198962= max

x [119877 (x) minus 119871 (x)]

= [119877 (0 1) minus 119871 (0 1)] = 119886119877 + 119887119877 minus 119886119871

1198963= max

x [119871 (x) minus 119872 (x)]

= [119871 (1 119909119877) minus119872(1 119909

119877)] = 119886

119871minus 119889 forall119909

119877isin [0 1]

1198964= max

x [119872 (x) minus 119871 (x)]

= [119872(0 119909119877) minus 119871 (0 119909

119877)] = 119889 + 119887

119877minus 119886119871 forall119909

119877isin [0 1]

1198965= max

x [119877 (x) minus 119872 (x)]

= [119877 (119909119871 1) minus 119872(119909

119871 1)] = 119886

119877minus 119889 forall119909

119871isin [0 1]

1198966= max

x [119872 (x) minus 119877 (x)]

= [119872(119909119871 0) minus 119877 (119909

119871 0)] = 119889 + 119887

119871minus 119886119877 forall119909

119871isin [0 1]

(25)

The resulting dynamics are described by a two-dimensionalmap x

119905+1= F(x

119905) defined as

F x119905+1=

F119871(x119905) if x

119905isin 119877119871

F119872(x119905) if x

119905isin 119877119872

F119877(x119905) if x

119905isin 119877119877

(26)

where

F119871

119909119871

119905+1= 119909119871

119905+ 120575119871(119877 (x119905) minus 119871 (x

119905)

119886119877+ 119887119877minus 119886119871

119909119877

119905

+119872(x119905) minus 119871 (x

119905)

119889 + 119887119877minus 119886119871

119909119872

119905)

119909119877

119905+1= (1 minus 120575

119871

119877 (x119905) minus 119871 (x

119905)

119886119877+ 119887119877minus 119886119871

)119909119877

119905

F119872

119909119871

119905+1= (1 minus 120575

119872

119871 (x119905) minus 119872(x

119905)

119886119871minus 119889

)119909119871

119905

119909119877

119905+1= (1 minus 120575

119872

119877 (x119905) minus 119872(x

119905)

119886119877minus 119889

)119909119877

119905

F119877

119909119871

119905+1= (1 minus 120575

119877

119871 (x119905) minus 119877 (x

119905)

119886119871+ 119887119871minus 119886119877

)119909119871

119905

119909119877

119905+1= 119909119877

119905+ 120575119877(119871 (x119905) minus 119877 (x

119905)

119886119871+ 119887119871minus 119886119877

119909119871

119905

+119872(x119905) minus 119877 (x

119905)

119889 + 119887119871minus 119886119877

119909119872

119905)

(27)

By construction we have that F maps 1198632 into itself In factto show that F 1198632 rarr 119863

2 it is enough to show that

given (119909119871119905 119909119877

119905) isin 119863

2 then (119909119871119905+1 119909119877

119905+1) = F(119909119871

119905 119909119877

119905) isin 119863

2From (119909119871

119905 119909119877

119905) isin 119863

2 we have that 0 le 119909119871119905+ 119909119877

119905le 1 and

119909119871

119905+ 119909119877

119905+ 119909119872

119905= 1 Then if (119909119871

119905+1 119909119877

119905+1) = F119871(119909119871

119905 119909119877

119905) we have

119909119871

119905+1+ 119909119877

119905+1= 119909119871

119905+ 119909119877

119905+ 120575119871

119872(x119905) minus 119871 (x

119905)

119889 + 119887119877minus 119886119871

119909119872

119905

le 119909119871

119905+ 119909119877

119905+ 119909119872

119905= 1

(28)

119909119871

119905+1ge 0 and 119909119877

119905+1ge 0 thus (119909119871

119905+1 119909119877

119905+1) isin 1198632 If (119909119871

119905+1 119909119877

119905+1) =

F119872(119909119871

119905 119909119877

119905) we have 0 le 119909119871

119905+1le 119909119871

119905and 0 le 119909119877

119905+1le 119909119877

119905 thus

(119909119871

119905+1 119909119877

119905+1) isin 1198632 If (119909119871

119905+1 119909119877

119905+1) = F119877(119909119871

119905 119909119877

119905) we have

119909119871

119905+1+ 119909119877

119905+1= 119909119871

119905+ 119909119877

119905+ 120575119877

119872(x119905) minus 119877 (x

119905)

119889 + 119887119871minus 119886119877

119909119872

119905

le 119909119871

119905+ 119909119877

119905+ 119909119872

119905= 1

(29)

119909119871

119905+1ge 0 and 119909119877

119905+1ge 0 thus (119909119871

119905+1 119909119877

119905+1) isin 1198632

Substituting from (18) we can rewrite the functions inexplicit form as

F119871

119909119871

119905+1= 120575119871+ (1 minus 120575

119871minus 120575119871

119887119877

1198964

)119909119871

119905

minus120575119871

119887119871

1198962

119909119877

119905+ 120575119871

119887119871

1198962

(119909119877

119905)2

+120575119871119887119877(1

1198964

minus1

1198962

)119909119871

119905119909119877

119905+ 120575119871

119887119877

1198964

(119909119871

119905)2

119909119877

119905+1= (1 minus 120575

119871minus 120575119871

119887119871

1198962

)119909119877

119905

+120575119871

119887119877

1198962

119909119871

119905119909119877

119905minus 120575119871

119887119871

1198962

(119909119877

119905)2

F119872

119909119871

119905+1= (1 + 120575

119872

1198964

1198963

)119909119871

119905minus 120575119872

119887119877

1198963

(119909119871

119905)2

119909119877

119905+1= (1 + 120575

119872

1198966

1198965

)119909119877

119905minus 120575119872

119887119871

1198965

(119909119877

119905)2

F119877

119909119871

119905+1= (1 minus 120575

119877minus 120575119877

119887119877

1198961

)119909119871

119905

+120575119877

119887119871

1198961

119909119871

119905119909119877

119905minus 120575119877

119887119877

1198961

(119909119871

119905)2

119909119877

119905+1= 120575119877+ (1 minus 120575

119877minus 120575119877

119887119871

1198966

)119909119877

119905

minus120575119877

119887119877

1198961

119909119871

119905+ 120575119877

119887119871

1198961

(119909119877

119905)2

+120575119877119887119871(1

1198966

minus1

1198961

)119909119871

119905119909119877

119905+ 120575119877

119887119877

1198966

(119909119871

119905)2

(30)

Differently from the one-dimensional map 119865 considered inthe previous section the two-dimensional map F is notcontinuous in 1198632 More precisely the functions F

119871 F119872

and F119877are continuous and thus F is continuous in the

interior of each region inwhich the different definitions applyHowever in general except for the point xlowast the map F is notcontinuous along the borders 119881

119894 119894 = 1 2 3 of the regions

119877119871 119877119872 and 119877

119877 In fact it is easy to see that considering

Discrete Dynamics in Nature and Society 9

xL

t+1

xL

t

xR

t

00

1

1

(a)

xL

t+1

xL

t

xR

t

00

1

1

(b)

Figure 5 Graphics of the three functions defining (a) 119909119871119905+1

and (b) 119909119877119905+1

with parameters 119886119871= 29 119887

119871= 16 119886

119877= 19 119887

119877= 24 119889 = 8 120575

119871= 05

120575119872= 08 and 120575

119877= 06

a point (119909119871 119909119877) belonging to one of the lines defined in (22)at the boundary of two regions only we have that the twofunctions defined on the two different regions lead to twodifferent points of1198632 An example of how the three functionsdefining 119909119871

119905+1behave is shown in Figure 5(a) and the three

functions defining 119909119877119905+1

are illustrated in Figure 5(b) showingthat F is continuous only in xlowast

The point xlowast in which the three regions are in contactand F is continuous is clearly particular In fact solving theequation 119871(x) = 119872(x) we have the unique solution 119909119871 = 119909lowast119871while solving the equation 119877(x) = 119872(x) we have the uniquesolution119909119877 = 119909lowast119877 It follows that at the point xlowast = (119909lowast119871 119909lowast119877)we have the same cost in each choice as 119871(xlowast) = 119872(xlowast) =119877(xlowast) so that at the point xlowast the commuters are indifferent toany available choice andnone of themwill switch choiceThusF(xlowast) = xlowast and xlowast is a stationary state Under the conditionsin (23) the point xlowast is feasible belonging to 1198632 We can nowprove that this is indeed the unique fixed point of map F

Proposition 3 The point xlowast = (119909lowast119871 119909lowast119877) given in (21) is theunique fixed point of the map F belonging to 1198632 when theconditions in (23) hold

Proof The proof of this proposition follows directly from theobservation that

F (x) = x if and only if 119871 (x) = 119872 (x) = 119877 (x) (31)One way is already proved as we have detected the fixed pointstarting from the solution of 119871(x) = 119872(x) and 119877(x) = 119872(x)showing that it is the unique point xlowast Vice versa let us lookfor the solutions of F(x) = x We have the following systems(where 119909119872 = 1 minus 119909119871 minus 119909119877)

F119871

119909119871= 119909119871+ 120575119871(119877 (x) minus 119871 (x)

1198962

119909119877

+119872(x) minus 119871 (x)

1198964

119909119872)

119909119877= (1 minus 120575

119871

119877 (x) minus 119871 (x)1198962

)119909119877

if x isin 119877119871

F119872

119909119871= (1 minus 120575

119872

119871 (x) minus 119872 (x)119886119871minus 119889

)119909119871

119909119877= (1 minus 120575

119872

119877 (x) minus 119872 (x)119886119877minus 119889

)119909119877

if x isin 119877119872

F119877

119909119871= (1 minus 120575

119877

119871 (x) minus 119877 (x)1198961

)119909119871

119909119877= 119909119877+ 120575119877(119871 (x) minus 119877 (x)

1198961

119909119871

+119872(x) minus 119877 (x)

1198966

119909119872)

if x isin 119877119877

(32)

or equivalently (we recall that 120575119871 120575119872 and 120575

119877are different

from zero)

F119871

119877 (x) minus 119871 (x)1198962

119909119877+119872(x) minus 119871 (x)

1198964

119909119872= 0

119877 (x) minus 119871 (x)1198962

119909119877= 0

if x isin 119877119871

F119872

119871 (x) minus 119872 (x)119886119871minus 119889

119909119871= 0

119877 (x) minus 119872 (x)119886119877minus 119889

119909119877= 0

if x isin 119877119872

F119877

119871 (x) minus 119877 (x)1198961

119909119871= 0

119871 (x) minus 119877 (x)1198961

119909119871+119872(x) minus 119877 (x)

1198966

119909119872= 0

if x isin 119877119877

(33)

Considering the equation F119871(x) = x we see that it is certainly

satisfied for 119909119877 = 0 and 119909119872 = 0 which leads to thecorner point with 119909119871 = 1 However if the fixed point is an

10 Discrete Dynamics in Nature and Society

interior point of 1198632 then the point (1 0) does not belong tothe proper region 119877

119871 that is it is a virtual fixed point Clearly

this equation is satisfied when 119871(x) = 119872(x) and 119871(x) = 119877(x)which occurs only at the point xlowast It is possible to have xlowast =(1 0) and this occurs for (119886

119871minus 119889)119887

119877= 0 and (119886

119877minus 119889)119887

119871= 1

Thus we can state that the only solution of F119871(x) = x in 119877

119871

is given by the fixed point xlowast Similarly we can reason in theother regions

It is worth to mention that particular cases are obtainedwhen the fixed point xlowast belongs to a boundary of the region1198632 For example when 119909lowast119877+119909lowast119871 = 1 and 119909lowast119872 = 0 then only

the two regions119877119871and 119877

119877are involvedThis occurs when the

parameters satisfy

119889 =119886119871119887119871+ 119886119877119887119877minus 119887119871119887119877

119887119877+ 119887119871

0 le119886119871minus 119889

119887119877

le 1

0 le119886119877minus 119889

119887119871

le 1

(34)

From (119886119877minus 119889)119887

119871= (119886119877+ 119887119877minus 119886119871)(119887119877+ 119887119871) we have 119909lowast119877 =

1minus(119886119877minus119889)119887

119871= (119886119871+119887119871minus119886119877)(119887119877+119887119871) = 1198961(119887119877+119887119871) = 119909lowast as

in the one-dimensional map studied in Section 2 and 119909lowast119871 =1 minus 119909lowast119877= 1 minus (119886

119871minus 119889)119887

119877= 1198962(119887119877+ 119887119871) = 1 minus 119896

1(119887119877+ 119887119871) =

1minus119909lowast From themapsF

119871andF119877in (27) and for the parameters

occurring in this case it can be seen that 119909119871119905+1= 1minus119909

119877

119905+1holds

in both so that the dynamics can be studied by using onlythe variable 119909119877

119905and the resulting one-dimensional map is as

follows

119865 (119909119877)

=

119891119871(119909119877)

= 119909119877+ 120575119877

119871 (119909119877)minus119877 (119909

119877)

1198961

(1 minus 119909119877)

if 0 le 119909119877 le 119909lowast

119891119877(119909119877)

= 119909119877minus 120575119871

119877 (119909119877) minus 119871 (119909

119877)

1198962

119909119877 if 119909lowast le 119909119877 le 1

(35)

which corresponds to the system already studied in Section 2As we have seen this fixed point is globally attracting even iflocally we can have the eigenvalue on some side larger than 1in modulus

The case with 119909lowast119871 = 0 so that 0 le 119909lowast119877 le 1 and 119909lowast119872 =1 minus 119909lowast119877 leaves only the two regions 119877

119877and 119877

119872 This occurs

when the parameters satisfy

119886119871minus 119889

119887119877

= 1 0 le119886119877minus 119889

119887119871

le 1 (36)

From the maps F119872

and F119877in (27) it can be seen that 119909119871

119905

converges to119909lowast119871 = 0 so that the resulting asymptotic dynamic

behavior can be studied via the one-dimensional map givenby 119909119877119905+1= 119879(119909

119877

119905) with 119909119877

119905isin [0 1] (and 119909119872

119905= 1 minus 119909

119877

119905) given by

119879 (119909119877)

=

119879119871(119909119877)

= 119909119877+ 120575119877

119872(119909119877) minus 119877 (119909

119877)

1198966

(1 minus 119909119877)

if 0 le 119909119877 le 119909lowast

119879119877(119909119877)

= 119909119877minus 120575119872

119877 (119909119877) minus119872(119909

119877)

119886119877minus 119889

119909119877 if 119909lowast le 119909119877 le 1

(37)

where119872(119909119877) minus 119877(119909119877) = 119886119871+ 119887119871(1 minus 119909

119877) minus 119886119877minus 119887119877

The case with 119909lowast119877 = 0 so that 0 le 119909lowast119871 le 1 and119909lowast119872= 1 minus 119909

lowast119871 leaves only the two regions 119877119871and 119877

119872 This

occurs when the parameters satisfy 0 le (119886119871minus 119889)119887

119877le 1

(119886119877minus 119889)119887

119871= 1 From the maps F

119872and F

119871in (27) it can

be seen that 119909119877119905converges to 119909lowast119877 = 0 so that the resulting

asymptotic dynamic behavior can be studied via the one-dimensional map given by 119909119871

119905+1= 119879(119909

119871

119905) similar to the one

obtained in (37) with obvious changesBy using arguments as in Section 2 it can be shown that

these one-dimensional systems have a unique fixed pointwhich is globally attracting (although it can be locally stableor unstable)

In order to study the local stability of the fixed point ofthe two-dimensional map F we can use the linearization ofthe different definitions in the fixed point which leads to thefollowing Jacobian matrices one for each region 119877

119871 119877119872 and

119877119877 that we denote respectively with 119869

119871 119869119872 and 119869

119877

119869119871(xlowast) = (

1 + 120575119871minus 120575119871

119887119877

119887119871

1198965

1198964

minus 120575119871

119887119877

119887119871

1198966

1198961

120575119871

1198966

1198961

120575119871

119887119877

119887119871

1198966

1198961

1 minus 120575119871

1198966

1198961

)

119869119872(xlowast) = (

1 minus 120575119872

1198964

1198963

0

0 1 minus 120575119872

1198966

1198965

)

119869119877(xlowast) = (

1 minus 120575119877

1198965

1198962

120575119877

119887119871

119887119877

1198965

1198962

120575119877

1198965

1198962

1 + 120575119877minus 120575119877

119887119871

119887119877

1198963

1198966

minus 120575119877

119887119871

119887119877

1198965

1198962

)

(38)

The two real eigenvalues of the Jacobian matrix in 119877119872are

in explicit form and both are smaller than +1 However theycan be larger than minus1 or not Also the eigenvalues of the otherjacobian matrices can be in modulus smaller or larger than1 However the global dynamics observed numerically givealways trajectories which are convergent to the fixed point xlowastat any set of allowed parameters values

Discrete Dynamics in Nature and Society 11

1

1

075

05

025

00 025 05 075

xR

xL

(a)

1

1

075

05

025

00 025 05 075

xR

xL

(b)

Figure 6 Examples of trajectories converging to the unique fixed point with parameters (a) 119886119871= 28 119887

119871= 24 119886

119877= 8 119887

119877= 24 119889 = 5 120575

119871= 04

120575119872= 03 120575

119877= 07 and ic 119909119871 = 08 119909119877 = 005 (b) 119886

119871= 29 119887

119871= 16 119886

119877= 15 119887

119877= 25 119889 = 7 120575

119871= 05 120575

119872= 08 120575

119877= 06 and ic 119909119871 = 005

119909119877= 09 Dots are linked by a continuous line for illustrative purpose only

This global attractivity in the two-dimensional map isdifficult to be proved rigorously however the results in theparticular cases commented above that is when the fixedpoint xlowast belongs to the boundary of 1198632 suggests that thesame result is true also when the fixed point is in the interiorof1198632

In Figure 6 we show some examples in Figure 6(a) wecan see a trajectory starting at 119877

119877 in Figure 6(b) a trajectory

whose initial point in 119877119871is mapped into 119877

119872 The shape of

the functions in the case of Figure 6(b) is similar to the oneshown in Figure 5

4 Conclusion

In this paper we have analyzed a dynamical version of theBraess paradox with nonimpulsive commuters who changeroad proportionally to the cost differenceWe were interestedto prove that with nonimpulsive commuters there exists aunique equilibrium both in the original network with twochoices and in the ternary onewhen a single link is addedWehave shown that both models are quite robust We were ableto provide an analytical proof of the global attractivity of theunique fixed point in the one-dimensional case when onlytwo roads are available andweusednumerical techniques andsimulations to give evidence in the two-dimensional case (iewhen a new road is added) as a rigorous proof can be doneonly for particular border cases

A limitation of our contribution is considering homo-geneous populations in spite of the fact that the evidencesupports heterogeneity Nevertheless heterogeneity of com-mutersrsquo behavior cannot be easily classified yet even when

conducting experiments common individual patterns arejust identified and classified On the other hand whenobserving the aggregate behavior inmany cases the literatureprovides evidence in support of the paradox outcome thatis the convergence to the Nash equilibrium The impulsiveagents in [30] react faster than the proportional agentspresented in this paper yet as a population they fail toachieve the Nash equilibria in certain cases while in thispaper the aggregate behavior does not We proposed thisaggregate behavior because it replies the convergence to theNash equilibria and at the same time it may be the resultof modeling some observed individual behaviors such asimitation and free riding

Taking into account the limitation of homogeneous pop-ulation analysis a natural extension of our contribution isconsidering heterogeneous populations with both impulsiveand nonimpulsive commuters We have reasons to believethat in such a case some properties of the different behaviorsare inherited and some are lost Therefore the results weprovide in this paper match those for impulsive populationsand are paramount to extend the analysis to more realisticcases In factmdashas the empirical results showmdashheterogeneouspopulations dynamics provide a better fit of what has beenobserved when considering human participants interaction

Furthermore it will be interesting to consider general-izations to networks with richer architectures Finally weobserve that since with the class of behavior we considerin this paper that dynamics depends on the shape of thecost functions it will be interesting to extend the analysis tononlinear cost functions

12 Discrete Dynamics in Nature and Society

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

V Avrutin is supported by the European Community withinthe scope of the project ldquoMultiple-discontinuity inducedbifurcations in theory and applicationsrdquo (Marie Curie Actionof the 7th Framework Programme Contract Agreement noPIEF-GA-2011-300281) U Merlone has worked under theauspices of COST Action IS1104 ldquoThe EU in the new complexgeography of economic systems models tools and policyevaluationrdquo

References

[1] R Arnott and K Small ldquoThe economics of traffic congestionrdquoAmerican Scientist vol 82 no 5 pp 446ndash455 1994

[2] T Bass ldquoRoad to ruinrdquo Discover pp 56ndash61 1992[3] B Calvert and G Keady ldquoBraessrsquos paradox and power-law non-

linearities in networksrdquoThe Journal of the AustralianMathemat-ical Society B Applied Mathematics vol 35 no 1 pp 1ndash22 1993

[4] J E Cohen andC Jeffries ldquoCongestion resulting from increasedcapacity in single-server queueing networksrdquo IEEEACMTrans-actions on Networking vol 5 no 2 pp 305ndash310 1997

[5] J E Cohen and F P Kelly ldquoA paradox of congestion in a queuingnetworkrdquo Journal of Applied Probability vol 27 no 3 pp 730ndash734 1990

[6] S Dafermos and A Nagurney ldquoOn some traffic equilibriumtheory paradoxesrdquo Transportation Research B Methodologicalvol 18 no 2 pp 101ndash110 1984

[7] C Fisk ldquoMore paradoxes in the equilibrium assignment prob-lemrdquo Transportation Research B Methodological vol 13 no 4pp 305ndash309 1979

[8] W H K Lam ldquoEffects of road pricing on system performancerdquoTraffic Engineering amp Control vol 29 no 12 pp 631ndash635 1988

[9] J D Murchland ldquoBraessrsquos paradox of traffic flowrdquo Transporta-tion Research vol 4 no 4 pp 391ndash394 1970

[10] R Steinberg and W I Zangwill ldquoThe prevalence o f BraessrsquoparadoxrdquoTransportation Science vol 17 no 3 pp 301ndash318 1983

[11] M Frank ldquoThe Braess paradoxrdquo Mathematical Programmingvol 20 no 1 pp 283ndash302 1981

[12] T Roughgarden Selfish Routing and the Price of Anarchy MITpress 2005

[13] B B Fu C X Zhao and S B Li ldquoThe analysis of Braessrsquo paradoxand robustness based on dynamic traffic assignment modelsrdquoDiscrete Dynamics in Nature and Society vol 2013 Article ID796842 8 pages 2013

[14] C Fisk and S Pallottino ldquoEmpirical evidence for equilibriumparadoxes with implications for optimal planning strategiesrdquoTransportation Research Part A General vol 15 no 3 pp 245ndash248 1981

[15] H Youn M T Gastner and H Jeong ldquoPrice of anarchy intransportation networks efficiency and optimality controlrdquoPhysical Review Letters vol 101 no 12 Article ID 128701 2008

[16] K Aoki Y Ohtsubo A Rapoport and T Saijo ldquoEffects of priorinvestment and personal responsibility in a simple network

gamerdquo Current Research in Social Psychology vol 13 no 2 pp10ndash21 2007

[17] J Morgan H Orzen and M Sefton ldquoNetwork architectureand traffic flows experiments on the Pigou-Knight-Downs andBraess paradoxesrdquoGames and Economic Behavior vol 66 no 1pp 348ndash372 2009

[18] A Rapoport V Mak and R Zwick ldquoNavigating congested net-works with variable demand experimental evidencerdquo Journal ofEconomic Psychology vol 27 no 5 pp 648ndash666 2006

[19] A Rapoport T Kugler S Dugar and E J Gisches ldquoBraessparadox in the laboratory an experimental study of route choicein traffic networks with asymmetric costsrdquo inDecisionModelingand Behavior in Uncertain and Complex Environment T KuglerJ C Smith T Connolly and Y J Son Eds pp 309ndash337Springer New York NY USA 2008

[20] A Rapoport T Kugler S Dugar and E J Gisches ldquoChoice ofroutes in congested traffic networks experimental tests of theBraess paradoxrdquo Games and Economic Behavior vol 65 no 2pp 538ndash571 2009

[21] E J Gisches and A Rapoport ldquoDegrading network capacitymay improve performance private versus public monitoring inthe Braess paradoxrdquoTheory and Decision vol 73 no 2 pp 267ndash293 2012

[22] A Dal Forno and U Merlone ldquoReplicating human interactionin Braess paradoxrdquo in Proceedings of the Simulation Conference(WSC rsquo13) pp 1754ndash1765 IEEE 2013

[23] M Ausloos H Dawid and U Merlone ldquoSpatial interactionsin agent-based modelingrdquo in Complexity and GeographicalEconomics Topics and Tools P Commendatore S S Kayamand I Kubin Eds Springer Berlin Germany 2014

[24] G-I Bischi and U Merlone ldquoGlobal dynamics in binary choicemodels with social influencerdquo The Journal of MathematicalSociology vol 33 no 4 pp 277ndash302 2009

[25] G I Bischi L Gardini and U Merlone ldquoImpulsivity in binarychoices and the emergence of periodicityrdquoDiscrete Dynamics inNature and Society vol 2009 Article ID 407913 22 pages 2009

[26] JH PattonM S Stanford andE S Barratt ldquoFactor structure ofthe Barratt Impulsiveness Scalerdquo Journal of Clinical Psychologyvol 51 no 6 pp 768ndash774 1995

[27] G I Bischi L Gardini and U Merlone ldquoPeriodic cycles andbifurcation curves for one-dimensional maps with two discon-tinuitiesrdquo Journal of Dynamical Systems GeometricTheories vol7 no 2 pp 101ndash123 2009

[28] G-I Bischi and U Merlone ldquoBinary choices in small and largegroups a unifiedmodelrdquo Physica A vol 389 no 4 pp 843ndash8532010

[29] A Dal Forno L Gardini and U Merlone ldquoTernary choicesin repeated games and border collision bifurcationsrdquo ChaosSolitons amp Fractals vol 45 no 3 pp 294ndash305 2012

[30] A Dal Forno and U Merlone ldquoBorder-collision bifurcationsin a model of Braess paradoxrdquo Mathematics and Computers inSimulation vol 87 pp 1ndash18 2013

[31] A Dal Forno V Giorgino and U Merlone ldquoAgent-based mod-eling with GTmethodology An examplerdquoWorking Paper 2014

[32] L Gardini U Merlone and F Tramontana ldquoInertia in binarychoices continuity breaking and big-bang bifurcation pointsrdquoJournal of Economic Behavior amp Organization vol 80 no 1 pp153ndash167 2011

[33] W H Sandholm Population Games and Evolutionary Dynam-ics The MIT Press Cambridge Mass USA 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 5xR t+1

xR

t

00

025

025 05

05

075

1

1075

(a)xR t+1

xR

t

00

025

025 05

05

075

1

1075

(b)

Figure 2 Qualitative shapes of the map In (a) case (i) when 0 lt 11989621198961le 1205751198712 the fixed point is locally unstable on the right side In (b)

case (iii) when 2120575119877le 11989621198961 the fixed point is locally unstable on the left side

0

1

0 1

xR t

xL

t

(a)

0

1

0 1

xR t

xL

t

(b)

0

1

0 1

xR t

xL

t

(c)

Figure 3 Qualitative shapes of the map in case (ii) when 1205751198712 lt 119896

21198961le 2120575

119877and the fixed point is locally stable on both sides

easy to see that for any 119909119877 isin [0 119909lowast] it is 1198911015840119871(119909lowast) lt 1198911015840

119871(119909lowast) and

thus the function on the left side of the fixed point is strictlydecreasing and expansive each point on the left side of thefixed point is mapped to the right side of the fixed point inone iteration While on the right side of the fixed point theconcave function 119891

119877(119909119877) has 1198911015840

119877(119909lowast) = 1 minus 120575

119871(11989611198962) gt 0 (as

this holds for 11989621198961gt 120575119871which is satisfied in this interval)

This implies that the point of local maximum on the rightside say 119909119877

119903119872 is larger than the fixed point 119909119877

119903119872gt 119909lowast Then

the map is a contraction in the interval 119882119904loc = [119909lowast 119909119877

119903119872]

and an absorbing interval which is mapped into119882119904loc in oneiteration is given by 119869 = [119891minus1

119871(119909119877

119903119872) 119909119877

119903119872] when 119891

119871(0) =

120575119877gt 119909119877

119903119872 or 119869 = [0 119909119877

119903119872] when 119891

119871(0) = 120575

119877le 119909119877

119903119872 Any

initial condition belonging to [0 1] 119869 has a trajectory whichis mapped into 119869 in a finite number of iterations Thus 119909lowast isglobally attracting An example is shown in Figure 2(b)

When 11989621198961belongs to the interval in (ii) 120575

1198712 lt 119896

21198961lt

2120575119877 we have that the fixed point is locally stable on both

sides Moreover inside this interval we can distinguish threeregions as already showed in (15) that is

(a) 1205751198712 lt 119896

21198961le 120575119871with 0 lt 1198911015840

119871(119909lowast) lt 1 and

minus1 lt 1198911015840

119877(119909lowast) lt 0 as in Figure 3(a)

(b) 120575119871lt 11989621198961lt 1120575

119877with 0 lt 1198911015840

119871(119909lowast) lt 1 and

0 lt 1198911015840

119877(119909lowast) lt 1 as in Figure 3(b)

6 Discrete Dynamics in Nature and Society

(c) 1120575119877le 11989621198961lt 2120575

119877with minus1 lt 1198911015840

119871(119909lowast) lt 0 and

0 lt 1198911015840

119877(119909lowast) lt 1 as in Figure 3(c)

In case (a) we have that the point of local minimum onthe left side say 119909119877

119897119898 is smaller than the fixed point119909119877

119897119898lt 119909lowast

Then themap is a contraction in the interval119882119904loc = [119909119877

119897119898 119909lowast]

and on the right side the map is decreasing so that anabsorbing interval which is mapped into119882119904loc in one iterationis given by 119869 = [119909119877

119897119898 119891minus1

119877(119909119877

119897119898)]when119891

119877(1) = 1minus120575

119871lt 119909119877

119897119898 or

119869 = [119909119877

119897119898 1]when119891

119877(1) = 1minus120575

119871ge 119909119877

119897119898 Any initial condition

belonging to [0 1] 119869 has a trajectory which is mapped into 119869in a finite number of iterationsThus 119909lowast is globally attracting

In case (c) we can reason similarly we have that the pointof local maximum on the right side say 119909119877

119903119872 is larger than

the fixed point 119909119877119903119872gt 119909lowast Then the map is a contraction in

the interval 119882119904loc = [119909lowast 119909119877

119903119872] and on the left side the map

is decreasing so that an absorbing interval which is mappedinto 119882119904loc in one iteration is given by 119869 = [119891minus1

119871(119909119877

119903119872) 119909119877

119903119872]

when 119891119871(0) = 120575

119877gt 119909119877

119903119872 or 119869 = [0 119909119877

119903119872] when 119891

119871(0) =

120575119877le 119909119877

119903119872 Any initial condition belonging to [0 1] 119869 has

a trajectory which is mapped into 119869 in a finite number ofiterations Thus 119909lowast is globally attracting

In case (b) we have that the point of local minimum onthe left side say 119909119877

119897119898 is smaller than the fixed point 119909119877

119897119898lt

119909lowast and also that the point of local maximum on the right

side say 119909119877119903119872

is larger than the fixed point 119909119877119903119872gt 119909lowast

Then the map is a contraction in the interval 119882119904loc =

[119909119877

119897119898 119909119877

119903119872] which includes the fixed point and an absorbing

interval which is mapped into119882119904loc in one iteration is givenby 119869 = [119891minus1

119871(119909119877

119903119872) 119891minus1

119877(119909119877

119897119898)] or 119869 = [0 119891minus1

119877(119909119877

119897119898)] or 119869 =

[119891minus1

119871(119909119877

119903119872) 1] or [0 1] depending under obvious conditions

on the values of 120575119871and 120575

119877 When 119869 sub [0 1] then any initial

condition belonging to [0 1] 119869 has a trajectory which ismapped into 119869 in a finite number of iterations Thus 119909lowast isglobally attracting

We notice that in Proposition 2 we prove that when thefixed point 119909lowast is internal to the interval [0 1] then it isglobally attracting Let us turn to comment the two extremawhich occur when 119909lowast = 0 or 119909lowast = 1 These solutions clearlyexist and are also globally attracting In fact the case 119909lowast = 0corresponds to 119896

1= 119871(0)minus119877(0) = 119886

119871+119887119871minus119886119877= 0 and only the

function 119891119877(119909119877) is applied in the interval [0 1] and from the

concavity of the function we have that this fixed point is glob-ally attracting even if the eigenvalue is equal to 1 The othercase 119909lowast = 1 corresponds to 119896

2= 119877(1)minus119871(1) = 119886

119877+119887119877minus119886119871= 0

and only the function 119891119871(119909119877) is applied in the interval [0 1]

and from the convexity of the function we have that this fixedpoint is globally attracting even if the eigenvalue is equal to 1

3 The Dynamics in the ExpandedNetwork with Three Roads

As it is well known the Braess paradox introduces a new linkand shows how this opportunity worsens both the individualand collective payoffs This link introduces a new alternativenow the commuters are facing three choices where the new

one consists of path 119878-119871-119877-119864 (see Figure 1(b)) which will bedenoted as 119872 in the following Models with three choiceswere introduced in [29] as it concerns discrete time and [33]as it concerns continuous time In this paper we analyze thedynamics when the travel time 119889 for the new resulting linkvaries in a range that makes this link a dominant choice forsome values and a dominated choice for others It is worth toobserve that in particular when the travel time on link 119871-119877 iszero the paradoxical result still holds

Now assume that commuters switch roadwhenever traveltime has become larger that is the other road is less costlyand therefore more attractive We indicate by 119909119871 ge 0 thefraction of population choosing action 119871 by 119909119872 ge 0 thefraction choosing action 119872 and by 119909119877 ge 0 the fractionchoosing action 119877 with the constraint 119909119871 + 119909119877 + 119909119872 = 1The travel times are given respectively as

119871 = 119886119871+ 119887119871(119909119871+ 119909119872)

119872 = 119887119871(119909119871+ 119909119872) + 119887119877(119909119872+ 119909119877) + 119889

119877 = 119886119877+ 119887119877(119909119872+ 119909119877)

(16)

By using the constraint let us define the fraction of populationchoosing119872 as 119909119872 = 1 minus 119909119871 minus 119909119877 ge 0 so to consider onlytwo variables x = (119909119871 119909119877) isin R2

+ and the phase space is the

triangle

1198632= x = (119909119871 119909119877) isin R2

+ 0 le 119909

119871+ 119909119877le 1 (17)

After substituting the expression of 119909119872 in the travel times weget

119871 (x) = 119886119871 + 119887119871 (1 minus 119909119877)

119872 (x) = 119887119871 (1 minus 119909119877) + 119887119877(1 minus 119909

119871) + 119889

119877 (x) = 119886119877 + 119887119877 (1 minus 119909119871)

(18)

The vertices of the phase space119875lowast119871(1 0)119875lowast

119872(0 0) and119875lowast

119877(0 1)

represent states in which the whole population choosesrespectively actions 119871119872 and 119877

Since commuters are interested in minimizing traveltime we denote by 119877

119860the region in which the choice 119860

(where119860 is 119871119872 or 119877) is preferable These regions are calleddominance regions of 119860 We have the following definition ofthe dominance regions

119877119871= x isin 1198632 119871 (x) le 119872 (x) 119871 (x) le 119877 (x)

119877119872= x isin 1198632 119872 (x) le 119871 (x) 119872 (x) le 119877 (x)

119877119877= x isin 1198632 119877 (x) le 119871 (x) 119877 (x) le 119872 (x)

(19)

Discrete Dynamics in Nature and Society 7

1

100

xR

RL

RM

RR

xL

(a)

1

100

xR

RL

RM

RR

xL

(b)

1

100

xR

RL

RM

RR

xL

(c)

Figure 4 Dominance regions in the three choices network with different values of the cost parameter 119889 of the map F at 119886119871= 27 119887

119871= 30

119886119877= 18 119887

119877= 30 and (a) 119889 = 6 (b) 119889 = 3 (c) 119889 = 0

By using the expressions in (18) we get the following defini-tions for the regions

119877119871= x isin 1198632 119909119871 le 119909lowast119871

119886119871+ 119887119871(1 minus 119909

119877) le 119886119877+ 119887119877(1 minus 119909

119871)

119877119872= x isin 1198632 119909119871 ge 119909lowast119871 119909119877 ge 119909lowast119877

119877119877= x isin 1198632 119886

119877+ 119887119877(1 minus 119909

119871) le 119886119871+ 119887119871(1 minus 119909

119877)

119909119877le 119909lowast119877

(20)

where we have introduced

119909lowast119871= 1 minus

119886119871minus 119889

119887119877

=119889 minus 119886119871+ 119887119877

119887119877

119909lowast119877= 1 minus

119886119877minus 119889

119887119871

=119889 minus 119886119877+ 119887119871

119887119871

(21)

The boundaries of the three regions are given by segments ofstraight lines of equations

1198811 119909119871= 119909lowast119871

1198812 119909119877= 119909lowast119877

1198813 119886119871+ 119887119871(1 minus 119909

119877) = 119886119877+ 119887119877(1 minus 119909

119871)

(22)

and it is immediate to see that the point xlowast = (119909lowast119871 119909lowast119877)satisfies also the third equation that is it belongs to all thethree straight lines xlowast isin 119881

119894for 119894 = 1 2 3

Using the geometric representation in the triangle 1198632the dominance regions are illustrated in Figure 4 (via anexample) using different colors Three cases are shown fordifferent values of the parameter 119889 in order to illustrate howthe region119877

119872depends on the parameters of the third branch

The point xlowast = (119909lowast119871 119909lowast119877) is the unique point on theboundaries 119881

119894of all the three regions Thus for our model we

have to require 119909lowast119871 ge 0 119909lowast119877 ge 0 and 119909lowast119871 + 119909lowast119877 le 1 leadingto 119909lowast119872 = 1 minus 119909lowast119871 minus 119909lowast119877 In order to have three regions in1198632

it must be 119909lowast119871 gt 0 119909lowast119877 gt 0 and 119909lowast119871 + 119909lowast119877 lt 1 The equalityin one of these three conditions implies the disappearance ofone region For example when 119909lowast119871 = 0 then 0 le 119909lowast119877 le 1and 119909lowast119872 = 1 minus 119909lowast119877 leaving only the two regions 119877

119877and 119877

119872

and so onThe conditions 119909lowast119871 ge 0 119909lowast119877 ge 0 and 119909lowast119871 + 119909lowast119877 le 1 lead

to the constraints that will be assumed satisfied henceforth

0 le119886119871minus 119889

119887119877

le 1 0 le119886119877minus 119889

119887119871

le 1

119886119871minus 119889

119887119877

+119886119877minus 119889

119887119871

ge 1

(23)

We notice that from the third condition in (23) we have thevalues of 119889 for which the third region is really present in 1198632that is

0 le 119889 le119886119871119887119871+ 119886119877119887119877minus 119887119871119887119877

119887119877+ 119887119871

(24)

Commuters are homogeneous and minimize their nextperiod travel cost At time 119905+1 the vector x

119905becomes common

knowledge and each commuter can observe the travel costs119871(x119905)119872(x

119905) and 119877(x

119905) We assume that if at time 119905 a fraction

119909119871

119905chooses path 119871 a fraction 119909119877

119905chooses action 119877 and travel

costs are such that 119877(x119905) lt 119871(x

119905) and 119877(x

119905) lt 119872(x

119905) then

both a fraction of the 119909119871119905commuters who chose action 119871 and

a fraction of the (1 minus 119909119871119905minus 119909119877

119905) commuters who chose action

119872 will switch to path 119877 in the next time period 119905 + 1 Thefraction of switching commuters is given by the switchingpropensities 120575

119871 120575119872 120575119877modulated by the relative differences

in payoffs This happens whenever a path gives a smallertravel cost In other words at any time 119905 all the commutersdecide their future action at time 119905 + 1 comparing the costs119871(x119905)119872(x

119905) and 119877(x

119905) As in the one-dimensional case here

also the differences of travel costs are normalized dividing

8 Discrete Dynamics in Nature and Society

each difference by the constants giving the related maximumvalue given in the different cases as follows

1198961= max

x [119871 (x) minus 119877 (x)]

= [119871 (1 0) minus 119877 (1 0)] = 119886119871 + 119887119871 minus 119886119877

1198962= max

x [119877 (x) minus 119871 (x)]

= [119877 (0 1) minus 119871 (0 1)] = 119886119877 + 119887119877 minus 119886119871

1198963= max

x [119871 (x) minus 119872 (x)]

= [119871 (1 119909119877) minus119872(1 119909

119877)] = 119886

119871minus 119889 forall119909

119877isin [0 1]

1198964= max

x [119872 (x) minus 119871 (x)]

= [119872(0 119909119877) minus 119871 (0 119909

119877)] = 119889 + 119887

119877minus 119886119871 forall119909

119877isin [0 1]

1198965= max

x [119877 (x) minus 119872 (x)]

= [119877 (119909119871 1) minus 119872(119909

119871 1)] = 119886

119877minus 119889 forall119909

119871isin [0 1]

1198966= max

x [119872 (x) minus 119877 (x)]

= [119872(119909119871 0) minus 119877 (119909

119871 0)] = 119889 + 119887

119871minus 119886119877 forall119909

119871isin [0 1]

(25)

The resulting dynamics are described by a two-dimensionalmap x

119905+1= F(x

119905) defined as

F x119905+1=

F119871(x119905) if x

119905isin 119877119871

F119872(x119905) if x

119905isin 119877119872

F119877(x119905) if x

119905isin 119877119877

(26)

where

F119871

119909119871

119905+1= 119909119871

119905+ 120575119871(119877 (x119905) minus 119871 (x

119905)

119886119877+ 119887119877minus 119886119871

119909119877

119905

+119872(x119905) minus 119871 (x

119905)

119889 + 119887119877minus 119886119871

119909119872

119905)

119909119877

119905+1= (1 minus 120575

119871

119877 (x119905) minus 119871 (x

119905)

119886119877+ 119887119877minus 119886119871

)119909119877

119905

F119872

119909119871

119905+1= (1 minus 120575

119872

119871 (x119905) minus 119872(x

119905)

119886119871minus 119889

)119909119871

119905

119909119877

119905+1= (1 minus 120575

119872

119877 (x119905) minus 119872(x

119905)

119886119877minus 119889

)119909119877

119905

F119877

119909119871

119905+1= (1 minus 120575

119877

119871 (x119905) minus 119877 (x

119905)

119886119871+ 119887119871minus 119886119877

)119909119871

119905

119909119877

119905+1= 119909119877

119905+ 120575119877(119871 (x119905) minus 119877 (x

119905)

119886119871+ 119887119871minus 119886119877

119909119871

119905

+119872(x119905) minus 119877 (x

119905)

119889 + 119887119871minus 119886119877

119909119872

119905)

(27)

By construction we have that F maps 1198632 into itself In factto show that F 1198632 rarr 119863

2 it is enough to show that

given (119909119871119905 119909119877

119905) isin 119863

2 then (119909119871119905+1 119909119877

119905+1) = F(119909119871

119905 119909119877

119905) isin 119863

2From (119909119871

119905 119909119877

119905) isin 119863

2 we have that 0 le 119909119871119905+ 119909119877

119905le 1 and

119909119871

119905+ 119909119877

119905+ 119909119872

119905= 1 Then if (119909119871

119905+1 119909119877

119905+1) = F119871(119909119871

119905 119909119877

119905) we have

119909119871

119905+1+ 119909119877

119905+1= 119909119871

119905+ 119909119877

119905+ 120575119871

119872(x119905) minus 119871 (x

119905)

119889 + 119887119877minus 119886119871

119909119872

119905

le 119909119871

119905+ 119909119877

119905+ 119909119872

119905= 1

(28)

119909119871

119905+1ge 0 and 119909119877

119905+1ge 0 thus (119909119871

119905+1 119909119877

119905+1) isin 1198632 If (119909119871

119905+1 119909119877

119905+1) =

F119872(119909119871

119905 119909119877

119905) we have 0 le 119909119871

119905+1le 119909119871

119905and 0 le 119909119877

119905+1le 119909119877

119905 thus

(119909119871

119905+1 119909119877

119905+1) isin 1198632 If (119909119871

119905+1 119909119877

119905+1) = F119877(119909119871

119905 119909119877

119905) we have

119909119871

119905+1+ 119909119877

119905+1= 119909119871

119905+ 119909119877

119905+ 120575119877

119872(x119905) minus 119877 (x

119905)

119889 + 119887119871minus 119886119877

119909119872

119905

le 119909119871

119905+ 119909119877

119905+ 119909119872

119905= 1

(29)

119909119871

119905+1ge 0 and 119909119877

119905+1ge 0 thus (119909119871

119905+1 119909119877

119905+1) isin 1198632

Substituting from (18) we can rewrite the functions inexplicit form as

F119871

119909119871

119905+1= 120575119871+ (1 minus 120575

119871minus 120575119871

119887119877

1198964

)119909119871

119905

minus120575119871

119887119871

1198962

119909119877

119905+ 120575119871

119887119871

1198962

(119909119877

119905)2

+120575119871119887119877(1

1198964

minus1

1198962

)119909119871

119905119909119877

119905+ 120575119871

119887119877

1198964

(119909119871

119905)2

119909119877

119905+1= (1 minus 120575

119871minus 120575119871

119887119871

1198962

)119909119877

119905

+120575119871

119887119877

1198962

119909119871

119905119909119877

119905minus 120575119871

119887119871

1198962

(119909119877

119905)2

F119872

119909119871

119905+1= (1 + 120575

119872

1198964

1198963

)119909119871

119905minus 120575119872

119887119877

1198963

(119909119871

119905)2

119909119877

119905+1= (1 + 120575

119872

1198966

1198965

)119909119877

119905minus 120575119872

119887119871

1198965

(119909119877

119905)2

F119877

119909119871

119905+1= (1 minus 120575

119877minus 120575119877

119887119877

1198961

)119909119871

119905

+120575119877

119887119871

1198961

119909119871

119905119909119877

119905minus 120575119877

119887119877

1198961

(119909119871

119905)2

119909119877

119905+1= 120575119877+ (1 minus 120575

119877minus 120575119877

119887119871

1198966

)119909119877

119905

minus120575119877

119887119877

1198961

119909119871

119905+ 120575119877

119887119871

1198961

(119909119877

119905)2

+120575119877119887119871(1

1198966

minus1

1198961

)119909119871

119905119909119877

119905+ 120575119877

119887119877

1198966

(119909119871

119905)2

(30)

Differently from the one-dimensional map 119865 considered inthe previous section the two-dimensional map F is notcontinuous in 1198632 More precisely the functions F

119871 F119872

and F119877are continuous and thus F is continuous in the

interior of each region inwhich the different definitions applyHowever in general except for the point xlowast the map F is notcontinuous along the borders 119881

119894 119894 = 1 2 3 of the regions

119877119871 119877119872 and 119877

119877 In fact it is easy to see that considering

Discrete Dynamics in Nature and Society 9

xL

t+1

xL

t

xR

t

00

1

1

(a)

xL

t+1

xL

t

xR

t

00

1

1

(b)

Figure 5 Graphics of the three functions defining (a) 119909119871119905+1

and (b) 119909119877119905+1

with parameters 119886119871= 29 119887

119871= 16 119886

119877= 19 119887

119877= 24 119889 = 8 120575

119871= 05

120575119872= 08 and 120575

119877= 06

a point (119909119871 119909119877) belonging to one of the lines defined in (22)at the boundary of two regions only we have that the twofunctions defined on the two different regions lead to twodifferent points of1198632 An example of how the three functionsdefining 119909119871

119905+1behave is shown in Figure 5(a) and the three

functions defining 119909119877119905+1

are illustrated in Figure 5(b) showingthat F is continuous only in xlowast

The point xlowast in which the three regions are in contactand F is continuous is clearly particular In fact solving theequation 119871(x) = 119872(x) we have the unique solution 119909119871 = 119909lowast119871while solving the equation 119877(x) = 119872(x) we have the uniquesolution119909119877 = 119909lowast119877 It follows that at the point xlowast = (119909lowast119871 119909lowast119877)we have the same cost in each choice as 119871(xlowast) = 119872(xlowast) =119877(xlowast) so that at the point xlowast the commuters are indifferent toany available choice andnone of themwill switch choiceThusF(xlowast) = xlowast and xlowast is a stationary state Under the conditionsin (23) the point xlowast is feasible belonging to 1198632 We can nowprove that this is indeed the unique fixed point of map F

Proposition 3 The point xlowast = (119909lowast119871 119909lowast119877) given in (21) is theunique fixed point of the map F belonging to 1198632 when theconditions in (23) hold

Proof The proof of this proposition follows directly from theobservation that

F (x) = x if and only if 119871 (x) = 119872 (x) = 119877 (x) (31)One way is already proved as we have detected the fixed pointstarting from the solution of 119871(x) = 119872(x) and 119877(x) = 119872(x)showing that it is the unique point xlowast Vice versa let us lookfor the solutions of F(x) = x We have the following systems(where 119909119872 = 1 minus 119909119871 minus 119909119877)

F119871

119909119871= 119909119871+ 120575119871(119877 (x) minus 119871 (x)

1198962

119909119877

+119872(x) minus 119871 (x)

1198964

119909119872)

119909119877= (1 minus 120575

119871

119877 (x) minus 119871 (x)1198962

)119909119877

if x isin 119877119871

F119872

119909119871= (1 minus 120575

119872

119871 (x) minus 119872 (x)119886119871minus 119889

)119909119871

119909119877= (1 minus 120575

119872

119877 (x) minus 119872 (x)119886119877minus 119889

)119909119877

if x isin 119877119872

F119877

119909119871= (1 minus 120575

119877

119871 (x) minus 119877 (x)1198961

)119909119871

119909119877= 119909119877+ 120575119877(119871 (x) minus 119877 (x)

1198961

119909119871

+119872(x) minus 119877 (x)

1198966

119909119872)

if x isin 119877119877

(32)

or equivalently (we recall that 120575119871 120575119872 and 120575

119877are different

from zero)

F119871

119877 (x) minus 119871 (x)1198962

119909119877+119872(x) minus 119871 (x)

1198964

119909119872= 0

119877 (x) minus 119871 (x)1198962

119909119877= 0

if x isin 119877119871

F119872

119871 (x) minus 119872 (x)119886119871minus 119889

119909119871= 0

119877 (x) minus 119872 (x)119886119877minus 119889

119909119877= 0

if x isin 119877119872

F119877

119871 (x) minus 119877 (x)1198961

119909119871= 0

119871 (x) minus 119877 (x)1198961

119909119871+119872(x) minus 119877 (x)

1198966

119909119872= 0

if x isin 119877119877

(33)

Considering the equation F119871(x) = x we see that it is certainly

satisfied for 119909119877 = 0 and 119909119872 = 0 which leads to thecorner point with 119909119871 = 1 However if the fixed point is an

10 Discrete Dynamics in Nature and Society

interior point of 1198632 then the point (1 0) does not belong tothe proper region 119877

119871 that is it is a virtual fixed point Clearly

this equation is satisfied when 119871(x) = 119872(x) and 119871(x) = 119877(x)which occurs only at the point xlowast It is possible to have xlowast =(1 0) and this occurs for (119886

119871minus 119889)119887

119877= 0 and (119886

119877minus 119889)119887

119871= 1

Thus we can state that the only solution of F119871(x) = x in 119877

119871

is given by the fixed point xlowast Similarly we can reason in theother regions

It is worth to mention that particular cases are obtainedwhen the fixed point xlowast belongs to a boundary of the region1198632 For example when 119909lowast119877+119909lowast119871 = 1 and 119909lowast119872 = 0 then only

the two regions119877119871and 119877

119877are involvedThis occurs when the

parameters satisfy

119889 =119886119871119887119871+ 119886119877119887119877minus 119887119871119887119877

119887119877+ 119887119871

0 le119886119871minus 119889

119887119877

le 1

0 le119886119877minus 119889

119887119871

le 1

(34)

From (119886119877minus 119889)119887

119871= (119886119877+ 119887119877minus 119886119871)(119887119877+ 119887119871) we have 119909lowast119877 =

1minus(119886119877minus119889)119887

119871= (119886119871+119887119871minus119886119877)(119887119877+119887119871) = 1198961(119887119877+119887119871) = 119909lowast as

in the one-dimensional map studied in Section 2 and 119909lowast119871 =1 minus 119909lowast119877= 1 minus (119886

119871minus 119889)119887

119877= 1198962(119887119877+ 119887119871) = 1 minus 119896

1(119887119877+ 119887119871) =

1minus119909lowast From themapsF

119871andF119877in (27) and for the parameters

occurring in this case it can be seen that 119909119871119905+1= 1minus119909

119877

119905+1holds

in both so that the dynamics can be studied by using onlythe variable 119909119877

119905and the resulting one-dimensional map is as

follows

119865 (119909119877)

=

119891119871(119909119877)

= 119909119877+ 120575119877

119871 (119909119877)minus119877 (119909

119877)

1198961

(1 minus 119909119877)

if 0 le 119909119877 le 119909lowast

119891119877(119909119877)

= 119909119877minus 120575119871

119877 (119909119877) minus 119871 (119909

119877)

1198962

119909119877 if 119909lowast le 119909119877 le 1

(35)

which corresponds to the system already studied in Section 2As we have seen this fixed point is globally attracting even iflocally we can have the eigenvalue on some side larger than 1in modulus

The case with 119909lowast119871 = 0 so that 0 le 119909lowast119877 le 1 and 119909lowast119872 =1 minus 119909lowast119877 leaves only the two regions 119877

119877and 119877

119872 This occurs

when the parameters satisfy

119886119871minus 119889

119887119877

= 1 0 le119886119877minus 119889

119887119871

le 1 (36)

From the maps F119872

and F119877in (27) it can be seen that 119909119871

119905

converges to119909lowast119871 = 0 so that the resulting asymptotic dynamic

behavior can be studied via the one-dimensional map givenby 119909119877119905+1= 119879(119909

119877

119905) with 119909119877

119905isin [0 1] (and 119909119872

119905= 1 minus 119909

119877

119905) given by

119879 (119909119877)

=

119879119871(119909119877)

= 119909119877+ 120575119877

119872(119909119877) minus 119877 (119909

119877)

1198966

(1 minus 119909119877)

if 0 le 119909119877 le 119909lowast

119879119877(119909119877)

= 119909119877minus 120575119872

119877 (119909119877) minus119872(119909

119877)

119886119877minus 119889

119909119877 if 119909lowast le 119909119877 le 1

(37)

where119872(119909119877) minus 119877(119909119877) = 119886119871+ 119887119871(1 minus 119909

119877) minus 119886119877minus 119887119877

The case with 119909lowast119877 = 0 so that 0 le 119909lowast119871 le 1 and119909lowast119872= 1 minus 119909

lowast119871 leaves only the two regions 119877119871and 119877

119872 This

occurs when the parameters satisfy 0 le (119886119871minus 119889)119887

119877le 1

(119886119877minus 119889)119887

119871= 1 From the maps F

119872and F

119871in (27) it can

be seen that 119909119877119905converges to 119909lowast119877 = 0 so that the resulting

asymptotic dynamic behavior can be studied via the one-dimensional map given by 119909119871

119905+1= 119879(119909

119871

119905) similar to the one

obtained in (37) with obvious changesBy using arguments as in Section 2 it can be shown that

these one-dimensional systems have a unique fixed pointwhich is globally attracting (although it can be locally stableor unstable)

In order to study the local stability of the fixed point ofthe two-dimensional map F we can use the linearization ofthe different definitions in the fixed point which leads to thefollowing Jacobian matrices one for each region 119877

119871 119877119872 and

119877119877 that we denote respectively with 119869

119871 119869119872 and 119869

119877

119869119871(xlowast) = (

1 + 120575119871minus 120575119871

119887119877

119887119871

1198965

1198964

minus 120575119871

119887119877

119887119871

1198966

1198961

120575119871

1198966

1198961

120575119871

119887119877

119887119871

1198966

1198961

1 minus 120575119871

1198966

1198961

)

119869119872(xlowast) = (

1 minus 120575119872

1198964

1198963

0

0 1 minus 120575119872

1198966

1198965

)

119869119877(xlowast) = (

1 minus 120575119877

1198965

1198962

120575119877

119887119871

119887119877

1198965

1198962

120575119877

1198965

1198962

1 + 120575119877minus 120575119877

119887119871

119887119877

1198963

1198966

minus 120575119877

119887119871

119887119877

1198965

1198962

)

(38)

The two real eigenvalues of the Jacobian matrix in 119877119872are

in explicit form and both are smaller than +1 However theycan be larger than minus1 or not Also the eigenvalues of the otherjacobian matrices can be in modulus smaller or larger than1 However the global dynamics observed numerically givealways trajectories which are convergent to the fixed point xlowastat any set of allowed parameters values

Discrete Dynamics in Nature and Society 11

1

1

075

05

025

00 025 05 075

xR

xL

(a)

1

1

075

05

025

00 025 05 075

xR

xL

(b)

Figure 6 Examples of trajectories converging to the unique fixed point with parameters (a) 119886119871= 28 119887

119871= 24 119886

119877= 8 119887

119877= 24 119889 = 5 120575

119871= 04

120575119872= 03 120575

119877= 07 and ic 119909119871 = 08 119909119877 = 005 (b) 119886

119871= 29 119887

119871= 16 119886

119877= 15 119887

119877= 25 119889 = 7 120575

119871= 05 120575

119872= 08 120575

119877= 06 and ic 119909119871 = 005

119909119877= 09 Dots are linked by a continuous line for illustrative purpose only

This global attractivity in the two-dimensional map isdifficult to be proved rigorously however the results in theparticular cases commented above that is when the fixedpoint xlowast belongs to the boundary of 1198632 suggests that thesame result is true also when the fixed point is in the interiorof1198632

In Figure 6 we show some examples in Figure 6(a) wecan see a trajectory starting at 119877

119877 in Figure 6(b) a trajectory

whose initial point in 119877119871is mapped into 119877

119872 The shape of

the functions in the case of Figure 6(b) is similar to the oneshown in Figure 5

4 Conclusion

In this paper we have analyzed a dynamical version of theBraess paradox with nonimpulsive commuters who changeroad proportionally to the cost differenceWe were interestedto prove that with nonimpulsive commuters there exists aunique equilibrium both in the original network with twochoices and in the ternary onewhen a single link is addedWehave shown that both models are quite robust We were ableto provide an analytical proof of the global attractivity of theunique fixed point in the one-dimensional case when onlytwo roads are available andweusednumerical techniques andsimulations to give evidence in the two-dimensional case (iewhen a new road is added) as a rigorous proof can be doneonly for particular border cases

A limitation of our contribution is considering homo-geneous populations in spite of the fact that the evidencesupports heterogeneity Nevertheless heterogeneity of com-mutersrsquo behavior cannot be easily classified yet even when

conducting experiments common individual patterns arejust identified and classified On the other hand whenobserving the aggregate behavior inmany cases the literatureprovides evidence in support of the paradox outcome thatis the convergence to the Nash equilibrium The impulsiveagents in [30] react faster than the proportional agentspresented in this paper yet as a population they fail toachieve the Nash equilibria in certain cases while in thispaper the aggregate behavior does not We proposed thisaggregate behavior because it replies the convergence to theNash equilibria and at the same time it may be the resultof modeling some observed individual behaviors such asimitation and free riding

Taking into account the limitation of homogeneous pop-ulation analysis a natural extension of our contribution isconsidering heterogeneous populations with both impulsiveand nonimpulsive commuters We have reasons to believethat in such a case some properties of the different behaviorsare inherited and some are lost Therefore the results weprovide in this paper match those for impulsive populationsand are paramount to extend the analysis to more realisticcases In factmdashas the empirical results showmdashheterogeneouspopulations dynamics provide a better fit of what has beenobserved when considering human participants interaction

Furthermore it will be interesting to consider general-izations to networks with richer architectures Finally weobserve that since with the class of behavior we considerin this paper that dynamics depends on the shape of thecost functions it will be interesting to extend the analysis tononlinear cost functions

12 Discrete Dynamics in Nature and Society

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

V Avrutin is supported by the European Community withinthe scope of the project ldquoMultiple-discontinuity inducedbifurcations in theory and applicationsrdquo (Marie Curie Actionof the 7th Framework Programme Contract Agreement noPIEF-GA-2011-300281) U Merlone has worked under theauspices of COST Action IS1104 ldquoThe EU in the new complexgeography of economic systems models tools and policyevaluationrdquo

References

[1] R Arnott and K Small ldquoThe economics of traffic congestionrdquoAmerican Scientist vol 82 no 5 pp 446ndash455 1994

[2] T Bass ldquoRoad to ruinrdquo Discover pp 56ndash61 1992[3] B Calvert and G Keady ldquoBraessrsquos paradox and power-law non-

linearities in networksrdquoThe Journal of the AustralianMathemat-ical Society B Applied Mathematics vol 35 no 1 pp 1ndash22 1993

[4] J E Cohen andC Jeffries ldquoCongestion resulting from increasedcapacity in single-server queueing networksrdquo IEEEACMTrans-actions on Networking vol 5 no 2 pp 305ndash310 1997

[5] J E Cohen and F P Kelly ldquoA paradox of congestion in a queuingnetworkrdquo Journal of Applied Probability vol 27 no 3 pp 730ndash734 1990

[6] S Dafermos and A Nagurney ldquoOn some traffic equilibriumtheory paradoxesrdquo Transportation Research B Methodologicalvol 18 no 2 pp 101ndash110 1984

[7] C Fisk ldquoMore paradoxes in the equilibrium assignment prob-lemrdquo Transportation Research B Methodological vol 13 no 4pp 305ndash309 1979

[8] W H K Lam ldquoEffects of road pricing on system performancerdquoTraffic Engineering amp Control vol 29 no 12 pp 631ndash635 1988

[9] J D Murchland ldquoBraessrsquos paradox of traffic flowrdquo Transporta-tion Research vol 4 no 4 pp 391ndash394 1970

[10] R Steinberg and W I Zangwill ldquoThe prevalence o f BraessrsquoparadoxrdquoTransportation Science vol 17 no 3 pp 301ndash318 1983

[11] M Frank ldquoThe Braess paradoxrdquo Mathematical Programmingvol 20 no 1 pp 283ndash302 1981

[12] T Roughgarden Selfish Routing and the Price of Anarchy MITpress 2005

[13] B B Fu C X Zhao and S B Li ldquoThe analysis of Braessrsquo paradoxand robustness based on dynamic traffic assignment modelsrdquoDiscrete Dynamics in Nature and Society vol 2013 Article ID796842 8 pages 2013

[14] C Fisk and S Pallottino ldquoEmpirical evidence for equilibriumparadoxes with implications for optimal planning strategiesrdquoTransportation Research Part A General vol 15 no 3 pp 245ndash248 1981

[15] H Youn M T Gastner and H Jeong ldquoPrice of anarchy intransportation networks efficiency and optimality controlrdquoPhysical Review Letters vol 101 no 12 Article ID 128701 2008

[16] K Aoki Y Ohtsubo A Rapoport and T Saijo ldquoEffects of priorinvestment and personal responsibility in a simple network

gamerdquo Current Research in Social Psychology vol 13 no 2 pp10ndash21 2007

[17] J Morgan H Orzen and M Sefton ldquoNetwork architectureand traffic flows experiments on the Pigou-Knight-Downs andBraess paradoxesrdquoGames and Economic Behavior vol 66 no 1pp 348ndash372 2009

[18] A Rapoport V Mak and R Zwick ldquoNavigating congested net-works with variable demand experimental evidencerdquo Journal ofEconomic Psychology vol 27 no 5 pp 648ndash666 2006

[19] A Rapoport T Kugler S Dugar and E J Gisches ldquoBraessparadox in the laboratory an experimental study of route choicein traffic networks with asymmetric costsrdquo inDecisionModelingand Behavior in Uncertain and Complex Environment T KuglerJ C Smith T Connolly and Y J Son Eds pp 309ndash337Springer New York NY USA 2008

[20] A Rapoport T Kugler S Dugar and E J Gisches ldquoChoice ofroutes in congested traffic networks experimental tests of theBraess paradoxrdquo Games and Economic Behavior vol 65 no 2pp 538ndash571 2009

[21] E J Gisches and A Rapoport ldquoDegrading network capacitymay improve performance private versus public monitoring inthe Braess paradoxrdquoTheory and Decision vol 73 no 2 pp 267ndash293 2012

[22] A Dal Forno and U Merlone ldquoReplicating human interactionin Braess paradoxrdquo in Proceedings of the Simulation Conference(WSC rsquo13) pp 1754ndash1765 IEEE 2013

[23] M Ausloos H Dawid and U Merlone ldquoSpatial interactionsin agent-based modelingrdquo in Complexity and GeographicalEconomics Topics and Tools P Commendatore S S Kayamand I Kubin Eds Springer Berlin Germany 2014

[24] G-I Bischi and U Merlone ldquoGlobal dynamics in binary choicemodels with social influencerdquo The Journal of MathematicalSociology vol 33 no 4 pp 277ndash302 2009

[25] G I Bischi L Gardini and U Merlone ldquoImpulsivity in binarychoices and the emergence of periodicityrdquoDiscrete Dynamics inNature and Society vol 2009 Article ID 407913 22 pages 2009

[26] JH PattonM S Stanford andE S Barratt ldquoFactor structure ofthe Barratt Impulsiveness Scalerdquo Journal of Clinical Psychologyvol 51 no 6 pp 768ndash774 1995

[27] G I Bischi L Gardini and U Merlone ldquoPeriodic cycles andbifurcation curves for one-dimensional maps with two discon-tinuitiesrdquo Journal of Dynamical Systems GeometricTheories vol7 no 2 pp 101ndash123 2009

[28] G-I Bischi and U Merlone ldquoBinary choices in small and largegroups a unifiedmodelrdquo Physica A vol 389 no 4 pp 843ndash8532010

[29] A Dal Forno L Gardini and U Merlone ldquoTernary choicesin repeated games and border collision bifurcationsrdquo ChaosSolitons amp Fractals vol 45 no 3 pp 294ndash305 2012

[30] A Dal Forno and U Merlone ldquoBorder-collision bifurcationsin a model of Braess paradoxrdquo Mathematics and Computers inSimulation vol 87 pp 1ndash18 2013

[31] A Dal Forno V Giorgino and U Merlone ldquoAgent-based mod-eling with GTmethodology An examplerdquoWorking Paper 2014

[32] L Gardini U Merlone and F Tramontana ldquoInertia in binarychoices continuity breaking and big-bang bifurcation pointsrdquoJournal of Economic Behavior amp Organization vol 80 no 1 pp153ndash167 2011

[33] W H Sandholm Population Games and Evolutionary Dynam-ics The MIT Press Cambridge Mass USA 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Discrete Dynamics in Nature and Society

(c) 1120575119877le 11989621198961lt 2120575

119877with minus1 lt 1198911015840

119871(119909lowast) lt 0 and

0 lt 1198911015840

119877(119909lowast) lt 1 as in Figure 3(c)

In case (a) we have that the point of local minimum onthe left side say 119909119877

119897119898 is smaller than the fixed point119909119877

119897119898lt 119909lowast

Then themap is a contraction in the interval119882119904loc = [119909119877

119897119898 119909lowast]

and on the right side the map is decreasing so that anabsorbing interval which is mapped into119882119904loc in one iterationis given by 119869 = [119909119877

119897119898 119891minus1

119877(119909119877

119897119898)]when119891

119877(1) = 1minus120575

119871lt 119909119877

119897119898 or

119869 = [119909119877

119897119898 1]when119891

119877(1) = 1minus120575

119871ge 119909119877

119897119898 Any initial condition

belonging to [0 1] 119869 has a trajectory which is mapped into 119869in a finite number of iterationsThus 119909lowast is globally attracting

In case (c) we can reason similarly we have that the pointof local maximum on the right side say 119909119877

119903119872 is larger than

the fixed point 119909119877119903119872gt 119909lowast Then the map is a contraction in

the interval 119882119904loc = [119909lowast 119909119877

119903119872] and on the left side the map

is decreasing so that an absorbing interval which is mappedinto 119882119904loc in one iteration is given by 119869 = [119891minus1

119871(119909119877

119903119872) 119909119877

119903119872]

when 119891119871(0) = 120575

119877gt 119909119877

119903119872 or 119869 = [0 119909119877

119903119872] when 119891

119871(0) =

120575119877le 119909119877

119903119872 Any initial condition belonging to [0 1] 119869 has

a trajectory which is mapped into 119869 in a finite number ofiterations Thus 119909lowast is globally attracting

In case (b) we have that the point of local minimum onthe left side say 119909119877

119897119898 is smaller than the fixed point 119909119877

119897119898lt

119909lowast and also that the point of local maximum on the right

side say 119909119877119903119872

is larger than the fixed point 119909119877119903119872gt 119909lowast

Then the map is a contraction in the interval 119882119904loc =

[119909119877

119897119898 119909119877

119903119872] which includes the fixed point and an absorbing

interval which is mapped into119882119904loc in one iteration is givenby 119869 = [119891minus1

119871(119909119877

119903119872) 119891minus1

119877(119909119877

119897119898)] or 119869 = [0 119891minus1

119877(119909119877

119897119898)] or 119869 =

[119891minus1

119871(119909119877

119903119872) 1] or [0 1] depending under obvious conditions

on the values of 120575119871and 120575

119877 When 119869 sub [0 1] then any initial

condition belonging to [0 1] 119869 has a trajectory which ismapped into 119869 in a finite number of iterations Thus 119909lowast isglobally attracting

We notice that in Proposition 2 we prove that when thefixed point 119909lowast is internal to the interval [0 1] then it isglobally attracting Let us turn to comment the two extremawhich occur when 119909lowast = 0 or 119909lowast = 1 These solutions clearlyexist and are also globally attracting In fact the case 119909lowast = 0corresponds to 119896

1= 119871(0)minus119877(0) = 119886

119871+119887119871minus119886119877= 0 and only the

function 119891119877(119909119877) is applied in the interval [0 1] and from the

concavity of the function we have that this fixed point is glob-ally attracting even if the eigenvalue is equal to 1 The othercase 119909lowast = 1 corresponds to 119896

2= 119877(1)minus119871(1) = 119886

119877+119887119877minus119886119871= 0

and only the function 119891119871(119909119877) is applied in the interval [0 1]

and from the convexity of the function we have that this fixedpoint is globally attracting even if the eigenvalue is equal to 1

3 The Dynamics in the ExpandedNetwork with Three Roads

As it is well known the Braess paradox introduces a new linkand shows how this opportunity worsens both the individualand collective payoffs This link introduces a new alternativenow the commuters are facing three choices where the new

one consists of path 119878-119871-119877-119864 (see Figure 1(b)) which will bedenoted as 119872 in the following Models with three choiceswere introduced in [29] as it concerns discrete time and [33]as it concerns continuous time In this paper we analyze thedynamics when the travel time 119889 for the new resulting linkvaries in a range that makes this link a dominant choice forsome values and a dominated choice for others It is worth toobserve that in particular when the travel time on link 119871-119877 iszero the paradoxical result still holds

Now assume that commuters switch roadwhenever traveltime has become larger that is the other road is less costlyand therefore more attractive We indicate by 119909119871 ge 0 thefraction of population choosing action 119871 by 119909119872 ge 0 thefraction choosing action 119872 and by 119909119877 ge 0 the fractionchoosing action 119877 with the constraint 119909119871 + 119909119877 + 119909119872 = 1The travel times are given respectively as

119871 = 119886119871+ 119887119871(119909119871+ 119909119872)

119872 = 119887119871(119909119871+ 119909119872) + 119887119877(119909119872+ 119909119877) + 119889

119877 = 119886119877+ 119887119877(119909119872+ 119909119877)

(16)

By using the constraint let us define the fraction of populationchoosing119872 as 119909119872 = 1 minus 119909119871 minus 119909119877 ge 0 so to consider onlytwo variables x = (119909119871 119909119877) isin R2

+ and the phase space is the

triangle

1198632= x = (119909119871 119909119877) isin R2

+ 0 le 119909

119871+ 119909119877le 1 (17)

After substituting the expression of 119909119872 in the travel times weget

119871 (x) = 119886119871 + 119887119871 (1 minus 119909119877)

119872 (x) = 119887119871 (1 minus 119909119877) + 119887119877(1 minus 119909

119871) + 119889

119877 (x) = 119886119877 + 119887119877 (1 minus 119909119871)

(18)

The vertices of the phase space119875lowast119871(1 0)119875lowast

119872(0 0) and119875lowast

119877(0 1)

represent states in which the whole population choosesrespectively actions 119871119872 and 119877

Since commuters are interested in minimizing traveltime we denote by 119877

119860the region in which the choice 119860

(where119860 is 119871119872 or 119877) is preferable These regions are calleddominance regions of 119860 We have the following definition ofthe dominance regions

119877119871= x isin 1198632 119871 (x) le 119872 (x) 119871 (x) le 119877 (x)

119877119872= x isin 1198632 119872 (x) le 119871 (x) 119872 (x) le 119877 (x)

119877119877= x isin 1198632 119877 (x) le 119871 (x) 119877 (x) le 119872 (x)

(19)

Discrete Dynamics in Nature and Society 7

1

100

xR

RL

RM

RR

xL

(a)

1

100

xR

RL

RM

RR

xL

(b)

1

100

xR

RL

RM

RR

xL

(c)

Figure 4 Dominance regions in the three choices network with different values of the cost parameter 119889 of the map F at 119886119871= 27 119887

119871= 30

119886119877= 18 119887

119877= 30 and (a) 119889 = 6 (b) 119889 = 3 (c) 119889 = 0

By using the expressions in (18) we get the following defini-tions for the regions

119877119871= x isin 1198632 119909119871 le 119909lowast119871

119886119871+ 119887119871(1 minus 119909

119877) le 119886119877+ 119887119877(1 minus 119909

119871)

119877119872= x isin 1198632 119909119871 ge 119909lowast119871 119909119877 ge 119909lowast119877

119877119877= x isin 1198632 119886

119877+ 119887119877(1 minus 119909

119871) le 119886119871+ 119887119871(1 minus 119909

119877)

119909119877le 119909lowast119877

(20)

where we have introduced

119909lowast119871= 1 minus

119886119871minus 119889

119887119877

=119889 minus 119886119871+ 119887119877

119887119877

119909lowast119877= 1 minus

119886119877minus 119889

119887119871

=119889 minus 119886119877+ 119887119871

119887119871

(21)

The boundaries of the three regions are given by segments ofstraight lines of equations

1198811 119909119871= 119909lowast119871

1198812 119909119877= 119909lowast119877

1198813 119886119871+ 119887119871(1 minus 119909

119877) = 119886119877+ 119887119877(1 minus 119909

119871)

(22)

and it is immediate to see that the point xlowast = (119909lowast119871 119909lowast119877)satisfies also the third equation that is it belongs to all thethree straight lines xlowast isin 119881

119894for 119894 = 1 2 3

Using the geometric representation in the triangle 1198632the dominance regions are illustrated in Figure 4 (via anexample) using different colors Three cases are shown fordifferent values of the parameter 119889 in order to illustrate howthe region119877

119872depends on the parameters of the third branch

The point xlowast = (119909lowast119871 119909lowast119877) is the unique point on theboundaries 119881

119894of all the three regions Thus for our model we

have to require 119909lowast119871 ge 0 119909lowast119877 ge 0 and 119909lowast119871 + 119909lowast119877 le 1 leadingto 119909lowast119872 = 1 minus 119909lowast119871 minus 119909lowast119877 In order to have three regions in1198632

it must be 119909lowast119871 gt 0 119909lowast119877 gt 0 and 119909lowast119871 + 119909lowast119877 lt 1 The equalityin one of these three conditions implies the disappearance ofone region For example when 119909lowast119871 = 0 then 0 le 119909lowast119877 le 1and 119909lowast119872 = 1 minus 119909lowast119877 leaving only the two regions 119877

119877and 119877

119872

and so onThe conditions 119909lowast119871 ge 0 119909lowast119877 ge 0 and 119909lowast119871 + 119909lowast119877 le 1 lead

to the constraints that will be assumed satisfied henceforth

0 le119886119871minus 119889

119887119877

le 1 0 le119886119877minus 119889

119887119871

le 1

119886119871minus 119889

119887119877

+119886119877minus 119889

119887119871

ge 1

(23)

We notice that from the third condition in (23) we have thevalues of 119889 for which the third region is really present in 1198632that is

0 le 119889 le119886119871119887119871+ 119886119877119887119877minus 119887119871119887119877

119887119877+ 119887119871

(24)

Commuters are homogeneous and minimize their nextperiod travel cost At time 119905+1 the vector x

119905becomes common

knowledge and each commuter can observe the travel costs119871(x119905)119872(x

119905) and 119877(x

119905) We assume that if at time 119905 a fraction

119909119871

119905chooses path 119871 a fraction 119909119877

119905chooses action 119877 and travel

costs are such that 119877(x119905) lt 119871(x

119905) and 119877(x

119905) lt 119872(x

119905) then

both a fraction of the 119909119871119905commuters who chose action 119871 and

a fraction of the (1 minus 119909119871119905minus 119909119877

119905) commuters who chose action

119872 will switch to path 119877 in the next time period 119905 + 1 Thefraction of switching commuters is given by the switchingpropensities 120575

119871 120575119872 120575119877modulated by the relative differences

in payoffs This happens whenever a path gives a smallertravel cost In other words at any time 119905 all the commutersdecide their future action at time 119905 + 1 comparing the costs119871(x119905)119872(x

119905) and 119877(x

119905) As in the one-dimensional case here

also the differences of travel costs are normalized dividing

8 Discrete Dynamics in Nature and Society

each difference by the constants giving the related maximumvalue given in the different cases as follows

1198961= max

x [119871 (x) minus 119877 (x)]

= [119871 (1 0) minus 119877 (1 0)] = 119886119871 + 119887119871 minus 119886119877

1198962= max

x [119877 (x) minus 119871 (x)]

= [119877 (0 1) minus 119871 (0 1)] = 119886119877 + 119887119877 minus 119886119871

1198963= max

x [119871 (x) minus 119872 (x)]

= [119871 (1 119909119877) minus119872(1 119909

119877)] = 119886

119871minus 119889 forall119909

119877isin [0 1]

1198964= max

x [119872 (x) minus 119871 (x)]

= [119872(0 119909119877) minus 119871 (0 119909

119877)] = 119889 + 119887

119877minus 119886119871 forall119909

119877isin [0 1]

1198965= max

x [119877 (x) minus 119872 (x)]

= [119877 (119909119871 1) minus 119872(119909

119871 1)] = 119886

119877minus 119889 forall119909

119871isin [0 1]

1198966= max

x [119872 (x) minus 119877 (x)]

= [119872(119909119871 0) minus 119877 (119909

119871 0)] = 119889 + 119887

119871minus 119886119877 forall119909

119871isin [0 1]

(25)

The resulting dynamics are described by a two-dimensionalmap x

119905+1= F(x

119905) defined as

F x119905+1=

F119871(x119905) if x

119905isin 119877119871

F119872(x119905) if x

119905isin 119877119872

F119877(x119905) if x

119905isin 119877119877

(26)

where

F119871

119909119871

119905+1= 119909119871

119905+ 120575119871(119877 (x119905) minus 119871 (x

119905)

119886119877+ 119887119877minus 119886119871

119909119877

119905

+119872(x119905) minus 119871 (x

119905)

119889 + 119887119877minus 119886119871

119909119872

119905)

119909119877

119905+1= (1 minus 120575

119871

119877 (x119905) minus 119871 (x

119905)

119886119877+ 119887119877minus 119886119871

)119909119877

119905

F119872

119909119871

119905+1= (1 minus 120575

119872

119871 (x119905) minus 119872(x

119905)

119886119871minus 119889

)119909119871

119905

119909119877

119905+1= (1 minus 120575

119872

119877 (x119905) minus 119872(x

119905)

119886119877minus 119889

)119909119877

119905

F119877

119909119871

119905+1= (1 minus 120575

119877

119871 (x119905) minus 119877 (x

119905)

119886119871+ 119887119871minus 119886119877

)119909119871

119905

119909119877

119905+1= 119909119877

119905+ 120575119877(119871 (x119905) minus 119877 (x

119905)

119886119871+ 119887119871minus 119886119877

119909119871

119905

+119872(x119905) minus 119877 (x

119905)

119889 + 119887119871minus 119886119877

119909119872

119905)

(27)

By construction we have that F maps 1198632 into itself In factto show that F 1198632 rarr 119863

2 it is enough to show that

given (119909119871119905 119909119877

119905) isin 119863

2 then (119909119871119905+1 119909119877

119905+1) = F(119909119871

119905 119909119877

119905) isin 119863

2From (119909119871

119905 119909119877

119905) isin 119863

2 we have that 0 le 119909119871119905+ 119909119877

119905le 1 and

119909119871

119905+ 119909119877

119905+ 119909119872

119905= 1 Then if (119909119871

119905+1 119909119877

119905+1) = F119871(119909119871

119905 119909119877

119905) we have

119909119871

119905+1+ 119909119877

119905+1= 119909119871

119905+ 119909119877

119905+ 120575119871

119872(x119905) minus 119871 (x

119905)

119889 + 119887119877minus 119886119871

119909119872

119905

le 119909119871

119905+ 119909119877

119905+ 119909119872

119905= 1

(28)

119909119871

119905+1ge 0 and 119909119877

119905+1ge 0 thus (119909119871

119905+1 119909119877

119905+1) isin 1198632 If (119909119871

119905+1 119909119877

119905+1) =

F119872(119909119871

119905 119909119877

119905) we have 0 le 119909119871

119905+1le 119909119871

119905and 0 le 119909119877

119905+1le 119909119877

119905 thus

(119909119871

119905+1 119909119877

119905+1) isin 1198632 If (119909119871

119905+1 119909119877

119905+1) = F119877(119909119871

119905 119909119877

119905) we have

119909119871

119905+1+ 119909119877

119905+1= 119909119871

119905+ 119909119877

119905+ 120575119877

119872(x119905) minus 119877 (x

119905)

119889 + 119887119871minus 119886119877

119909119872

119905

le 119909119871

119905+ 119909119877

119905+ 119909119872

119905= 1

(29)

119909119871

119905+1ge 0 and 119909119877

119905+1ge 0 thus (119909119871

119905+1 119909119877

119905+1) isin 1198632

Substituting from (18) we can rewrite the functions inexplicit form as

F119871

119909119871

119905+1= 120575119871+ (1 minus 120575

119871minus 120575119871

119887119877

1198964

)119909119871

119905

minus120575119871

119887119871

1198962

119909119877

119905+ 120575119871

119887119871

1198962

(119909119877

119905)2

+120575119871119887119877(1

1198964

minus1

1198962

)119909119871

119905119909119877

119905+ 120575119871

119887119877

1198964

(119909119871

119905)2

119909119877

119905+1= (1 minus 120575

119871minus 120575119871

119887119871

1198962

)119909119877

119905

+120575119871

119887119877

1198962

119909119871

119905119909119877

119905minus 120575119871

119887119871

1198962

(119909119877

119905)2

F119872

119909119871

119905+1= (1 + 120575

119872

1198964

1198963

)119909119871

119905minus 120575119872

119887119877

1198963

(119909119871

119905)2

119909119877

119905+1= (1 + 120575

119872

1198966

1198965

)119909119877

119905minus 120575119872

119887119871

1198965

(119909119877

119905)2

F119877

119909119871

119905+1= (1 minus 120575

119877minus 120575119877

119887119877

1198961

)119909119871

119905

+120575119877

119887119871

1198961

119909119871

119905119909119877

119905minus 120575119877

119887119877

1198961

(119909119871

119905)2

119909119877

119905+1= 120575119877+ (1 minus 120575

119877minus 120575119877

119887119871

1198966

)119909119877

119905

minus120575119877

119887119877

1198961

119909119871

119905+ 120575119877

119887119871

1198961

(119909119877

119905)2

+120575119877119887119871(1

1198966

minus1

1198961

)119909119871

119905119909119877

119905+ 120575119877

119887119877

1198966

(119909119871

119905)2

(30)

Differently from the one-dimensional map 119865 considered inthe previous section the two-dimensional map F is notcontinuous in 1198632 More precisely the functions F

119871 F119872

and F119877are continuous and thus F is continuous in the

interior of each region inwhich the different definitions applyHowever in general except for the point xlowast the map F is notcontinuous along the borders 119881

119894 119894 = 1 2 3 of the regions

119877119871 119877119872 and 119877

119877 In fact it is easy to see that considering

Discrete Dynamics in Nature and Society 9

xL

t+1

xL

t

xR

t

00

1

1

(a)

xL

t+1

xL

t

xR

t

00

1

1

(b)

Figure 5 Graphics of the three functions defining (a) 119909119871119905+1

and (b) 119909119877119905+1

with parameters 119886119871= 29 119887

119871= 16 119886

119877= 19 119887

119877= 24 119889 = 8 120575

119871= 05

120575119872= 08 and 120575

119877= 06

a point (119909119871 119909119877) belonging to one of the lines defined in (22)at the boundary of two regions only we have that the twofunctions defined on the two different regions lead to twodifferent points of1198632 An example of how the three functionsdefining 119909119871

119905+1behave is shown in Figure 5(a) and the three

functions defining 119909119877119905+1

are illustrated in Figure 5(b) showingthat F is continuous only in xlowast

The point xlowast in which the three regions are in contactand F is continuous is clearly particular In fact solving theequation 119871(x) = 119872(x) we have the unique solution 119909119871 = 119909lowast119871while solving the equation 119877(x) = 119872(x) we have the uniquesolution119909119877 = 119909lowast119877 It follows that at the point xlowast = (119909lowast119871 119909lowast119877)we have the same cost in each choice as 119871(xlowast) = 119872(xlowast) =119877(xlowast) so that at the point xlowast the commuters are indifferent toany available choice andnone of themwill switch choiceThusF(xlowast) = xlowast and xlowast is a stationary state Under the conditionsin (23) the point xlowast is feasible belonging to 1198632 We can nowprove that this is indeed the unique fixed point of map F

Proposition 3 The point xlowast = (119909lowast119871 119909lowast119877) given in (21) is theunique fixed point of the map F belonging to 1198632 when theconditions in (23) hold

Proof The proof of this proposition follows directly from theobservation that

F (x) = x if and only if 119871 (x) = 119872 (x) = 119877 (x) (31)One way is already proved as we have detected the fixed pointstarting from the solution of 119871(x) = 119872(x) and 119877(x) = 119872(x)showing that it is the unique point xlowast Vice versa let us lookfor the solutions of F(x) = x We have the following systems(where 119909119872 = 1 minus 119909119871 minus 119909119877)

F119871

119909119871= 119909119871+ 120575119871(119877 (x) minus 119871 (x)

1198962

119909119877

+119872(x) minus 119871 (x)

1198964

119909119872)

119909119877= (1 minus 120575

119871

119877 (x) minus 119871 (x)1198962

)119909119877

if x isin 119877119871

F119872

119909119871= (1 minus 120575

119872

119871 (x) minus 119872 (x)119886119871minus 119889

)119909119871

119909119877= (1 minus 120575

119872

119877 (x) minus 119872 (x)119886119877minus 119889

)119909119877

if x isin 119877119872

F119877

119909119871= (1 minus 120575

119877

119871 (x) minus 119877 (x)1198961

)119909119871

119909119877= 119909119877+ 120575119877(119871 (x) minus 119877 (x)

1198961

119909119871

+119872(x) minus 119877 (x)

1198966

119909119872)

if x isin 119877119877

(32)

or equivalently (we recall that 120575119871 120575119872 and 120575

119877are different

from zero)

F119871

119877 (x) minus 119871 (x)1198962

119909119877+119872(x) minus 119871 (x)

1198964

119909119872= 0

119877 (x) minus 119871 (x)1198962

119909119877= 0

if x isin 119877119871

F119872

119871 (x) minus 119872 (x)119886119871minus 119889

119909119871= 0

119877 (x) minus 119872 (x)119886119877minus 119889

119909119877= 0

if x isin 119877119872

F119877

119871 (x) minus 119877 (x)1198961

119909119871= 0

119871 (x) minus 119877 (x)1198961

119909119871+119872(x) minus 119877 (x)

1198966

119909119872= 0

if x isin 119877119877

(33)

Considering the equation F119871(x) = x we see that it is certainly

satisfied for 119909119877 = 0 and 119909119872 = 0 which leads to thecorner point with 119909119871 = 1 However if the fixed point is an

10 Discrete Dynamics in Nature and Society

interior point of 1198632 then the point (1 0) does not belong tothe proper region 119877

119871 that is it is a virtual fixed point Clearly

this equation is satisfied when 119871(x) = 119872(x) and 119871(x) = 119877(x)which occurs only at the point xlowast It is possible to have xlowast =(1 0) and this occurs for (119886

119871minus 119889)119887

119877= 0 and (119886

119877minus 119889)119887

119871= 1

Thus we can state that the only solution of F119871(x) = x in 119877

119871

is given by the fixed point xlowast Similarly we can reason in theother regions

It is worth to mention that particular cases are obtainedwhen the fixed point xlowast belongs to a boundary of the region1198632 For example when 119909lowast119877+119909lowast119871 = 1 and 119909lowast119872 = 0 then only

the two regions119877119871and 119877

119877are involvedThis occurs when the

parameters satisfy

119889 =119886119871119887119871+ 119886119877119887119877minus 119887119871119887119877

119887119877+ 119887119871

0 le119886119871minus 119889

119887119877

le 1

0 le119886119877minus 119889

119887119871

le 1

(34)

From (119886119877minus 119889)119887

119871= (119886119877+ 119887119877minus 119886119871)(119887119877+ 119887119871) we have 119909lowast119877 =

1minus(119886119877minus119889)119887

119871= (119886119871+119887119871minus119886119877)(119887119877+119887119871) = 1198961(119887119877+119887119871) = 119909lowast as

in the one-dimensional map studied in Section 2 and 119909lowast119871 =1 minus 119909lowast119877= 1 minus (119886

119871minus 119889)119887

119877= 1198962(119887119877+ 119887119871) = 1 minus 119896

1(119887119877+ 119887119871) =

1minus119909lowast From themapsF

119871andF119877in (27) and for the parameters

occurring in this case it can be seen that 119909119871119905+1= 1minus119909

119877

119905+1holds

in both so that the dynamics can be studied by using onlythe variable 119909119877

119905and the resulting one-dimensional map is as

follows

119865 (119909119877)

=

119891119871(119909119877)

= 119909119877+ 120575119877

119871 (119909119877)minus119877 (119909

119877)

1198961

(1 minus 119909119877)

if 0 le 119909119877 le 119909lowast

119891119877(119909119877)

= 119909119877minus 120575119871

119877 (119909119877) minus 119871 (119909

119877)

1198962

119909119877 if 119909lowast le 119909119877 le 1

(35)

which corresponds to the system already studied in Section 2As we have seen this fixed point is globally attracting even iflocally we can have the eigenvalue on some side larger than 1in modulus

The case with 119909lowast119871 = 0 so that 0 le 119909lowast119877 le 1 and 119909lowast119872 =1 minus 119909lowast119877 leaves only the two regions 119877

119877and 119877

119872 This occurs

when the parameters satisfy

119886119871minus 119889

119887119877

= 1 0 le119886119877minus 119889

119887119871

le 1 (36)

From the maps F119872

and F119877in (27) it can be seen that 119909119871

119905

converges to119909lowast119871 = 0 so that the resulting asymptotic dynamic

behavior can be studied via the one-dimensional map givenby 119909119877119905+1= 119879(119909

119877

119905) with 119909119877

119905isin [0 1] (and 119909119872

119905= 1 minus 119909

119877

119905) given by

119879 (119909119877)

=

119879119871(119909119877)

= 119909119877+ 120575119877

119872(119909119877) minus 119877 (119909

119877)

1198966

(1 minus 119909119877)

if 0 le 119909119877 le 119909lowast

119879119877(119909119877)

= 119909119877minus 120575119872

119877 (119909119877) minus119872(119909

119877)

119886119877minus 119889

119909119877 if 119909lowast le 119909119877 le 1

(37)

where119872(119909119877) minus 119877(119909119877) = 119886119871+ 119887119871(1 minus 119909

119877) minus 119886119877minus 119887119877

The case with 119909lowast119877 = 0 so that 0 le 119909lowast119871 le 1 and119909lowast119872= 1 minus 119909

lowast119871 leaves only the two regions 119877119871and 119877

119872 This

occurs when the parameters satisfy 0 le (119886119871minus 119889)119887

119877le 1

(119886119877minus 119889)119887

119871= 1 From the maps F

119872and F

119871in (27) it can

be seen that 119909119877119905converges to 119909lowast119877 = 0 so that the resulting

asymptotic dynamic behavior can be studied via the one-dimensional map given by 119909119871

119905+1= 119879(119909

119871

119905) similar to the one

obtained in (37) with obvious changesBy using arguments as in Section 2 it can be shown that

these one-dimensional systems have a unique fixed pointwhich is globally attracting (although it can be locally stableor unstable)

In order to study the local stability of the fixed point ofthe two-dimensional map F we can use the linearization ofthe different definitions in the fixed point which leads to thefollowing Jacobian matrices one for each region 119877

119871 119877119872 and

119877119877 that we denote respectively with 119869

119871 119869119872 and 119869

119877

119869119871(xlowast) = (

1 + 120575119871minus 120575119871

119887119877

119887119871

1198965

1198964

minus 120575119871

119887119877

119887119871

1198966

1198961

120575119871

1198966

1198961

120575119871

119887119877

119887119871

1198966

1198961

1 minus 120575119871

1198966

1198961

)

119869119872(xlowast) = (

1 minus 120575119872

1198964

1198963

0

0 1 minus 120575119872

1198966

1198965

)

119869119877(xlowast) = (

1 minus 120575119877

1198965

1198962

120575119877

119887119871

119887119877

1198965

1198962

120575119877

1198965

1198962

1 + 120575119877minus 120575119877

119887119871

119887119877

1198963

1198966

minus 120575119877

119887119871

119887119877

1198965

1198962

)

(38)

The two real eigenvalues of the Jacobian matrix in 119877119872are

in explicit form and both are smaller than +1 However theycan be larger than minus1 or not Also the eigenvalues of the otherjacobian matrices can be in modulus smaller or larger than1 However the global dynamics observed numerically givealways trajectories which are convergent to the fixed point xlowastat any set of allowed parameters values

Discrete Dynamics in Nature and Society 11

1

1

075

05

025

00 025 05 075

xR

xL

(a)

1

1

075

05

025

00 025 05 075

xR

xL

(b)

Figure 6 Examples of trajectories converging to the unique fixed point with parameters (a) 119886119871= 28 119887

119871= 24 119886

119877= 8 119887

119877= 24 119889 = 5 120575

119871= 04

120575119872= 03 120575

119877= 07 and ic 119909119871 = 08 119909119877 = 005 (b) 119886

119871= 29 119887

119871= 16 119886

119877= 15 119887

119877= 25 119889 = 7 120575

119871= 05 120575

119872= 08 120575

119877= 06 and ic 119909119871 = 005

119909119877= 09 Dots are linked by a continuous line for illustrative purpose only

This global attractivity in the two-dimensional map isdifficult to be proved rigorously however the results in theparticular cases commented above that is when the fixedpoint xlowast belongs to the boundary of 1198632 suggests that thesame result is true also when the fixed point is in the interiorof1198632

In Figure 6 we show some examples in Figure 6(a) wecan see a trajectory starting at 119877

119877 in Figure 6(b) a trajectory

whose initial point in 119877119871is mapped into 119877

119872 The shape of

the functions in the case of Figure 6(b) is similar to the oneshown in Figure 5

4 Conclusion

In this paper we have analyzed a dynamical version of theBraess paradox with nonimpulsive commuters who changeroad proportionally to the cost differenceWe were interestedto prove that with nonimpulsive commuters there exists aunique equilibrium both in the original network with twochoices and in the ternary onewhen a single link is addedWehave shown that both models are quite robust We were ableto provide an analytical proof of the global attractivity of theunique fixed point in the one-dimensional case when onlytwo roads are available andweusednumerical techniques andsimulations to give evidence in the two-dimensional case (iewhen a new road is added) as a rigorous proof can be doneonly for particular border cases

A limitation of our contribution is considering homo-geneous populations in spite of the fact that the evidencesupports heterogeneity Nevertheless heterogeneity of com-mutersrsquo behavior cannot be easily classified yet even when

conducting experiments common individual patterns arejust identified and classified On the other hand whenobserving the aggregate behavior inmany cases the literatureprovides evidence in support of the paradox outcome thatis the convergence to the Nash equilibrium The impulsiveagents in [30] react faster than the proportional agentspresented in this paper yet as a population they fail toachieve the Nash equilibria in certain cases while in thispaper the aggregate behavior does not We proposed thisaggregate behavior because it replies the convergence to theNash equilibria and at the same time it may be the resultof modeling some observed individual behaviors such asimitation and free riding

Taking into account the limitation of homogeneous pop-ulation analysis a natural extension of our contribution isconsidering heterogeneous populations with both impulsiveand nonimpulsive commuters We have reasons to believethat in such a case some properties of the different behaviorsare inherited and some are lost Therefore the results weprovide in this paper match those for impulsive populationsand are paramount to extend the analysis to more realisticcases In factmdashas the empirical results showmdashheterogeneouspopulations dynamics provide a better fit of what has beenobserved when considering human participants interaction

Furthermore it will be interesting to consider general-izations to networks with richer architectures Finally weobserve that since with the class of behavior we considerin this paper that dynamics depends on the shape of thecost functions it will be interesting to extend the analysis tononlinear cost functions

12 Discrete Dynamics in Nature and Society

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

V Avrutin is supported by the European Community withinthe scope of the project ldquoMultiple-discontinuity inducedbifurcations in theory and applicationsrdquo (Marie Curie Actionof the 7th Framework Programme Contract Agreement noPIEF-GA-2011-300281) U Merlone has worked under theauspices of COST Action IS1104 ldquoThe EU in the new complexgeography of economic systems models tools and policyevaluationrdquo

References

[1] R Arnott and K Small ldquoThe economics of traffic congestionrdquoAmerican Scientist vol 82 no 5 pp 446ndash455 1994

[2] T Bass ldquoRoad to ruinrdquo Discover pp 56ndash61 1992[3] B Calvert and G Keady ldquoBraessrsquos paradox and power-law non-

linearities in networksrdquoThe Journal of the AustralianMathemat-ical Society B Applied Mathematics vol 35 no 1 pp 1ndash22 1993

[4] J E Cohen andC Jeffries ldquoCongestion resulting from increasedcapacity in single-server queueing networksrdquo IEEEACMTrans-actions on Networking vol 5 no 2 pp 305ndash310 1997

[5] J E Cohen and F P Kelly ldquoA paradox of congestion in a queuingnetworkrdquo Journal of Applied Probability vol 27 no 3 pp 730ndash734 1990

[6] S Dafermos and A Nagurney ldquoOn some traffic equilibriumtheory paradoxesrdquo Transportation Research B Methodologicalvol 18 no 2 pp 101ndash110 1984

[7] C Fisk ldquoMore paradoxes in the equilibrium assignment prob-lemrdquo Transportation Research B Methodological vol 13 no 4pp 305ndash309 1979

[8] W H K Lam ldquoEffects of road pricing on system performancerdquoTraffic Engineering amp Control vol 29 no 12 pp 631ndash635 1988

[9] J D Murchland ldquoBraessrsquos paradox of traffic flowrdquo Transporta-tion Research vol 4 no 4 pp 391ndash394 1970

[10] R Steinberg and W I Zangwill ldquoThe prevalence o f BraessrsquoparadoxrdquoTransportation Science vol 17 no 3 pp 301ndash318 1983

[11] M Frank ldquoThe Braess paradoxrdquo Mathematical Programmingvol 20 no 1 pp 283ndash302 1981

[12] T Roughgarden Selfish Routing and the Price of Anarchy MITpress 2005

[13] B B Fu C X Zhao and S B Li ldquoThe analysis of Braessrsquo paradoxand robustness based on dynamic traffic assignment modelsrdquoDiscrete Dynamics in Nature and Society vol 2013 Article ID796842 8 pages 2013

[14] C Fisk and S Pallottino ldquoEmpirical evidence for equilibriumparadoxes with implications for optimal planning strategiesrdquoTransportation Research Part A General vol 15 no 3 pp 245ndash248 1981

[15] H Youn M T Gastner and H Jeong ldquoPrice of anarchy intransportation networks efficiency and optimality controlrdquoPhysical Review Letters vol 101 no 12 Article ID 128701 2008

[16] K Aoki Y Ohtsubo A Rapoport and T Saijo ldquoEffects of priorinvestment and personal responsibility in a simple network

gamerdquo Current Research in Social Psychology vol 13 no 2 pp10ndash21 2007

[17] J Morgan H Orzen and M Sefton ldquoNetwork architectureand traffic flows experiments on the Pigou-Knight-Downs andBraess paradoxesrdquoGames and Economic Behavior vol 66 no 1pp 348ndash372 2009

[18] A Rapoport V Mak and R Zwick ldquoNavigating congested net-works with variable demand experimental evidencerdquo Journal ofEconomic Psychology vol 27 no 5 pp 648ndash666 2006

[19] A Rapoport T Kugler S Dugar and E J Gisches ldquoBraessparadox in the laboratory an experimental study of route choicein traffic networks with asymmetric costsrdquo inDecisionModelingand Behavior in Uncertain and Complex Environment T KuglerJ C Smith T Connolly and Y J Son Eds pp 309ndash337Springer New York NY USA 2008

[20] A Rapoport T Kugler S Dugar and E J Gisches ldquoChoice ofroutes in congested traffic networks experimental tests of theBraess paradoxrdquo Games and Economic Behavior vol 65 no 2pp 538ndash571 2009

[21] E J Gisches and A Rapoport ldquoDegrading network capacitymay improve performance private versus public monitoring inthe Braess paradoxrdquoTheory and Decision vol 73 no 2 pp 267ndash293 2012

[22] A Dal Forno and U Merlone ldquoReplicating human interactionin Braess paradoxrdquo in Proceedings of the Simulation Conference(WSC rsquo13) pp 1754ndash1765 IEEE 2013

[23] M Ausloos H Dawid and U Merlone ldquoSpatial interactionsin agent-based modelingrdquo in Complexity and GeographicalEconomics Topics and Tools P Commendatore S S Kayamand I Kubin Eds Springer Berlin Germany 2014

[24] G-I Bischi and U Merlone ldquoGlobal dynamics in binary choicemodels with social influencerdquo The Journal of MathematicalSociology vol 33 no 4 pp 277ndash302 2009

[25] G I Bischi L Gardini and U Merlone ldquoImpulsivity in binarychoices and the emergence of periodicityrdquoDiscrete Dynamics inNature and Society vol 2009 Article ID 407913 22 pages 2009

[26] JH PattonM S Stanford andE S Barratt ldquoFactor structure ofthe Barratt Impulsiveness Scalerdquo Journal of Clinical Psychologyvol 51 no 6 pp 768ndash774 1995

[27] G I Bischi L Gardini and U Merlone ldquoPeriodic cycles andbifurcation curves for one-dimensional maps with two discon-tinuitiesrdquo Journal of Dynamical Systems GeometricTheories vol7 no 2 pp 101ndash123 2009

[28] G-I Bischi and U Merlone ldquoBinary choices in small and largegroups a unifiedmodelrdquo Physica A vol 389 no 4 pp 843ndash8532010

[29] A Dal Forno L Gardini and U Merlone ldquoTernary choicesin repeated games and border collision bifurcationsrdquo ChaosSolitons amp Fractals vol 45 no 3 pp 294ndash305 2012

[30] A Dal Forno and U Merlone ldquoBorder-collision bifurcationsin a model of Braess paradoxrdquo Mathematics and Computers inSimulation vol 87 pp 1ndash18 2013

[31] A Dal Forno V Giorgino and U Merlone ldquoAgent-based mod-eling with GTmethodology An examplerdquoWorking Paper 2014

[32] L Gardini U Merlone and F Tramontana ldquoInertia in binarychoices continuity breaking and big-bang bifurcation pointsrdquoJournal of Economic Behavior amp Organization vol 80 no 1 pp153ndash167 2011

[33] W H Sandholm Population Games and Evolutionary Dynam-ics The MIT Press Cambridge Mass USA 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 7

1

100

xR

RL

RM

RR

xL

(a)

1

100

xR

RL

RM

RR

xL

(b)

1

100

xR

RL

RM

RR

xL

(c)

Figure 4 Dominance regions in the three choices network with different values of the cost parameter 119889 of the map F at 119886119871= 27 119887

119871= 30

119886119877= 18 119887

119877= 30 and (a) 119889 = 6 (b) 119889 = 3 (c) 119889 = 0

By using the expressions in (18) we get the following defini-tions for the regions

119877119871= x isin 1198632 119909119871 le 119909lowast119871

119886119871+ 119887119871(1 minus 119909

119877) le 119886119877+ 119887119877(1 minus 119909

119871)

119877119872= x isin 1198632 119909119871 ge 119909lowast119871 119909119877 ge 119909lowast119877

119877119877= x isin 1198632 119886

119877+ 119887119877(1 minus 119909

119871) le 119886119871+ 119887119871(1 minus 119909

119877)

119909119877le 119909lowast119877

(20)

where we have introduced

119909lowast119871= 1 minus

119886119871minus 119889

119887119877

=119889 minus 119886119871+ 119887119877

119887119877

119909lowast119877= 1 minus

119886119877minus 119889

119887119871

=119889 minus 119886119877+ 119887119871

119887119871

(21)

The boundaries of the three regions are given by segments ofstraight lines of equations

1198811 119909119871= 119909lowast119871

1198812 119909119877= 119909lowast119877

1198813 119886119871+ 119887119871(1 minus 119909

119877) = 119886119877+ 119887119877(1 minus 119909

119871)

(22)

and it is immediate to see that the point xlowast = (119909lowast119871 119909lowast119877)satisfies also the third equation that is it belongs to all thethree straight lines xlowast isin 119881

119894for 119894 = 1 2 3

Using the geometric representation in the triangle 1198632the dominance regions are illustrated in Figure 4 (via anexample) using different colors Three cases are shown fordifferent values of the parameter 119889 in order to illustrate howthe region119877

119872depends on the parameters of the third branch

The point xlowast = (119909lowast119871 119909lowast119877) is the unique point on theboundaries 119881

119894of all the three regions Thus for our model we

have to require 119909lowast119871 ge 0 119909lowast119877 ge 0 and 119909lowast119871 + 119909lowast119877 le 1 leadingto 119909lowast119872 = 1 minus 119909lowast119871 minus 119909lowast119877 In order to have three regions in1198632

it must be 119909lowast119871 gt 0 119909lowast119877 gt 0 and 119909lowast119871 + 119909lowast119877 lt 1 The equalityin one of these three conditions implies the disappearance ofone region For example when 119909lowast119871 = 0 then 0 le 119909lowast119877 le 1and 119909lowast119872 = 1 minus 119909lowast119877 leaving only the two regions 119877

119877and 119877

119872

and so onThe conditions 119909lowast119871 ge 0 119909lowast119877 ge 0 and 119909lowast119871 + 119909lowast119877 le 1 lead

to the constraints that will be assumed satisfied henceforth

0 le119886119871minus 119889

119887119877

le 1 0 le119886119877minus 119889

119887119871

le 1

119886119871minus 119889

119887119877

+119886119877minus 119889

119887119871

ge 1

(23)

We notice that from the third condition in (23) we have thevalues of 119889 for which the third region is really present in 1198632that is

0 le 119889 le119886119871119887119871+ 119886119877119887119877minus 119887119871119887119877

119887119877+ 119887119871

(24)

Commuters are homogeneous and minimize their nextperiod travel cost At time 119905+1 the vector x

119905becomes common

knowledge and each commuter can observe the travel costs119871(x119905)119872(x

119905) and 119877(x

119905) We assume that if at time 119905 a fraction

119909119871

119905chooses path 119871 a fraction 119909119877

119905chooses action 119877 and travel

costs are such that 119877(x119905) lt 119871(x

119905) and 119877(x

119905) lt 119872(x

119905) then

both a fraction of the 119909119871119905commuters who chose action 119871 and

a fraction of the (1 minus 119909119871119905minus 119909119877

119905) commuters who chose action

119872 will switch to path 119877 in the next time period 119905 + 1 Thefraction of switching commuters is given by the switchingpropensities 120575

119871 120575119872 120575119877modulated by the relative differences

in payoffs This happens whenever a path gives a smallertravel cost In other words at any time 119905 all the commutersdecide their future action at time 119905 + 1 comparing the costs119871(x119905)119872(x

119905) and 119877(x

119905) As in the one-dimensional case here

also the differences of travel costs are normalized dividing

8 Discrete Dynamics in Nature and Society

each difference by the constants giving the related maximumvalue given in the different cases as follows

1198961= max

x [119871 (x) minus 119877 (x)]

= [119871 (1 0) minus 119877 (1 0)] = 119886119871 + 119887119871 minus 119886119877

1198962= max

x [119877 (x) minus 119871 (x)]

= [119877 (0 1) minus 119871 (0 1)] = 119886119877 + 119887119877 minus 119886119871

1198963= max

x [119871 (x) minus 119872 (x)]

= [119871 (1 119909119877) minus119872(1 119909

119877)] = 119886

119871minus 119889 forall119909

119877isin [0 1]

1198964= max

x [119872 (x) minus 119871 (x)]

= [119872(0 119909119877) minus 119871 (0 119909

119877)] = 119889 + 119887

119877minus 119886119871 forall119909

119877isin [0 1]

1198965= max

x [119877 (x) minus 119872 (x)]

= [119877 (119909119871 1) minus 119872(119909

119871 1)] = 119886

119877minus 119889 forall119909

119871isin [0 1]

1198966= max

x [119872 (x) minus 119877 (x)]

= [119872(119909119871 0) minus 119877 (119909

119871 0)] = 119889 + 119887

119871minus 119886119877 forall119909

119871isin [0 1]

(25)

The resulting dynamics are described by a two-dimensionalmap x

119905+1= F(x

119905) defined as

F x119905+1=

F119871(x119905) if x

119905isin 119877119871

F119872(x119905) if x

119905isin 119877119872

F119877(x119905) if x

119905isin 119877119877

(26)

where

F119871

119909119871

119905+1= 119909119871

119905+ 120575119871(119877 (x119905) minus 119871 (x

119905)

119886119877+ 119887119877minus 119886119871

119909119877

119905

+119872(x119905) minus 119871 (x

119905)

119889 + 119887119877minus 119886119871

119909119872

119905)

119909119877

119905+1= (1 minus 120575

119871

119877 (x119905) minus 119871 (x

119905)

119886119877+ 119887119877minus 119886119871

)119909119877

119905

F119872

119909119871

119905+1= (1 minus 120575

119872

119871 (x119905) minus 119872(x

119905)

119886119871minus 119889

)119909119871

119905

119909119877

119905+1= (1 minus 120575

119872

119877 (x119905) minus 119872(x

119905)

119886119877minus 119889

)119909119877

119905

F119877

119909119871

119905+1= (1 minus 120575

119877

119871 (x119905) minus 119877 (x

119905)

119886119871+ 119887119871minus 119886119877

)119909119871

119905

119909119877

119905+1= 119909119877

119905+ 120575119877(119871 (x119905) minus 119877 (x

119905)

119886119871+ 119887119871minus 119886119877

119909119871

119905

+119872(x119905) minus 119877 (x

119905)

119889 + 119887119871minus 119886119877

119909119872

119905)

(27)

By construction we have that F maps 1198632 into itself In factto show that F 1198632 rarr 119863

2 it is enough to show that

given (119909119871119905 119909119877

119905) isin 119863

2 then (119909119871119905+1 119909119877

119905+1) = F(119909119871

119905 119909119877

119905) isin 119863

2From (119909119871

119905 119909119877

119905) isin 119863

2 we have that 0 le 119909119871119905+ 119909119877

119905le 1 and

119909119871

119905+ 119909119877

119905+ 119909119872

119905= 1 Then if (119909119871

119905+1 119909119877

119905+1) = F119871(119909119871

119905 119909119877

119905) we have

119909119871

119905+1+ 119909119877

119905+1= 119909119871

119905+ 119909119877

119905+ 120575119871

119872(x119905) minus 119871 (x

119905)

119889 + 119887119877minus 119886119871

119909119872

119905

le 119909119871

119905+ 119909119877

119905+ 119909119872

119905= 1

(28)

119909119871

119905+1ge 0 and 119909119877

119905+1ge 0 thus (119909119871

119905+1 119909119877

119905+1) isin 1198632 If (119909119871

119905+1 119909119877

119905+1) =

F119872(119909119871

119905 119909119877

119905) we have 0 le 119909119871

119905+1le 119909119871

119905and 0 le 119909119877

119905+1le 119909119877

119905 thus

(119909119871

119905+1 119909119877

119905+1) isin 1198632 If (119909119871

119905+1 119909119877

119905+1) = F119877(119909119871

119905 119909119877

119905) we have

119909119871

119905+1+ 119909119877

119905+1= 119909119871

119905+ 119909119877

119905+ 120575119877

119872(x119905) minus 119877 (x

119905)

119889 + 119887119871minus 119886119877

119909119872

119905

le 119909119871

119905+ 119909119877

119905+ 119909119872

119905= 1

(29)

119909119871

119905+1ge 0 and 119909119877

119905+1ge 0 thus (119909119871

119905+1 119909119877

119905+1) isin 1198632

Substituting from (18) we can rewrite the functions inexplicit form as

F119871

119909119871

119905+1= 120575119871+ (1 minus 120575

119871minus 120575119871

119887119877

1198964

)119909119871

119905

minus120575119871

119887119871

1198962

119909119877

119905+ 120575119871

119887119871

1198962

(119909119877

119905)2

+120575119871119887119877(1

1198964

minus1

1198962

)119909119871

119905119909119877

119905+ 120575119871

119887119877

1198964

(119909119871

119905)2

119909119877

119905+1= (1 minus 120575

119871minus 120575119871

119887119871

1198962

)119909119877

119905

+120575119871

119887119877

1198962

119909119871

119905119909119877

119905minus 120575119871

119887119871

1198962

(119909119877

119905)2

F119872

119909119871

119905+1= (1 + 120575

119872

1198964

1198963

)119909119871

119905minus 120575119872

119887119877

1198963

(119909119871

119905)2

119909119877

119905+1= (1 + 120575

119872

1198966

1198965

)119909119877

119905minus 120575119872

119887119871

1198965

(119909119877

119905)2

F119877

119909119871

119905+1= (1 minus 120575

119877minus 120575119877

119887119877

1198961

)119909119871

119905

+120575119877

119887119871

1198961

119909119871

119905119909119877

119905minus 120575119877

119887119877

1198961

(119909119871

119905)2

119909119877

119905+1= 120575119877+ (1 minus 120575

119877minus 120575119877

119887119871

1198966

)119909119877

119905

minus120575119877

119887119877

1198961

119909119871

119905+ 120575119877

119887119871

1198961

(119909119877

119905)2

+120575119877119887119871(1

1198966

minus1

1198961

)119909119871

119905119909119877

119905+ 120575119877

119887119877

1198966

(119909119871

119905)2

(30)

Differently from the one-dimensional map 119865 considered inthe previous section the two-dimensional map F is notcontinuous in 1198632 More precisely the functions F

119871 F119872

and F119877are continuous and thus F is continuous in the

interior of each region inwhich the different definitions applyHowever in general except for the point xlowast the map F is notcontinuous along the borders 119881

119894 119894 = 1 2 3 of the regions

119877119871 119877119872 and 119877

119877 In fact it is easy to see that considering

Discrete Dynamics in Nature and Society 9

xL

t+1

xL

t

xR

t

00

1

1

(a)

xL

t+1

xL

t

xR

t

00

1

1

(b)

Figure 5 Graphics of the three functions defining (a) 119909119871119905+1

and (b) 119909119877119905+1

with parameters 119886119871= 29 119887

119871= 16 119886

119877= 19 119887

119877= 24 119889 = 8 120575

119871= 05

120575119872= 08 and 120575

119877= 06

a point (119909119871 119909119877) belonging to one of the lines defined in (22)at the boundary of two regions only we have that the twofunctions defined on the two different regions lead to twodifferent points of1198632 An example of how the three functionsdefining 119909119871

119905+1behave is shown in Figure 5(a) and the three

functions defining 119909119877119905+1

are illustrated in Figure 5(b) showingthat F is continuous only in xlowast

The point xlowast in which the three regions are in contactand F is continuous is clearly particular In fact solving theequation 119871(x) = 119872(x) we have the unique solution 119909119871 = 119909lowast119871while solving the equation 119877(x) = 119872(x) we have the uniquesolution119909119877 = 119909lowast119877 It follows that at the point xlowast = (119909lowast119871 119909lowast119877)we have the same cost in each choice as 119871(xlowast) = 119872(xlowast) =119877(xlowast) so that at the point xlowast the commuters are indifferent toany available choice andnone of themwill switch choiceThusF(xlowast) = xlowast and xlowast is a stationary state Under the conditionsin (23) the point xlowast is feasible belonging to 1198632 We can nowprove that this is indeed the unique fixed point of map F

Proposition 3 The point xlowast = (119909lowast119871 119909lowast119877) given in (21) is theunique fixed point of the map F belonging to 1198632 when theconditions in (23) hold

Proof The proof of this proposition follows directly from theobservation that

F (x) = x if and only if 119871 (x) = 119872 (x) = 119877 (x) (31)One way is already proved as we have detected the fixed pointstarting from the solution of 119871(x) = 119872(x) and 119877(x) = 119872(x)showing that it is the unique point xlowast Vice versa let us lookfor the solutions of F(x) = x We have the following systems(where 119909119872 = 1 minus 119909119871 minus 119909119877)

F119871

119909119871= 119909119871+ 120575119871(119877 (x) minus 119871 (x)

1198962

119909119877

+119872(x) minus 119871 (x)

1198964

119909119872)

119909119877= (1 minus 120575

119871

119877 (x) minus 119871 (x)1198962

)119909119877

if x isin 119877119871

F119872

119909119871= (1 minus 120575

119872

119871 (x) minus 119872 (x)119886119871minus 119889

)119909119871

119909119877= (1 minus 120575

119872

119877 (x) minus 119872 (x)119886119877minus 119889

)119909119877

if x isin 119877119872

F119877

119909119871= (1 minus 120575

119877

119871 (x) minus 119877 (x)1198961

)119909119871

119909119877= 119909119877+ 120575119877(119871 (x) minus 119877 (x)

1198961

119909119871

+119872(x) minus 119877 (x)

1198966

119909119872)

if x isin 119877119877

(32)

or equivalently (we recall that 120575119871 120575119872 and 120575

119877are different

from zero)

F119871

119877 (x) minus 119871 (x)1198962

119909119877+119872(x) minus 119871 (x)

1198964

119909119872= 0

119877 (x) minus 119871 (x)1198962

119909119877= 0

if x isin 119877119871

F119872

119871 (x) minus 119872 (x)119886119871minus 119889

119909119871= 0

119877 (x) minus 119872 (x)119886119877minus 119889

119909119877= 0

if x isin 119877119872

F119877

119871 (x) minus 119877 (x)1198961

119909119871= 0

119871 (x) minus 119877 (x)1198961

119909119871+119872(x) minus 119877 (x)

1198966

119909119872= 0

if x isin 119877119877

(33)

Considering the equation F119871(x) = x we see that it is certainly

satisfied for 119909119877 = 0 and 119909119872 = 0 which leads to thecorner point with 119909119871 = 1 However if the fixed point is an

10 Discrete Dynamics in Nature and Society

interior point of 1198632 then the point (1 0) does not belong tothe proper region 119877

119871 that is it is a virtual fixed point Clearly

this equation is satisfied when 119871(x) = 119872(x) and 119871(x) = 119877(x)which occurs only at the point xlowast It is possible to have xlowast =(1 0) and this occurs for (119886

119871minus 119889)119887

119877= 0 and (119886

119877minus 119889)119887

119871= 1

Thus we can state that the only solution of F119871(x) = x in 119877

119871

is given by the fixed point xlowast Similarly we can reason in theother regions

It is worth to mention that particular cases are obtainedwhen the fixed point xlowast belongs to a boundary of the region1198632 For example when 119909lowast119877+119909lowast119871 = 1 and 119909lowast119872 = 0 then only

the two regions119877119871and 119877

119877are involvedThis occurs when the

parameters satisfy

119889 =119886119871119887119871+ 119886119877119887119877minus 119887119871119887119877

119887119877+ 119887119871

0 le119886119871minus 119889

119887119877

le 1

0 le119886119877minus 119889

119887119871

le 1

(34)

From (119886119877minus 119889)119887

119871= (119886119877+ 119887119877minus 119886119871)(119887119877+ 119887119871) we have 119909lowast119877 =

1minus(119886119877minus119889)119887

119871= (119886119871+119887119871minus119886119877)(119887119877+119887119871) = 1198961(119887119877+119887119871) = 119909lowast as

in the one-dimensional map studied in Section 2 and 119909lowast119871 =1 minus 119909lowast119877= 1 minus (119886

119871minus 119889)119887

119877= 1198962(119887119877+ 119887119871) = 1 minus 119896

1(119887119877+ 119887119871) =

1minus119909lowast From themapsF

119871andF119877in (27) and for the parameters

occurring in this case it can be seen that 119909119871119905+1= 1minus119909

119877

119905+1holds

in both so that the dynamics can be studied by using onlythe variable 119909119877

119905and the resulting one-dimensional map is as

follows

119865 (119909119877)

=

119891119871(119909119877)

= 119909119877+ 120575119877

119871 (119909119877)minus119877 (119909

119877)

1198961

(1 minus 119909119877)

if 0 le 119909119877 le 119909lowast

119891119877(119909119877)

= 119909119877minus 120575119871

119877 (119909119877) minus 119871 (119909

119877)

1198962

119909119877 if 119909lowast le 119909119877 le 1

(35)

which corresponds to the system already studied in Section 2As we have seen this fixed point is globally attracting even iflocally we can have the eigenvalue on some side larger than 1in modulus

The case with 119909lowast119871 = 0 so that 0 le 119909lowast119877 le 1 and 119909lowast119872 =1 minus 119909lowast119877 leaves only the two regions 119877

119877and 119877

119872 This occurs

when the parameters satisfy

119886119871minus 119889

119887119877

= 1 0 le119886119877minus 119889

119887119871

le 1 (36)

From the maps F119872

and F119877in (27) it can be seen that 119909119871

119905

converges to119909lowast119871 = 0 so that the resulting asymptotic dynamic

behavior can be studied via the one-dimensional map givenby 119909119877119905+1= 119879(119909

119877

119905) with 119909119877

119905isin [0 1] (and 119909119872

119905= 1 minus 119909

119877

119905) given by

119879 (119909119877)

=

119879119871(119909119877)

= 119909119877+ 120575119877

119872(119909119877) minus 119877 (119909

119877)

1198966

(1 minus 119909119877)

if 0 le 119909119877 le 119909lowast

119879119877(119909119877)

= 119909119877minus 120575119872

119877 (119909119877) minus119872(119909

119877)

119886119877minus 119889

119909119877 if 119909lowast le 119909119877 le 1

(37)

where119872(119909119877) minus 119877(119909119877) = 119886119871+ 119887119871(1 minus 119909

119877) minus 119886119877minus 119887119877

The case with 119909lowast119877 = 0 so that 0 le 119909lowast119871 le 1 and119909lowast119872= 1 minus 119909

lowast119871 leaves only the two regions 119877119871and 119877

119872 This

occurs when the parameters satisfy 0 le (119886119871minus 119889)119887

119877le 1

(119886119877minus 119889)119887

119871= 1 From the maps F

119872and F

119871in (27) it can

be seen that 119909119877119905converges to 119909lowast119877 = 0 so that the resulting

asymptotic dynamic behavior can be studied via the one-dimensional map given by 119909119871

119905+1= 119879(119909

119871

119905) similar to the one

obtained in (37) with obvious changesBy using arguments as in Section 2 it can be shown that

these one-dimensional systems have a unique fixed pointwhich is globally attracting (although it can be locally stableor unstable)

In order to study the local stability of the fixed point ofthe two-dimensional map F we can use the linearization ofthe different definitions in the fixed point which leads to thefollowing Jacobian matrices one for each region 119877

119871 119877119872 and

119877119877 that we denote respectively with 119869

119871 119869119872 and 119869

119877

119869119871(xlowast) = (

1 + 120575119871minus 120575119871

119887119877

119887119871

1198965

1198964

minus 120575119871

119887119877

119887119871

1198966

1198961

120575119871

1198966

1198961

120575119871

119887119877

119887119871

1198966

1198961

1 minus 120575119871

1198966

1198961

)

119869119872(xlowast) = (

1 minus 120575119872

1198964

1198963

0

0 1 minus 120575119872

1198966

1198965

)

119869119877(xlowast) = (

1 minus 120575119877

1198965

1198962

120575119877

119887119871

119887119877

1198965

1198962

120575119877

1198965

1198962

1 + 120575119877minus 120575119877

119887119871

119887119877

1198963

1198966

minus 120575119877

119887119871

119887119877

1198965

1198962

)

(38)

The two real eigenvalues of the Jacobian matrix in 119877119872are

in explicit form and both are smaller than +1 However theycan be larger than minus1 or not Also the eigenvalues of the otherjacobian matrices can be in modulus smaller or larger than1 However the global dynamics observed numerically givealways trajectories which are convergent to the fixed point xlowastat any set of allowed parameters values

Discrete Dynamics in Nature and Society 11

1

1

075

05

025

00 025 05 075

xR

xL

(a)

1

1

075

05

025

00 025 05 075

xR

xL

(b)

Figure 6 Examples of trajectories converging to the unique fixed point with parameters (a) 119886119871= 28 119887

119871= 24 119886

119877= 8 119887

119877= 24 119889 = 5 120575

119871= 04

120575119872= 03 120575

119877= 07 and ic 119909119871 = 08 119909119877 = 005 (b) 119886

119871= 29 119887

119871= 16 119886

119877= 15 119887

119877= 25 119889 = 7 120575

119871= 05 120575

119872= 08 120575

119877= 06 and ic 119909119871 = 005

119909119877= 09 Dots are linked by a continuous line for illustrative purpose only

This global attractivity in the two-dimensional map isdifficult to be proved rigorously however the results in theparticular cases commented above that is when the fixedpoint xlowast belongs to the boundary of 1198632 suggests that thesame result is true also when the fixed point is in the interiorof1198632

In Figure 6 we show some examples in Figure 6(a) wecan see a trajectory starting at 119877

119877 in Figure 6(b) a trajectory

whose initial point in 119877119871is mapped into 119877

119872 The shape of

the functions in the case of Figure 6(b) is similar to the oneshown in Figure 5

4 Conclusion

In this paper we have analyzed a dynamical version of theBraess paradox with nonimpulsive commuters who changeroad proportionally to the cost differenceWe were interestedto prove that with nonimpulsive commuters there exists aunique equilibrium both in the original network with twochoices and in the ternary onewhen a single link is addedWehave shown that both models are quite robust We were ableto provide an analytical proof of the global attractivity of theunique fixed point in the one-dimensional case when onlytwo roads are available andweusednumerical techniques andsimulations to give evidence in the two-dimensional case (iewhen a new road is added) as a rigorous proof can be doneonly for particular border cases

A limitation of our contribution is considering homo-geneous populations in spite of the fact that the evidencesupports heterogeneity Nevertheless heterogeneity of com-mutersrsquo behavior cannot be easily classified yet even when

conducting experiments common individual patterns arejust identified and classified On the other hand whenobserving the aggregate behavior inmany cases the literatureprovides evidence in support of the paradox outcome thatis the convergence to the Nash equilibrium The impulsiveagents in [30] react faster than the proportional agentspresented in this paper yet as a population they fail toachieve the Nash equilibria in certain cases while in thispaper the aggregate behavior does not We proposed thisaggregate behavior because it replies the convergence to theNash equilibria and at the same time it may be the resultof modeling some observed individual behaviors such asimitation and free riding

Taking into account the limitation of homogeneous pop-ulation analysis a natural extension of our contribution isconsidering heterogeneous populations with both impulsiveand nonimpulsive commuters We have reasons to believethat in such a case some properties of the different behaviorsare inherited and some are lost Therefore the results weprovide in this paper match those for impulsive populationsand are paramount to extend the analysis to more realisticcases In factmdashas the empirical results showmdashheterogeneouspopulations dynamics provide a better fit of what has beenobserved when considering human participants interaction

Furthermore it will be interesting to consider general-izations to networks with richer architectures Finally weobserve that since with the class of behavior we considerin this paper that dynamics depends on the shape of thecost functions it will be interesting to extend the analysis tononlinear cost functions

12 Discrete Dynamics in Nature and Society

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

V Avrutin is supported by the European Community withinthe scope of the project ldquoMultiple-discontinuity inducedbifurcations in theory and applicationsrdquo (Marie Curie Actionof the 7th Framework Programme Contract Agreement noPIEF-GA-2011-300281) U Merlone has worked under theauspices of COST Action IS1104 ldquoThe EU in the new complexgeography of economic systems models tools and policyevaluationrdquo

References

[1] R Arnott and K Small ldquoThe economics of traffic congestionrdquoAmerican Scientist vol 82 no 5 pp 446ndash455 1994

[2] T Bass ldquoRoad to ruinrdquo Discover pp 56ndash61 1992[3] B Calvert and G Keady ldquoBraessrsquos paradox and power-law non-

linearities in networksrdquoThe Journal of the AustralianMathemat-ical Society B Applied Mathematics vol 35 no 1 pp 1ndash22 1993

[4] J E Cohen andC Jeffries ldquoCongestion resulting from increasedcapacity in single-server queueing networksrdquo IEEEACMTrans-actions on Networking vol 5 no 2 pp 305ndash310 1997

[5] J E Cohen and F P Kelly ldquoA paradox of congestion in a queuingnetworkrdquo Journal of Applied Probability vol 27 no 3 pp 730ndash734 1990

[6] S Dafermos and A Nagurney ldquoOn some traffic equilibriumtheory paradoxesrdquo Transportation Research B Methodologicalvol 18 no 2 pp 101ndash110 1984

[7] C Fisk ldquoMore paradoxes in the equilibrium assignment prob-lemrdquo Transportation Research B Methodological vol 13 no 4pp 305ndash309 1979

[8] W H K Lam ldquoEffects of road pricing on system performancerdquoTraffic Engineering amp Control vol 29 no 12 pp 631ndash635 1988

[9] J D Murchland ldquoBraessrsquos paradox of traffic flowrdquo Transporta-tion Research vol 4 no 4 pp 391ndash394 1970

[10] R Steinberg and W I Zangwill ldquoThe prevalence o f BraessrsquoparadoxrdquoTransportation Science vol 17 no 3 pp 301ndash318 1983

[11] M Frank ldquoThe Braess paradoxrdquo Mathematical Programmingvol 20 no 1 pp 283ndash302 1981

[12] T Roughgarden Selfish Routing and the Price of Anarchy MITpress 2005

[13] B B Fu C X Zhao and S B Li ldquoThe analysis of Braessrsquo paradoxand robustness based on dynamic traffic assignment modelsrdquoDiscrete Dynamics in Nature and Society vol 2013 Article ID796842 8 pages 2013

[14] C Fisk and S Pallottino ldquoEmpirical evidence for equilibriumparadoxes with implications for optimal planning strategiesrdquoTransportation Research Part A General vol 15 no 3 pp 245ndash248 1981

[15] H Youn M T Gastner and H Jeong ldquoPrice of anarchy intransportation networks efficiency and optimality controlrdquoPhysical Review Letters vol 101 no 12 Article ID 128701 2008

[16] K Aoki Y Ohtsubo A Rapoport and T Saijo ldquoEffects of priorinvestment and personal responsibility in a simple network

gamerdquo Current Research in Social Psychology vol 13 no 2 pp10ndash21 2007

[17] J Morgan H Orzen and M Sefton ldquoNetwork architectureand traffic flows experiments on the Pigou-Knight-Downs andBraess paradoxesrdquoGames and Economic Behavior vol 66 no 1pp 348ndash372 2009

[18] A Rapoport V Mak and R Zwick ldquoNavigating congested net-works with variable demand experimental evidencerdquo Journal ofEconomic Psychology vol 27 no 5 pp 648ndash666 2006

[19] A Rapoport T Kugler S Dugar and E J Gisches ldquoBraessparadox in the laboratory an experimental study of route choicein traffic networks with asymmetric costsrdquo inDecisionModelingand Behavior in Uncertain and Complex Environment T KuglerJ C Smith T Connolly and Y J Son Eds pp 309ndash337Springer New York NY USA 2008

[20] A Rapoport T Kugler S Dugar and E J Gisches ldquoChoice ofroutes in congested traffic networks experimental tests of theBraess paradoxrdquo Games and Economic Behavior vol 65 no 2pp 538ndash571 2009

[21] E J Gisches and A Rapoport ldquoDegrading network capacitymay improve performance private versus public monitoring inthe Braess paradoxrdquoTheory and Decision vol 73 no 2 pp 267ndash293 2012

[22] A Dal Forno and U Merlone ldquoReplicating human interactionin Braess paradoxrdquo in Proceedings of the Simulation Conference(WSC rsquo13) pp 1754ndash1765 IEEE 2013

[23] M Ausloos H Dawid and U Merlone ldquoSpatial interactionsin agent-based modelingrdquo in Complexity and GeographicalEconomics Topics and Tools P Commendatore S S Kayamand I Kubin Eds Springer Berlin Germany 2014

[24] G-I Bischi and U Merlone ldquoGlobal dynamics in binary choicemodels with social influencerdquo The Journal of MathematicalSociology vol 33 no 4 pp 277ndash302 2009

[25] G I Bischi L Gardini and U Merlone ldquoImpulsivity in binarychoices and the emergence of periodicityrdquoDiscrete Dynamics inNature and Society vol 2009 Article ID 407913 22 pages 2009

[26] JH PattonM S Stanford andE S Barratt ldquoFactor structure ofthe Barratt Impulsiveness Scalerdquo Journal of Clinical Psychologyvol 51 no 6 pp 768ndash774 1995

[27] G I Bischi L Gardini and U Merlone ldquoPeriodic cycles andbifurcation curves for one-dimensional maps with two discon-tinuitiesrdquo Journal of Dynamical Systems GeometricTheories vol7 no 2 pp 101ndash123 2009

[28] G-I Bischi and U Merlone ldquoBinary choices in small and largegroups a unifiedmodelrdquo Physica A vol 389 no 4 pp 843ndash8532010

[29] A Dal Forno L Gardini and U Merlone ldquoTernary choicesin repeated games and border collision bifurcationsrdquo ChaosSolitons amp Fractals vol 45 no 3 pp 294ndash305 2012

[30] A Dal Forno and U Merlone ldquoBorder-collision bifurcationsin a model of Braess paradoxrdquo Mathematics and Computers inSimulation vol 87 pp 1ndash18 2013

[31] A Dal Forno V Giorgino and U Merlone ldquoAgent-based mod-eling with GTmethodology An examplerdquoWorking Paper 2014

[32] L Gardini U Merlone and F Tramontana ldquoInertia in binarychoices continuity breaking and big-bang bifurcation pointsrdquoJournal of Economic Behavior amp Organization vol 80 no 1 pp153ndash167 2011

[33] W H Sandholm Population Games and Evolutionary Dynam-ics The MIT Press Cambridge Mass USA 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Stochastic AnalysisInternational Journal of

8 Discrete Dynamics in Nature and Society

each difference by the constants giving the related maximumvalue given in the different cases as follows

1198961= max

x [119871 (x) minus 119877 (x)]

= [119871 (1 0) minus 119877 (1 0)] = 119886119871 + 119887119871 minus 119886119877

1198962= max

x [119877 (x) minus 119871 (x)]

= [119877 (0 1) minus 119871 (0 1)] = 119886119877 + 119887119877 minus 119886119871

1198963= max

x [119871 (x) minus 119872 (x)]

= [119871 (1 119909119877) minus119872(1 119909

119877)] = 119886

119871minus 119889 forall119909

119877isin [0 1]

1198964= max

x [119872 (x) minus 119871 (x)]

= [119872(0 119909119877) minus 119871 (0 119909

119877)] = 119889 + 119887

119877minus 119886119871 forall119909

119877isin [0 1]

1198965= max

x [119877 (x) minus 119872 (x)]

= [119877 (119909119871 1) minus 119872(119909

119871 1)] = 119886

119877minus 119889 forall119909

119871isin [0 1]

1198966= max

x [119872 (x) minus 119877 (x)]

= [119872(119909119871 0) minus 119877 (119909

119871 0)] = 119889 + 119887

119871minus 119886119877 forall119909

119871isin [0 1]

(25)

The resulting dynamics are described by a two-dimensionalmap x

119905+1= F(x

119905) defined as

F x119905+1=

F119871(x119905) if x

119905isin 119877119871

F119872(x119905) if x

119905isin 119877119872

F119877(x119905) if x

119905isin 119877119877

(26)

where

F119871

119909119871

119905+1= 119909119871

119905+ 120575119871(119877 (x119905) minus 119871 (x

119905)

119886119877+ 119887119877minus 119886119871

119909119877

119905

+119872(x119905) minus 119871 (x

119905)

119889 + 119887119877minus 119886119871

119909119872

119905)

119909119877

119905+1= (1 minus 120575

119871

119877 (x119905) minus 119871 (x

119905)

119886119877+ 119887119877minus 119886119871

)119909119877

119905

F119872

119909119871

119905+1= (1 minus 120575

119872

119871 (x119905) minus 119872(x

119905)

119886119871minus 119889

)119909119871

119905

119909119877

119905+1= (1 minus 120575

119872

119877 (x119905) minus 119872(x

119905)

119886119877minus 119889

)119909119877

119905

F119877

119909119871

119905+1= (1 minus 120575

119877

119871 (x119905) minus 119877 (x

119905)

119886119871+ 119887119871minus 119886119877

)119909119871

119905

119909119877

119905+1= 119909119877

119905+ 120575119877(119871 (x119905) minus 119877 (x

119905)

119886119871+ 119887119871minus 119886119877

119909119871

119905

+119872(x119905) minus 119877 (x

119905)

119889 + 119887119871minus 119886119877

119909119872

119905)

(27)

By construction we have that F maps 1198632 into itself In factto show that F 1198632 rarr 119863

2 it is enough to show that

given (119909119871119905 119909119877

119905) isin 119863

2 then (119909119871119905+1 119909119877

119905+1) = F(119909119871

119905 119909119877

119905) isin 119863

2From (119909119871

119905 119909119877

119905) isin 119863

2 we have that 0 le 119909119871119905+ 119909119877

119905le 1 and

119909119871

119905+ 119909119877

119905+ 119909119872

119905= 1 Then if (119909119871

119905+1 119909119877

119905+1) = F119871(119909119871

119905 119909119877

119905) we have

119909119871

119905+1+ 119909119877

119905+1= 119909119871

119905+ 119909119877

119905+ 120575119871

119872(x119905) minus 119871 (x

119905)

119889 + 119887119877minus 119886119871

119909119872

119905

le 119909119871

119905+ 119909119877

119905+ 119909119872

119905= 1

(28)

119909119871

119905+1ge 0 and 119909119877

119905+1ge 0 thus (119909119871

119905+1 119909119877

119905+1) isin 1198632 If (119909119871

119905+1 119909119877

119905+1) =

F119872(119909119871

119905 119909119877

119905) we have 0 le 119909119871

119905+1le 119909119871

119905and 0 le 119909119877

119905+1le 119909119877

119905 thus

(119909119871

119905+1 119909119877

119905+1) isin 1198632 If (119909119871

119905+1 119909119877

119905+1) = F119877(119909119871

119905 119909119877

119905) we have

119909119871

119905+1+ 119909119877

119905+1= 119909119871

119905+ 119909119877

119905+ 120575119877

119872(x119905) minus 119877 (x

119905)

119889 + 119887119871minus 119886119877

119909119872

119905

le 119909119871

119905+ 119909119877

119905+ 119909119872

119905= 1

(29)

119909119871

119905+1ge 0 and 119909119877

119905+1ge 0 thus (119909119871

119905+1 119909119877

119905+1) isin 1198632

Substituting from (18) we can rewrite the functions inexplicit form as

F119871

119909119871

119905+1= 120575119871+ (1 minus 120575

119871minus 120575119871

119887119877

1198964

)119909119871

119905

minus120575119871

119887119871

1198962

119909119877

119905+ 120575119871

119887119871

1198962

(119909119877

119905)2

+120575119871119887119877(1

1198964

minus1

1198962

)119909119871

119905119909119877

119905+ 120575119871

119887119877

1198964

(119909119871

119905)2

119909119877

119905+1= (1 minus 120575

119871minus 120575119871

119887119871

1198962

)119909119877

119905

+120575119871

119887119877

1198962

119909119871

119905119909119877

119905minus 120575119871

119887119871

1198962

(119909119877

119905)2

F119872

119909119871

119905+1= (1 + 120575

119872

1198964

1198963

)119909119871

119905minus 120575119872

119887119877

1198963

(119909119871

119905)2

119909119877

119905+1= (1 + 120575

119872

1198966

1198965

)119909119877

119905minus 120575119872

119887119871

1198965

(119909119877

119905)2

F119877

119909119871

119905+1= (1 minus 120575

119877minus 120575119877

119887119877

1198961

)119909119871

119905

+120575119877

119887119871

1198961

119909119871

119905119909119877

119905minus 120575119877

119887119877

1198961

(119909119871

119905)2

119909119877

119905+1= 120575119877+ (1 minus 120575

119877minus 120575119877

119887119871

1198966

)119909119877

119905

minus120575119877

119887119877

1198961

119909119871

119905+ 120575119877

119887119871

1198961

(119909119877

119905)2

+120575119877119887119871(1

1198966

minus1

1198961

)119909119871

119905119909119877

119905+ 120575119877

119887119877

1198966

(119909119871

119905)2

(30)

Differently from the one-dimensional map 119865 considered inthe previous section the two-dimensional map F is notcontinuous in 1198632 More precisely the functions F

119871 F119872

and F119877are continuous and thus F is continuous in the

interior of each region inwhich the different definitions applyHowever in general except for the point xlowast the map F is notcontinuous along the borders 119881

119894 119894 = 1 2 3 of the regions

119877119871 119877119872 and 119877

119877 In fact it is easy to see that considering

Discrete Dynamics in Nature and Society 9

xL

t+1

xL

t

xR

t

00

1

1

(a)

xL

t+1

xL

t

xR

t

00

1

1

(b)

Figure 5 Graphics of the three functions defining (a) 119909119871119905+1

and (b) 119909119877119905+1

with parameters 119886119871= 29 119887

119871= 16 119886

119877= 19 119887

119877= 24 119889 = 8 120575

119871= 05

120575119872= 08 and 120575

119877= 06

a point (119909119871 119909119877) belonging to one of the lines defined in (22)at the boundary of two regions only we have that the twofunctions defined on the two different regions lead to twodifferent points of1198632 An example of how the three functionsdefining 119909119871

119905+1behave is shown in Figure 5(a) and the three

functions defining 119909119877119905+1

are illustrated in Figure 5(b) showingthat F is continuous only in xlowast

The point xlowast in which the three regions are in contactand F is continuous is clearly particular In fact solving theequation 119871(x) = 119872(x) we have the unique solution 119909119871 = 119909lowast119871while solving the equation 119877(x) = 119872(x) we have the uniquesolution119909119877 = 119909lowast119877 It follows that at the point xlowast = (119909lowast119871 119909lowast119877)we have the same cost in each choice as 119871(xlowast) = 119872(xlowast) =119877(xlowast) so that at the point xlowast the commuters are indifferent toany available choice andnone of themwill switch choiceThusF(xlowast) = xlowast and xlowast is a stationary state Under the conditionsin (23) the point xlowast is feasible belonging to 1198632 We can nowprove that this is indeed the unique fixed point of map F

Proposition 3 The point xlowast = (119909lowast119871 119909lowast119877) given in (21) is theunique fixed point of the map F belonging to 1198632 when theconditions in (23) hold

Proof The proof of this proposition follows directly from theobservation that

F (x) = x if and only if 119871 (x) = 119872 (x) = 119877 (x) (31)One way is already proved as we have detected the fixed pointstarting from the solution of 119871(x) = 119872(x) and 119877(x) = 119872(x)showing that it is the unique point xlowast Vice versa let us lookfor the solutions of F(x) = x We have the following systems(where 119909119872 = 1 minus 119909119871 minus 119909119877)

F119871

119909119871= 119909119871+ 120575119871(119877 (x) minus 119871 (x)

1198962

119909119877

+119872(x) minus 119871 (x)

1198964

119909119872)

119909119877= (1 minus 120575

119871

119877 (x) minus 119871 (x)1198962

)119909119877

if x isin 119877119871

F119872

119909119871= (1 minus 120575

119872

119871 (x) minus 119872 (x)119886119871minus 119889

)119909119871

119909119877= (1 minus 120575

119872

119877 (x) minus 119872 (x)119886119877minus 119889

)119909119877

if x isin 119877119872

F119877

119909119871= (1 minus 120575

119877

119871 (x) minus 119877 (x)1198961

)119909119871

119909119877= 119909119877+ 120575119877(119871 (x) minus 119877 (x)

1198961

119909119871

+119872(x) minus 119877 (x)

1198966

119909119872)

if x isin 119877119877

(32)

or equivalently (we recall that 120575119871 120575119872 and 120575

119877are different

from zero)

F119871

119877 (x) minus 119871 (x)1198962

119909119877+119872(x) minus 119871 (x)

1198964

119909119872= 0

119877 (x) minus 119871 (x)1198962

119909119877= 0

if x isin 119877119871

F119872

119871 (x) minus 119872 (x)119886119871minus 119889

119909119871= 0

119877 (x) minus 119872 (x)119886119877minus 119889

119909119877= 0

if x isin 119877119872

F119877

119871 (x) minus 119877 (x)1198961

119909119871= 0

119871 (x) minus 119877 (x)1198961

119909119871+119872(x) minus 119877 (x)

1198966

119909119872= 0

if x isin 119877119877

(33)

Considering the equation F119871(x) = x we see that it is certainly

satisfied for 119909119877 = 0 and 119909119872 = 0 which leads to thecorner point with 119909119871 = 1 However if the fixed point is an

10 Discrete Dynamics in Nature and Society

interior point of 1198632 then the point (1 0) does not belong tothe proper region 119877

119871 that is it is a virtual fixed point Clearly

this equation is satisfied when 119871(x) = 119872(x) and 119871(x) = 119877(x)which occurs only at the point xlowast It is possible to have xlowast =(1 0) and this occurs for (119886

119871minus 119889)119887

119877= 0 and (119886

119877minus 119889)119887

119871= 1

Thus we can state that the only solution of F119871(x) = x in 119877

119871

is given by the fixed point xlowast Similarly we can reason in theother regions

It is worth to mention that particular cases are obtainedwhen the fixed point xlowast belongs to a boundary of the region1198632 For example when 119909lowast119877+119909lowast119871 = 1 and 119909lowast119872 = 0 then only

the two regions119877119871and 119877

119877are involvedThis occurs when the

parameters satisfy

119889 =119886119871119887119871+ 119886119877119887119877minus 119887119871119887119877

119887119877+ 119887119871

0 le119886119871minus 119889

119887119877

le 1

0 le119886119877minus 119889

119887119871

le 1

(34)

From (119886119877minus 119889)119887

119871= (119886119877+ 119887119877minus 119886119871)(119887119877+ 119887119871) we have 119909lowast119877 =

1minus(119886119877minus119889)119887

119871= (119886119871+119887119871minus119886119877)(119887119877+119887119871) = 1198961(119887119877+119887119871) = 119909lowast as

in the one-dimensional map studied in Section 2 and 119909lowast119871 =1 minus 119909lowast119877= 1 minus (119886

119871minus 119889)119887

119877= 1198962(119887119877+ 119887119871) = 1 minus 119896

1(119887119877+ 119887119871) =

1minus119909lowast From themapsF

119871andF119877in (27) and for the parameters

occurring in this case it can be seen that 119909119871119905+1= 1minus119909

119877

119905+1holds

in both so that the dynamics can be studied by using onlythe variable 119909119877

119905and the resulting one-dimensional map is as

follows

119865 (119909119877)

=

119891119871(119909119877)

= 119909119877+ 120575119877

119871 (119909119877)minus119877 (119909

119877)

1198961

(1 minus 119909119877)

if 0 le 119909119877 le 119909lowast

119891119877(119909119877)

= 119909119877minus 120575119871

119877 (119909119877) minus 119871 (119909

119877)

1198962

119909119877 if 119909lowast le 119909119877 le 1

(35)

which corresponds to the system already studied in Section 2As we have seen this fixed point is globally attracting even iflocally we can have the eigenvalue on some side larger than 1in modulus

The case with 119909lowast119871 = 0 so that 0 le 119909lowast119877 le 1 and 119909lowast119872 =1 minus 119909lowast119877 leaves only the two regions 119877

119877and 119877

119872 This occurs

when the parameters satisfy

119886119871minus 119889

119887119877

= 1 0 le119886119877minus 119889

119887119871

le 1 (36)

From the maps F119872

and F119877in (27) it can be seen that 119909119871

119905

converges to119909lowast119871 = 0 so that the resulting asymptotic dynamic

behavior can be studied via the one-dimensional map givenby 119909119877119905+1= 119879(119909

119877

119905) with 119909119877

119905isin [0 1] (and 119909119872

119905= 1 minus 119909

119877

119905) given by

119879 (119909119877)

=

119879119871(119909119877)

= 119909119877+ 120575119877

119872(119909119877) minus 119877 (119909

119877)

1198966

(1 minus 119909119877)

if 0 le 119909119877 le 119909lowast

119879119877(119909119877)

= 119909119877minus 120575119872

119877 (119909119877) minus119872(119909

119877)

119886119877minus 119889

119909119877 if 119909lowast le 119909119877 le 1

(37)

where119872(119909119877) minus 119877(119909119877) = 119886119871+ 119887119871(1 minus 119909

119877) minus 119886119877minus 119887119877

The case with 119909lowast119877 = 0 so that 0 le 119909lowast119871 le 1 and119909lowast119872= 1 minus 119909

lowast119871 leaves only the two regions 119877119871and 119877

119872 This

occurs when the parameters satisfy 0 le (119886119871minus 119889)119887

119877le 1

(119886119877minus 119889)119887

119871= 1 From the maps F

119872and F

119871in (27) it can

be seen that 119909119877119905converges to 119909lowast119877 = 0 so that the resulting

asymptotic dynamic behavior can be studied via the one-dimensional map given by 119909119871

119905+1= 119879(119909

119871

119905) similar to the one

obtained in (37) with obvious changesBy using arguments as in Section 2 it can be shown that

these one-dimensional systems have a unique fixed pointwhich is globally attracting (although it can be locally stableor unstable)

In order to study the local stability of the fixed point ofthe two-dimensional map F we can use the linearization ofthe different definitions in the fixed point which leads to thefollowing Jacobian matrices one for each region 119877

119871 119877119872 and

119877119877 that we denote respectively with 119869

119871 119869119872 and 119869

119877

119869119871(xlowast) = (

1 + 120575119871minus 120575119871

119887119877

119887119871

1198965

1198964

minus 120575119871

119887119877

119887119871

1198966

1198961

120575119871

1198966

1198961

120575119871

119887119877

119887119871

1198966

1198961

1 minus 120575119871

1198966

1198961

)

119869119872(xlowast) = (

1 minus 120575119872

1198964

1198963

0

0 1 minus 120575119872

1198966

1198965

)

119869119877(xlowast) = (

1 minus 120575119877

1198965

1198962

120575119877

119887119871

119887119877

1198965

1198962

120575119877

1198965

1198962

1 + 120575119877minus 120575119877

119887119871

119887119877

1198963

1198966

minus 120575119877

119887119871

119887119877

1198965

1198962

)

(38)

The two real eigenvalues of the Jacobian matrix in 119877119872are

in explicit form and both are smaller than +1 However theycan be larger than minus1 or not Also the eigenvalues of the otherjacobian matrices can be in modulus smaller or larger than1 However the global dynamics observed numerically givealways trajectories which are convergent to the fixed point xlowastat any set of allowed parameters values

Discrete Dynamics in Nature and Society 11

1

1

075

05

025

00 025 05 075

xR

xL

(a)

1

1

075

05

025

00 025 05 075

xR

xL

(b)

Figure 6 Examples of trajectories converging to the unique fixed point with parameters (a) 119886119871= 28 119887

119871= 24 119886

119877= 8 119887

119877= 24 119889 = 5 120575

119871= 04

120575119872= 03 120575

119877= 07 and ic 119909119871 = 08 119909119877 = 005 (b) 119886

119871= 29 119887

119871= 16 119886

119877= 15 119887

119877= 25 119889 = 7 120575

119871= 05 120575

119872= 08 120575

119877= 06 and ic 119909119871 = 005

119909119877= 09 Dots are linked by a continuous line for illustrative purpose only

This global attractivity in the two-dimensional map isdifficult to be proved rigorously however the results in theparticular cases commented above that is when the fixedpoint xlowast belongs to the boundary of 1198632 suggests that thesame result is true also when the fixed point is in the interiorof1198632

In Figure 6 we show some examples in Figure 6(a) wecan see a trajectory starting at 119877

119877 in Figure 6(b) a trajectory

whose initial point in 119877119871is mapped into 119877

119872 The shape of

the functions in the case of Figure 6(b) is similar to the oneshown in Figure 5

4 Conclusion

In this paper we have analyzed a dynamical version of theBraess paradox with nonimpulsive commuters who changeroad proportionally to the cost differenceWe were interestedto prove that with nonimpulsive commuters there exists aunique equilibrium both in the original network with twochoices and in the ternary onewhen a single link is addedWehave shown that both models are quite robust We were ableto provide an analytical proof of the global attractivity of theunique fixed point in the one-dimensional case when onlytwo roads are available andweusednumerical techniques andsimulations to give evidence in the two-dimensional case (iewhen a new road is added) as a rigorous proof can be doneonly for particular border cases

A limitation of our contribution is considering homo-geneous populations in spite of the fact that the evidencesupports heterogeneity Nevertheless heterogeneity of com-mutersrsquo behavior cannot be easily classified yet even when

conducting experiments common individual patterns arejust identified and classified On the other hand whenobserving the aggregate behavior inmany cases the literatureprovides evidence in support of the paradox outcome thatis the convergence to the Nash equilibrium The impulsiveagents in [30] react faster than the proportional agentspresented in this paper yet as a population they fail toachieve the Nash equilibria in certain cases while in thispaper the aggregate behavior does not We proposed thisaggregate behavior because it replies the convergence to theNash equilibria and at the same time it may be the resultof modeling some observed individual behaviors such asimitation and free riding

Taking into account the limitation of homogeneous pop-ulation analysis a natural extension of our contribution isconsidering heterogeneous populations with both impulsiveand nonimpulsive commuters We have reasons to believethat in such a case some properties of the different behaviorsare inherited and some are lost Therefore the results weprovide in this paper match those for impulsive populationsand are paramount to extend the analysis to more realisticcases In factmdashas the empirical results showmdashheterogeneouspopulations dynamics provide a better fit of what has beenobserved when considering human participants interaction

Furthermore it will be interesting to consider general-izations to networks with richer architectures Finally weobserve that since with the class of behavior we considerin this paper that dynamics depends on the shape of thecost functions it will be interesting to extend the analysis tononlinear cost functions

12 Discrete Dynamics in Nature and Society

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

V Avrutin is supported by the European Community withinthe scope of the project ldquoMultiple-discontinuity inducedbifurcations in theory and applicationsrdquo (Marie Curie Actionof the 7th Framework Programme Contract Agreement noPIEF-GA-2011-300281) U Merlone has worked under theauspices of COST Action IS1104 ldquoThe EU in the new complexgeography of economic systems models tools and policyevaluationrdquo

References

[1] R Arnott and K Small ldquoThe economics of traffic congestionrdquoAmerican Scientist vol 82 no 5 pp 446ndash455 1994

[2] T Bass ldquoRoad to ruinrdquo Discover pp 56ndash61 1992[3] B Calvert and G Keady ldquoBraessrsquos paradox and power-law non-

linearities in networksrdquoThe Journal of the AustralianMathemat-ical Society B Applied Mathematics vol 35 no 1 pp 1ndash22 1993

[4] J E Cohen andC Jeffries ldquoCongestion resulting from increasedcapacity in single-server queueing networksrdquo IEEEACMTrans-actions on Networking vol 5 no 2 pp 305ndash310 1997

[5] J E Cohen and F P Kelly ldquoA paradox of congestion in a queuingnetworkrdquo Journal of Applied Probability vol 27 no 3 pp 730ndash734 1990

[6] S Dafermos and A Nagurney ldquoOn some traffic equilibriumtheory paradoxesrdquo Transportation Research B Methodologicalvol 18 no 2 pp 101ndash110 1984

[7] C Fisk ldquoMore paradoxes in the equilibrium assignment prob-lemrdquo Transportation Research B Methodological vol 13 no 4pp 305ndash309 1979

[8] W H K Lam ldquoEffects of road pricing on system performancerdquoTraffic Engineering amp Control vol 29 no 12 pp 631ndash635 1988

[9] J D Murchland ldquoBraessrsquos paradox of traffic flowrdquo Transporta-tion Research vol 4 no 4 pp 391ndash394 1970

[10] R Steinberg and W I Zangwill ldquoThe prevalence o f BraessrsquoparadoxrdquoTransportation Science vol 17 no 3 pp 301ndash318 1983

[11] M Frank ldquoThe Braess paradoxrdquo Mathematical Programmingvol 20 no 1 pp 283ndash302 1981

[12] T Roughgarden Selfish Routing and the Price of Anarchy MITpress 2005

[13] B B Fu C X Zhao and S B Li ldquoThe analysis of Braessrsquo paradoxand robustness based on dynamic traffic assignment modelsrdquoDiscrete Dynamics in Nature and Society vol 2013 Article ID796842 8 pages 2013

[14] C Fisk and S Pallottino ldquoEmpirical evidence for equilibriumparadoxes with implications for optimal planning strategiesrdquoTransportation Research Part A General vol 15 no 3 pp 245ndash248 1981

[15] H Youn M T Gastner and H Jeong ldquoPrice of anarchy intransportation networks efficiency and optimality controlrdquoPhysical Review Letters vol 101 no 12 Article ID 128701 2008

[16] K Aoki Y Ohtsubo A Rapoport and T Saijo ldquoEffects of priorinvestment and personal responsibility in a simple network

gamerdquo Current Research in Social Psychology vol 13 no 2 pp10ndash21 2007

[17] J Morgan H Orzen and M Sefton ldquoNetwork architectureand traffic flows experiments on the Pigou-Knight-Downs andBraess paradoxesrdquoGames and Economic Behavior vol 66 no 1pp 348ndash372 2009

[18] A Rapoport V Mak and R Zwick ldquoNavigating congested net-works with variable demand experimental evidencerdquo Journal ofEconomic Psychology vol 27 no 5 pp 648ndash666 2006

[19] A Rapoport T Kugler S Dugar and E J Gisches ldquoBraessparadox in the laboratory an experimental study of route choicein traffic networks with asymmetric costsrdquo inDecisionModelingand Behavior in Uncertain and Complex Environment T KuglerJ C Smith T Connolly and Y J Son Eds pp 309ndash337Springer New York NY USA 2008

[20] A Rapoport T Kugler S Dugar and E J Gisches ldquoChoice ofroutes in congested traffic networks experimental tests of theBraess paradoxrdquo Games and Economic Behavior vol 65 no 2pp 538ndash571 2009

[21] E J Gisches and A Rapoport ldquoDegrading network capacitymay improve performance private versus public monitoring inthe Braess paradoxrdquoTheory and Decision vol 73 no 2 pp 267ndash293 2012

[22] A Dal Forno and U Merlone ldquoReplicating human interactionin Braess paradoxrdquo in Proceedings of the Simulation Conference(WSC rsquo13) pp 1754ndash1765 IEEE 2013

[23] M Ausloos H Dawid and U Merlone ldquoSpatial interactionsin agent-based modelingrdquo in Complexity and GeographicalEconomics Topics and Tools P Commendatore S S Kayamand I Kubin Eds Springer Berlin Germany 2014

[24] G-I Bischi and U Merlone ldquoGlobal dynamics in binary choicemodels with social influencerdquo The Journal of MathematicalSociology vol 33 no 4 pp 277ndash302 2009

[25] G I Bischi L Gardini and U Merlone ldquoImpulsivity in binarychoices and the emergence of periodicityrdquoDiscrete Dynamics inNature and Society vol 2009 Article ID 407913 22 pages 2009

[26] JH PattonM S Stanford andE S Barratt ldquoFactor structure ofthe Barratt Impulsiveness Scalerdquo Journal of Clinical Psychologyvol 51 no 6 pp 768ndash774 1995

[27] G I Bischi L Gardini and U Merlone ldquoPeriodic cycles andbifurcation curves for one-dimensional maps with two discon-tinuitiesrdquo Journal of Dynamical Systems GeometricTheories vol7 no 2 pp 101ndash123 2009

[28] G-I Bischi and U Merlone ldquoBinary choices in small and largegroups a unifiedmodelrdquo Physica A vol 389 no 4 pp 843ndash8532010

[29] A Dal Forno L Gardini and U Merlone ldquoTernary choicesin repeated games and border collision bifurcationsrdquo ChaosSolitons amp Fractals vol 45 no 3 pp 294ndash305 2012

[30] A Dal Forno and U Merlone ldquoBorder-collision bifurcationsin a model of Braess paradoxrdquo Mathematics and Computers inSimulation vol 87 pp 1ndash18 2013

[31] A Dal Forno V Giorgino and U Merlone ldquoAgent-based mod-eling with GTmethodology An examplerdquoWorking Paper 2014

[32] L Gardini U Merlone and F Tramontana ldquoInertia in binarychoices continuity breaking and big-bang bifurcation pointsrdquoJournal of Economic Behavior amp Organization vol 80 no 1 pp153ndash167 2011

[33] W H Sandholm Population Games and Evolutionary Dynam-ics The MIT Press Cambridge Mass USA 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 9

xL

t+1

xL

t

xR

t

00

1

1

(a)

xL

t+1

xL

t

xR

t

00

1

1

(b)

Figure 5 Graphics of the three functions defining (a) 119909119871119905+1

and (b) 119909119877119905+1

with parameters 119886119871= 29 119887

119871= 16 119886

119877= 19 119887

119877= 24 119889 = 8 120575

119871= 05

120575119872= 08 and 120575

119877= 06

a point (119909119871 119909119877) belonging to one of the lines defined in (22)at the boundary of two regions only we have that the twofunctions defined on the two different regions lead to twodifferent points of1198632 An example of how the three functionsdefining 119909119871

119905+1behave is shown in Figure 5(a) and the three

functions defining 119909119877119905+1

are illustrated in Figure 5(b) showingthat F is continuous only in xlowast

The point xlowast in which the three regions are in contactand F is continuous is clearly particular In fact solving theequation 119871(x) = 119872(x) we have the unique solution 119909119871 = 119909lowast119871while solving the equation 119877(x) = 119872(x) we have the uniquesolution119909119877 = 119909lowast119877 It follows that at the point xlowast = (119909lowast119871 119909lowast119877)we have the same cost in each choice as 119871(xlowast) = 119872(xlowast) =119877(xlowast) so that at the point xlowast the commuters are indifferent toany available choice andnone of themwill switch choiceThusF(xlowast) = xlowast and xlowast is a stationary state Under the conditionsin (23) the point xlowast is feasible belonging to 1198632 We can nowprove that this is indeed the unique fixed point of map F

Proposition 3 The point xlowast = (119909lowast119871 119909lowast119877) given in (21) is theunique fixed point of the map F belonging to 1198632 when theconditions in (23) hold

Proof The proof of this proposition follows directly from theobservation that

F (x) = x if and only if 119871 (x) = 119872 (x) = 119877 (x) (31)One way is already proved as we have detected the fixed pointstarting from the solution of 119871(x) = 119872(x) and 119877(x) = 119872(x)showing that it is the unique point xlowast Vice versa let us lookfor the solutions of F(x) = x We have the following systems(where 119909119872 = 1 minus 119909119871 minus 119909119877)

F119871

119909119871= 119909119871+ 120575119871(119877 (x) minus 119871 (x)

1198962

119909119877

+119872(x) minus 119871 (x)

1198964

119909119872)

119909119877= (1 minus 120575

119871

119877 (x) minus 119871 (x)1198962

)119909119877

if x isin 119877119871

F119872

119909119871= (1 minus 120575

119872

119871 (x) minus 119872 (x)119886119871minus 119889

)119909119871

119909119877= (1 minus 120575

119872

119877 (x) minus 119872 (x)119886119877minus 119889

)119909119877

if x isin 119877119872

F119877

119909119871= (1 minus 120575

119877

119871 (x) minus 119877 (x)1198961

)119909119871

119909119877= 119909119877+ 120575119877(119871 (x) minus 119877 (x)

1198961

119909119871

+119872(x) minus 119877 (x)

1198966

119909119872)

if x isin 119877119877

(32)

or equivalently (we recall that 120575119871 120575119872 and 120575

119877are different

from zero)

F119871

119877 (x) minus 119871 (x)1198962

119909119877+119872(x) minus 119871 (x)

1198964

119909119872= 0

119877 (x) minus 119871 (x)1198962

119909119877= 0

if x isin 119877119871

F119872

119871 (x) minus 119872 (x)119886119871minus 119889

119909119871= 0

119877 (x) minus 119872 (x)119886119877minus 119889

119909119877= 0

if x isin 119877119872

F119877

119871 (x) minus 119877 (x)1198961

119909119871= 0

119871 (x) minus 119877 (x)1198961

119909119871+119872(x) minus 119877 (x)

1198966

119909119872= 0

if x isin 119877119877

(33)

Considering the equation F119871(x) = x we see that it is certainly

satisfied for 119909119877 = 0 and 119909119872 = 0 which leads to thecorner point with 119909119871 = 1 However if the fixed point is an

10 Discrete Dynamics in Nature and Society

interior point of 1198632 then the point (1 0) does not belong tothe proper region 119877

119871 that is it is a virtual fixed point Clearly

this equation is satisfied when 119871(x) = 119872(x) and 119871(x) = 119877(x)which occurs only at the point xlowast It is possible to have xlowast =(1 0) and this occurs for (119886

119871minus 119889)119887

119877= 0 and (119886

119877minus 119889)119887

119871= 1

Thus we can state that the only solution of F119871(x) = x in 119877

119871

is given by the fixed point xlowast Similarly we can reason in theother regions

It is worth to mention that particular cases are obtainedwhen the fixed point xlowast belongs to a boundary of the region1198632 For example when 119909lowast119877+119909lowast119871 = 1 and 119909lowast119872 = 0 then only

the two regions119877119871and 119877

119877are involvedThis occurs when the

parameters satisfy

119889 =119886119871119887119871+ 119886119877119887119877minus 119887119871119887119877

119887119877+ 119887119871

0 le119886119871minus 119889

119887119877

le 1

0 le119886119877minus 119889

119887119871

le 1

(34)

From (119886119877minus 119889)119887

119871= (119886119877+ 119887119877minus 119886119871)(119887119877+ 119887119871) we have 119909lowast119877 =

1minus(119886119877minus119889)119887

119871= (119886119871+119887119871minus119886119877)(119887119877+119887119871) = 1198961(119887119877+119887119871) = 119909lowast as

in the one-dimensional map studied in Section 2 and 119909lowast119871 =1 minus 119909lowast119877= 1 minus (119886

119871minus 119889)119887

119877= 1198962(119887119877+ 119887119871) = 1 minus 119896

1(119887119877+ 119887119871) =

1minus119909lowast From themapsF

119871andF119877in (27) and for the parameters

occurring in this case it can be seen that 119909119871119905+1= 1minus119909

119877

119905+1holds

in both so that the dynamics can be studied by using onlythe variable 119909119877

119905and the resulting one-dimensional map is as

follows

119865 (119909119877)

=

119891119871(119909119877)

= 119909119877+ 120575119877

119871 (119909119877)minus119877 (119909

119877)

1198961

(1 minus 119909119877)

if 0 le 119909119877 le 119909lowast

119891119877(119909119877)

= 119909119877minus 120575119871

119877 (119909119877) minus 119871 (119909

119877)

1198962

119909119877 if 119909lowast le 119909119877 le 1

(35)

which corresponds to the system already studied in Section 2As we have seen this fixed point is globally attracting even iflocally we can have the eigenvalue on some side larger than 1in modulus

The case with 119909lowast119871 = 0 so that 0 le 119909lowast119877 le 1 and 119909lowast119872 =1 minus 119909lowast119877 leaves only the two regions 119877

119877and 119877

119872 This occurs

when the parameters satisfy

119886119871minus 119889

119887119877

= 1 0 le119886119877minus 119889

119887119871

le 1 (36)

From the maps F119872

and F119877in (27) it can be seen that 119909119871

119905

converges to119909lowast119871 = 0 so that the resulting asymptotic dynamic

behavior can be studied via the one-dimensional map givenby 119909119877119905+1= 119879(119909

119877

119905) with 119909119877

119905isin [0 1] (and 119909119872

119905= 1 minus 119909

119877

119905) given by

119879 (119909119877)

=

119879119871(119909119877)

= 119909119877+ 120575119877

119872(119909119877) minus 119877 (119909

119877)

1198966

(1 minus 119909119877)

if 0 le 119909119877 le 119909lowast

119879119877(119909119877)

= 119909119877minus 120575119872

119877 (119909119877) minus119872(119909

119877)

119886119877minus 119889

119909119877 if 119909lowast le 119909119877 le 1

(37)

where119872(119909119877) minus 119877(119909119877) = 119886119871+ 119887119871(1 minus 119909

119877) minus 119886119877minus 119887119877

The case with 119909lowast119877 = 0 so that 0 le 119909lowast119871 le 1 and119909lowast119872= 1 minus 119909

lowast119871 leaves only the two regions 119877119871and 119877

119872 This

occurs when the parameters satisfy 0 le (119886119871minus 119889)119887

119877le 1

(119886119877minus 119889)119887

119871= 1 From the maps F

119872and F

119871in (27) it can

be seen that 119909119877119905converges to 119909lowast119877 = 0 so that the resulting

asymptotic dynamic behavior can be studied via the one-dimensional map given by 119909119871

119905+1= 119879(119909

119871

119905) similar to the one

obtained in (37) with obvious changesBy using arguments as in Section 2 it can be shown that

these one-dimensional systems have a unique fixed pointwhich is globally attracting (although it can be locally stableor unstable)

In order to study the local stability of the fixed point ofthe two-dimensional map F we can use the linearization ofthe different definitions in the fixed point which leads to thefollowing Jacobian matrices one for each region 119877

119871 119877119872 and

119877119877 that we denote respectively with 119869

119871 119869119872 and 119869

119877

119869119871(xlowast) = (

1 + 120575119871minus 120575119871

119887119877

119887119871

1198965

1198964

minus 120575119871

119887119877

119887119871

1198966

1198961

120575119871

1198966

1198961

120575119871

119887119877

119887119871

1198966

1198961

1 minus 120575119871

1198966

1198961

)

119869119872(xlowast) = (

1 minus 120575119872

1198964

1198963

0

0 1 minus 120575119872

1198966

1198965

)

119869119877(xlowast) = (

1 minus 120575119877

1198965

1198962

120575119877

119887119871

119887119877

1198965

1198962

120575119877

1198965

1198962

1 + 120575119877minus 120575119877

119887119871

119887119877

1198963

1198966

minus 120575119877

119887119871

119887119877

1198965

1198962

)

(38)

The two real eigenvalues of the Jacobian matrix in 119877119872are

in explicit form and both are smaller than +1 However theycan be larger than minus1 or not Also the eigenvalues of the otherjacobian matrices can be in modulus smaller or larger than1 However the global dynamics observed numerically givealways trajectories which are convergent to the fixed point xlowastat any set of allowed parameters values

Discrete Dynamics in Nature and Society 11

1

1

075

05

025

00 025 05 075

xR

xL

(a)

1

1

075

05

025

00 025 05 075

xR

xL

(b)

Figure 6 Examples of trajectories converging to the unique fixed point with parameters (a) 119886119871= 28 119887

119871= 24 119886

119877= 8 119887

119877= 24 119889 = 5 120575

119871= 04

120575119872= 03 120575

119877= 07 and ic 119909119871 = 08 119909119877 = 005 (b) 119886

119871= 29 119887

119871= 16 119886

119877= 15 119887

119877= 25 119889 = 7 120575

119871= 05 120575

119872= 08 120575

119877= 06 and ic 119909119871 = 005

119909119877= 09 Dots are linked by a continuous line for illustrative purpose only

This global attractivity in the two-dimensional map isdifficult to be proved rigorously however the results in theparticular cases commented above that is when the fixedpoint xlowast belongs to the boundary of 1198632 suggests that thesame result is true also when the fixed point is in the interiorof1198632

In Figure 6 we show some examples in Figure 6(a) wecan see a trajectory starting at 119877

119877 in Figure 6(b) a trajectory

whose initial point in 119877119871is mapped into 119877

119872 The shape of

the functions in the case of Figure 6(b) is similar to the oneshown in Figure 5

4 Conclusion

In this paper we have analyzed a dynamical version of theBraess paradox with nonimpulsive commuters who changeroad proportionally to the cost differenceWe were interestedto prove that with nonimpulsive commuters there exists aunique equilibrium both in the original network with twochoices and in the ternary onewhen a single link is addedWehave shown that both models are quite robust We were ableto provide an analytical proof of the global attractivity of theunique fixed point in the one-dimensional case when onlytwo roads are available andweusednumerical techniques andsimulations to give evidence in the two-dimensional case (iewhen a new road is added) as a rigorous proof can be doneonly for particular border cases

A limitation of our contribution is considering homo-geneous populations in spite of the fact that the evidencesupports heterogeneity Nevertheless heterogeneity of com-mutersrsquo behavior cannot be easily classified yet even when

conducting experiments common individual patterns arejust identified and classified On the other hand whenobserving the aggregate behavior inmany cases the literatureprovides evidence in support of the paradox outcome thatis the convergence to the Nash equilibrium The impulsiveagents in [30] react faster than the proportional agentspresented in this paper yet as a population they fail toachieve the Nash equilibria in certain cases while in thispaper the aggregate behavior does not We proposed thisaggregate behavior because it replies the convergence to theNash equilibria and at the same time it may be the resultof modeling some observed individual behaviors such asimitation and free riding

Taking into account the limitation of homogeneous pop-ulation analysis a natural extension of our contribution isconsidering heterogeneous populations with both impulsiveand nonimpulsive commuters We have reasons to believethat in such a case some properties of the different behaviorsare inherited and some are lost Therefore the results weprovide in this paper match those for impulsive populationsand are paramount to extend the analysis to more realisticcases In factmdashas the empirical results showmdashheterogeneouspopulations dynamics provide a better fit of what has beenobserved when considering human participants interaction

Furthermore it will be interesting to consider general-izations to networks with richer architectures Finally weobserve that since with the class of behavior we considerin this paper that dynamics depends on the shape of thecost functions it will be interesting to extend the analysis tononlinear cost functions

12 Discrete Dynamics in Nature and Society

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

V Avrutin is supported by the European Community withinthe scope of the project ldquoMultiple-discontinuity inducedbifurcations in theory and applicationsrdquo (Marie Curie Actionof the 7th Framework Programme Contract Agreement noPIEF-GA-2011-300281) U Merlone has worked under theauspices of COST Action IS1104 ldquoThe EU in the new complexgeography of economic systems models tools and policyevaluationrdquo

References

[1] R Arnott and K Small ldquoThe economics of traffic congestionrdquoAmerican Scientist vol 82 no 5 pp 446ndash455 1994

[2] T Bass ldquoRoad to ruinrdquo Discover pp 56ndash61 1992[3] B Calvert and G Keady ldquoBraessrsquos paradox and power-law non-

linearities in networksrdquoThe Journal of the AustralianMathemat-ical Society B Applied Mathematics vol 35 no 1 pp 1ndash22 1993

[4] J E Cohen andC Jeffries ldquoCongestion resulting from increasedcapacity in single-server queueing networksrdquo IEEEACMTrans-actions on Networking vol 5 no 2 pp 305ndash310 1997

[5] J E Cohen and F P Kelly ldquoA paradox of congestion in a queuingnetworkrdquo Journal of Applied Probability vol 27 no 3 pp 730ndash734 1990

[6] S Dafermos and A Nagurney ldquoOn some traffic equilibriumtheory paradoxesrdquo Transportation Research B Methodologicalvol 18 no 2 pp 101ndash110 1984

[7] C Fisk ldquoMore paradoxes in the equilibrium assignment prob-lemrdquo Transportation Research B Methodological vol 13 no 4pp 305ndash309 1979

[8] W H K Lam ldquoEffects of road pricing on system performancerdquoTraffic Engineering amp Control vol 29 no 12 pp 631ndash635 1988

[9] J D Murchland ldquoBraessrsquos paradox of traffic flowrdquo Transporta-tion Research vol 4 no 4 pp 391ndash394 1970

[10] R Steinberg and W I Zangwill ldquoThe prevalence o f BraessrsquoparadoxrdquoTransportation Science vol 17 no 3 pp 301ndash318 1983

[11] M Frank ldquoThe Braess paradoxrdquo Mathematical Programmingvol 20 no 1 pp 283ndash302 1981

[12] T Roughgarden Selfish Routing and the Price of Anarchy MITpress 2005

[13] B B Fu C X Zhao and S B Li ldquoThe analysis of Braessrsquo paradoxand robustness based on dynamic traffic assignment modelsrdquoDiscrete Dynamics in Nature and Society vol 2013 Article ID796842 8 pages 2013

[14] C Fisk and S Pallottino ldquoEmpirical evidence for equilibriumparadoxes with implications for optimal planning strategiesrdquoTransportation Research Part A General vol 15 no 3 pp 245ndash248 1981

[15] H Youn M T Gastner and H Jeong ldquoPrice of anarchy intransportation networks efficiency and optimality controlrdquoPhysical Review Letters vol 101 no 12 Article ID 128701 2008

[16] K Aoki Y Ohtsubo A Rapoport and T Saijo ldquoEffects of priorinvestment and personal responsibility in a simple network

gamerdquo Current Research in Social Psychology vol 13 no 2 pp10ndash21 2007

[17] J Morgan H Orzen and M Sefton ldquoNetwork architectureand traffic flows experiments on the Pigou-Knight-Downs andBraess paradoxesrdquoGames and Economic Behavior vol 66 no 1pp 348ndash372 2009

[18] A Rapoport V Mak and R Zwick ldquoNavigating congested net-works with variable demand experimental evidencerdquo Journal ofEconomic Psychology vol 27 no 5 pp 648ndash666 2006

[19] A Rapoport T Kugler S Dugar and E J Gisches ldquoBraessparadox in the laboratory an experimental study of route choicein traffic networks with asymmetric costsrdquo inDecisionModelingand Behavior in Uncertain and Complex Environment T KuglerJ C Smith T Connolly and Y J Son Eds pp 309ndash337Springer New York NY USA 2008

[20] A Rapoport T Kugler S Dugar and E J Gisches ldquoChoice ofroutes in congested traffic networks experimental tests of theBraess paradoxrdquo Games and Economic Behavior vol 65 no 2pp 538ndash571 2009

[21] E J Gisches and A Rapoport ldquoDegrading network capacitymay improve performance private versus public monitoring inthe Braess paradoxrdquoTheory and Decision vol 73 no 2 pp 267ndash293 2012

[22] A Dal Forno and U Merlone ldquoReplicating human interactionin Braess paradoxrdquo in Proceedings of the Simulation Conference(WSC rsquo13) pp 1754ndash1765 IEEE 2013

[23] M Ausloos H Dawid and U Merlone ldquoSpatial interactionsin agent-based modelingrdquo in Complexity and GeographicalEconomics Topics and Tools P Commendatore S S Kayamand I Kubin Eds Springer Berlin Germany 2014

[24] G-I Bischi and U Merlone ldquoGlobal dynamics in binary choicemodels with social influencerdquo The Journal of MathematicalSociology vol 33 no 4 pp 277ndash302 2009

[25] G I Bischi L Gardini and U Merlone ldquoImpulsivity in binarychoices and the emergence of periodicityrdquoDiscrete Dynamics inNature and Society vol 2009 Article ID 407913 22 pages 2009

[26] JH PattonM S Stanford andE S Barratt ldquoFactor structure ofthe Barratt Impulsiveness Scalerdquo Journal of Clinical Psychologyvol 51 no 6 pp 768ndash774 1995

[27] G I Bischi L Gardini and U Merlone ldquoPeriodic cycles andbifurcation curves for one-dimensional maps with two discon-tinuitiesrdquo Journal of Dynamical Systems GeometricTheories vol7 no 2 pp 101ndash123 2009

[28] G-I Bischi and U Merlone ldquoBinary choices in small and largegroups a unifiedmodelrdquo Physica A vol 389 no 4 pp 843ndash8532010

[29] A Dal Forno L Gardini and U Merlone ldquoTernary choicesin repeated games and border collision bifurcationsrdquo ChaosSolitons amp Fractals vol 45 no 3 pp 294ndash305 2012

[30] A Dal Forno and U Merlone ldquoBorder-collision bifurcationsin a model of Braess paradoxrdquo Mathematics and Computers inSimulation vol 87 pp 1ndash18 2013

[31] A Dal Forno V Giorgino and U Merlone ldquoAgent-based mod-eling with GTmethodology An examplerdquoWorking Paper 2014

[32] L Gardini U Merlone and F Tramontana ldquoInertia in binarychoices continuity breaking and big-bang bifurcation pointsrdquoJournal of Economic Behavior amp Organization vol 80 no 1 pp153ndash167 2011

[33] W H Sandholm Population Games and Evolutionary Dynam-ics The MIT Press Cambridge Mass USA 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Discrete Dynamics in Nature and Society

interior point of 1198632 then the point (1 0) does not belong tothe proper region 119877

119871 that is it is a virtual fixed point Clearly

this equation is satisfied when 119871(x) = 119872(x) and 119871(x) = 119877(x)which occurs only at the point xlowast It is possible to have xlowast =(1 0) and this occurs for (119886

119871minus 119889)119887

119877= 0 and (119886

119877minus 119889)119887

119871= 1

Thus we can state that the only solution of F119871(x) = x in 119877

119871

is given by the fixed point xlowast Similarly we can reason in theother regions

It is worth to mention that particular cases are obtainedwhen the fixed point xlowast belongs to a boundary of the region1198632 For example when 119909lowast119877+119909lowast119871 = 1 and 119909lowast119872 = 0 then only

the two regions119877119871and 119877

119877are involvedThis occurs when the

parameters satisfy

119889 =119886119871119887119871+ 119886119877119887119877minus 119887119871119887119877

119887119877+ 119887119871

0 le119886119871minus 119889

119887119877

le 1

0 le119886119877minus 119889

119887119871

le 1

(34)

From (119886119877minus 119889)119887

119871= (119886119877+ 119887119877minus 119886119871)(119887119877+ 119887119871) we have 119909lowast119877 =

1minus(119886119877minus119889)119887

119871= (119886119871+119887119871minus119886119877)(119887119877+119887119871) = 1198961(119887119877+119887119871) = 119909lowast as

in the one-dimensional map studied in Section 2 and 119909lowast119871 =1 minus 119909lowast119877= 1 minus (119886

119871minus 119889)119887

119877= 1198962(119887119877+ 119887119871) = 1 minus 119896

1(119887119877+ 119887119871) =

1minus119909lowast From themapsF

119871andF119877in (27) and for the parameters

occurring in this case it can be seen that 119909119871119905+1= 1minus119909

119877

119905+1holds

in both so that the dynamics can be studied by using onlythe variable 119909119877

119905and the resulting one-dimensional map is as

follows

119865 (119909119877)

=

119891119871(119909119877)

= 119909119877+ 120575119877

119871 (119909119877)minus119877 (119909

119877)

1198961

(1 minus 119909119877)

if 0 le 119909119877 le 119909lowast

119891119877(119909119877)

= 119909119877minus 120575119871

119877 (119909119877) minus 119871 (119909

119877)

1198962

119909119877 if 119909lowast le 119909119877 le 1

(35)

which corresponds to the system already studied in Section 2As we have seen this fixed point is globally attracting even iflocally we can have the eigenvalue on some side larger than 1in modulus

The case with 119909lowast119871 = 0 so that 0 le 119909lowast119877 le 1 and 119909lowast119872 =1 minus 119909lowast119877 leaves only the two regions 119877

119877and 119877

119872 This occurs

when the parameters satisfy

119886119871minus 119889

119887119877

= 1 0 le119886119877minus 119889

119887119871

le 1 (36)

From the maps F119872

and F119877in (27) it can be seen that 119909119871

119905

converges to119909lowast119871 = 0 so that the resulting asymptotic dynamic

behavior can be studied via the one-dimensional map givenby 119909119877119905+1= 119879(119909

119877

119905) with 119909119877

119905isin [0 1] (and 119909119872

119905= 1 minus 119909

119877

119905) given by

119879 (119909119877)

=

119879119871(119909119877)

= 119909119877+ 120575119877

119872(119909119877) minus 119877 (119909

119877)

1198966

(1 minus 119909119877)

if 0 le 119909119877 le 119909lowast

119879119877(119909119877)

= 119909119877minus 120575119872

119877 (119909119877) minus119872(119909

119877)

119886119877minus 119889

119909119877 if 119909lowast le 119909119877 le 1

(37)

where119872(119909119877) minus 119877(119909119877) = 119886119871+ 119887119871(1 minus 119909

119877) minus 119886119877minus 119887119877

The case with 119909lowast119877 = 0 so that 0 le 119909lowast119871 le 1 and119909lowast119872= 1 minus 119909

lowast119871 leaves only the two regions 119877119871and 119877

119872 This

occurs when the parameters satisfy 0 le (119886119871minus 119889)119887

119877le 1

(119886119877minus 119889)119887

119871= 1 From the maps F

119872and F

119871in (27) it can

be seen that 119909119877119905converges to 119909lowast119877 = 0 so that the resulting

asymptotic dynamic behavior can be studied via the one-dimensional map given by 119909119871

119905+1= 119879(119909

119871

119905) similar to the one

obtained in (37) with obvious changesBy using arguments as in Section 2 it can be shown that

these one-dimensional systems have a unique fixed pointwhich is globally attracting (although it can be locally stableor unstable)

In order to study the local stability of the fixed point ofthe two-dimensional map F we can use the linearization ofthe different definitions in the fixed point which leads to thefollowing Jacobian matrices one for each region 119877

119871 119877119872 and

119877119877 that we denote respectively with 119869

119871 119869119872 and 119869

119877

119869119871(xlowast) = (

1 + 120575119871minus 120575119871

119887119877

119887119871

1198965

1198964

minus 120575119871

119887119877

119887119871

1198966

1198961

120575119871

1198966

1198961

120575119871

119887119877

119887119871

1198966

1198961

1 minus 120575119871

1198966

1198961

)

119869119872(xlowast) = (

1 minus 120575119872

1198964

1198963

0

0 1 minus 120575119872

1198966

1198965

)

119869119877(xlowast) = (

1 minus 120575119877

1198965

1198962

120575119877

119887119871

119887119877

1198965

1198962

120575119877

1198965

1198962

1 + 120575119877minus 120575119877

119887119871

119887119877

1198963

1198966

minus 120575119877

119887119871

119887119877

1198965

1198962

)

(38)

The two real eigenvalues of the Jacobian matrix in 119877119872are

in explicit form and both are smaller than +1 However theycan be larger than minus1 or not Also the eigenvalues of the otherjacobian matrices can be in modulus smaller or larger than1 However the global dynamics observed numerically givealways trajectories which are convergent to the fixed point xlowastat any set of allowed parameters values

Discrete Dynamics in Nature and Society 11

1

1

075

05

025

00 025 05 075

xR

xL

(a)

1

1

075

05

025

00 025 05 075

xR

xL

(b)

Figure 6 Examples of trajectories converging to the unique fixed point with parameters (a) 119886119871= 28 119887

119871= 24 119886

119877= 8 119887

119877= 24 119889 = 5 120575

119871= 04

120575119872= 03 120575

119877= 07 and ic 119909119871 = 08 119909119877 = 005 (b) 119886

119871= 29 119887

119871= 16 119886

119877= 15 119887

119877= 25 119889 = 7 120575

119871= 05 120575

119872= 08 120575

119877= 06 and ic 119909119871 = 005

119909119877= 09 Dots are linked by a continuous line for illustrative purpose only

This global attractivity in the two-dimensional map isdifficult to be proved rigorously however the results in theparticular cases commented above that is when the fixedpoint xlowast belongs to the boundary of 1198632 suggests that thesame result is true also when the fixed point is in the interiorof1198632

In Figure 6 we show some examples in Figure 6(a) wecan see a trajectory starting at 119877

119877 in Figure 6(b) a trajectory

whose initial point in 119877119871is mapped into 119877

119872 The shape of

the functions in the case of Figure 6(b) is similar to the oneshown in Figure 5

4 Conclusion

In this paper we have analyzed a dynamical version of theBraess paradox with nonimpulsive commuters who changeroad proportionally to the cost differenceWe were interestedto prove that with nonimpulsive commuters there exists aunique equilibrium both in the original network with twochoices and in the ternary onewhen a single link is addedWehave shown that both models are quite robust We were ableto provide an analytical proof of the global attractivity of theunique fixed point in the one-dimensional case when onlytwo roads are available andweusednumerical techniques andsimulations to give evidence in the two-dimensional case (iewhen a new road is added) as a rigorous proof can be doneonly for particular border cases

A limitation of our contribution is considering homo-geneous populations in spite of the fact that the evidencesupports heterogeneity Nevertheless heterogeneity of com-mutersrsquo behavior cannot be easily classified yet even when

conducting experiments common individual patterns arejust identified and classified On the other hand whenobserving the aggregate behavior inmany cases the literatureprovides evidence in support of the paradox outcome thatis the convergence to the Nash equilibrium The impulsiveagents in [30] react faster than the proportional agentspresented in this paper yet as a population they fail toachieve the Nash equilibria in certain cases while in thispaper the aggregate behavior does not We proposed thisaggregate behavior because it replies the convergence to theNash equilibria and at the same time it may be the resultof modeling some observed individual behaviors such asimitation and free riding

Taking into account the limitation of homogeneous pop-ulation analysis a natural extension of our contribution isconsidering heterogeneous populations with both impulsiveand nonimpulsive commuters We have reasons to believethat in such a case some properties of the different behaviorsare inherited and some are lost Therefore the results weprovide in this paper match those for impulsive populationsand are paramount to extend the analysis to more realisticcases In factmdashas the empirical results showmdashheterogeneouspopulations dynamics provide a better fit of what has beenobserved when considering human participants interaction

Furthermore it will be interesting to consider general-izations to networks with richer architectures Finally weobserve that since with the class of behavior we considerin this paper that dynamics depends on the shape of thecost functions it will be interesting to extend the analysis tononlinear cost functions

12 Discrete Dynamics in Nature and Society

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

V Avrutin is supported by the European Community withinthe scope of the project ldquoMultiple-discontinuity inducedbifurcations in theory and applicationsrdquo (Marie Curie Actionof the 7th Framework Programme Contract Agreement noPIEF-GA-2011-300281) U Merlone has worked under theauspices of COST Action IS1104 ldquoThe EU in the new complexgeography of economic systems models tools and policyevaluationrdquo

References

[1] R Arnott and K Small ldquoThe economics of traffic congestionrdquoAmerican Scientist vol 82 no 5 pp 446ndash455 1994

[2] T Bass ldquoRoad to ruinrdquo Discover pp 56ndash61 1992[3] B Calvert and G Keady ldquoBraessrsquos paradox and power-law non-

linearities in networksrdquoThe Journal of the AustralianMathemat-ical Society B Applied Mathematics vol 35 no 1 pp 1ndash22 1993

[4] J E Cohen andC Jeffries ldquoCongestion resulting from increasedcapacity in single-server queueing networksrdquo IEEEACMTrans-actions on Networking vol 5 no 2 pp 305ndash310 1997

[5] J E Cohen and F P Kelly ldquoA paradox of congestion in a queuingnetworkrdquo Journal of Applied Probability vol 27 no 3 pp 730ndash734 1990

[6] S Dafermos and A Nagurney ldquoOn some traffic equilibriumtheory paradoxesrdquo Transportation Research B Methodologicalvol 18 no 2 pp 101ndash110 1984

[7] C Fisk ldquoMore paradoxes in the equilibrium assignment prob-lemrdquo Transportation Research B Methodological vol 13 no 4pp 305ndash309 1979

[8] W H K Lam ldquoEffects of road pricing on system performancerdquoTraffic Engineering amp Control vol 29 no 12 pp 631ndash635 1988

[9] J D Murchland ldquoBraessrsquos paradox of traffic flowrdquo Transporta-tion Research vol 4 no 4 pp 391ndash394 1970

[10] R Steinberg and W I Zangwill ldquoThe prevalence o f BraessrsquoparadoxrdquoTransportation Science vol 17 no 3 pp 301ndash318 1983

[11] M Frank ldquoThe Braess paradoxrdquo Mathematical Programmingvol 20 no 1 pp 283ndash302 1981

[12] T Roughgarden Selfish Routing and the Price of Anarchy MITpress 2005

[13] B B Fu C X Zhao and S B Li ldquoThe analysis of Braessrsquo paradoxand robustness based on dynamic traffic assignment modelsrdquoDiscrete Dynamics in Nature and Society vol 2013 Article ID796842 8 pages 2013

[14] C Fisk and S Pallottino ldquoEmpirical evidence for equilibriumparadoxes with implications for optimal planning strategiesrdquoTransportation Research Part A General vol 15 no 3 pp 245ndash248 1981

[15] H Youn M T Gastner and H Jeong ldquoPrice of anarchy intransportation networks efficiency and optimality controlrdquoPhysical Review Letters vol 101 no 12 Article ID 128701 2008

[16] K Aoki Y Ohtsubo A Rapoport and T Saijo ldquoEffects of priorinvestment and personal responsibility in a simple network

gamerdquo Current Research in Social Psychology vol 13 no 2 pp10ndash21 2007

[17] J Morgan H Orzen and M Sefton ldquoNetwork architectureand traffic flows experiments on the Pigou-Knight-Downs andBraess paradoxesrdquoGames and Economic Behavior vol 66 no 1pp 348ndash372 2009

[18] A Rapoport V Mak and R Zwick ldquoNavigating congested net-works with variable demand experimental evidencerdquo Journal ofEconomic Psychology vol 27 no 5 pp 648ndash666 2006

[19] A Rapoport T Kugler S Dugar and E J Gisches ldquoBraessparadox in the laboratory an experimental study of route choicein traffic networks with asymmetric costsrdquo inDecisionModelingand Behavior in Uncertain and Complex Environment T KuglerJ C Smith T Connolly and Y J Son Eds pp 309ndash337Springer New York NY USA 2008

[20] A Rapoport T Kugler S Dugar and E J Gisches ldquoChoice ofroutes in congested traffic networks experimental tests of theBraess paradoxrdquo Games and Economic Behavior vol 65 no 2pp 538ndash571 2009

[21] E J Gisches and A Rapoport ldquoDegrading network capacitymay improve performance private versus public monitoring inthe Braess paradoxrdquoTheory and Decision vol 73 no 2 pp 267ndash293 2012

[22] A Dal Forno and U Merlone ldquoReplicating human interactionin Braess paradoxrdquo in Proceedings of the Simulation Conference(WSC rsquo13) pp 1754ndash1765 IEEE 2013

[23] M Ausloos H Dawid and U Merlone ldquoSpatial interactionsin agent-based modelingrdquo in Complexity and GeographicalEconomics Topics and Tools P Commendatore S S Kayamand I Kubin Eds Springer Berlin Germany 2014

[24] G-I Bischi and U Merlone ldquoGlobal dynamics in binary choicemodels with social influencerdquo The Journal of MathematicalSociology vol 33 no 4 pp 277ndash302 2009

[25] G I Bischi L Gardini and U Merlone ldquoImpulsivity in binarychoices and the emergence of periodicityrdquoDiscrete Dynamics inNature and Society vol 2009 Article ID 407913 22 pages 2009

[26] JH PattonM S Stanford andE S Barratt ldquoFactor structure ofthe Barratt Impulsiveness Scalerdquo Journal of Clinical Psychologyvol 51 no 6 pp 768ndash774 1995

[27] G I Bischi L Gardini and U Merlone ldquoPeriodic cycles andbifurcation curves for one-dimensional maps with two discon-tinuitiesrdquo Journal of Dynamical Systems GeometricTheories vol7 no 2 pp 101ndash123 2009

[28] G-I Bischi and U Merlone ldquoBinary choices in small and largegroups a unifiedmodelrdquo Physica A vol 389 no 4 pp 843ndash8532010

[29] A Dal Forno L Gardini and U Merlone ldquoTernary choicesin repeated games and border collision bifurcationsrdquo ChaosSolitons amp Fractals vol 45 no 3 pp 294ndash305 2012

[30] A Dal Forno and U Merlone ldquoBorder-collision bifurcationsin a model of Braess paradoxrdquo Mathematics and Computers inSimulation vol 87 pp 1ndash18 2013

[31] A Dal Forno V Giorgino and U Merlone ldquoAgent-based mod-eling with GTmethodology An examplerdquoWorking Paper 2014

[32] L Gardini U Merlone and F Tramontana ldquoInertia in binarychoices continuity breaking and big-bang bifurcation pointsrdquoJournal of Economic Behavior amp Organization vol 80 no 1 pp153ndash167 2011

[33] W H Sandholm Population Games and Evolutionary Dynam-ics The MIT Press Cambridge Mass USA 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 11

1

1

075

05

025

00 025 05 075

xR

xL

(a)

1

1

075

05

025

00 025 05 075

xR

xL

(b)

Figure 6 Examples of trajectories converging to the unique fixed point with parameters (a) 119886119871= 28 119887

119871= 24 119886

119877= 8 119887

119877= 24 119889 = 5 120575

119871= 04

120575119872= 03 120575

119877= 07 and ic 119909119871 = 08 119909119877 = 005 (b) 119886

119871= 29 119887

119871= 16 119886

119877= 15 119887

119877= 25 119889 = 7 120575

119871= 05 120575

119872= 08 120575

119877= 06 and ic 119909119871 = 005

119909119877= 09 Dots are linked by a continuous line for illustrative purpose only

This global attractivity in the two-dimensional map isdifficult to be proved rigorously however the results in theparticular cases commented above that is when the fixedpoint xlowast belongs to the boundary of 1198632 suggests that thesame result is true also when the fixed point is in the interiorof1198632

In Figure 6 we show some examples in Figure 6(a) wecan see a trajectory starting at 119877

119877 in Figure 6(b) a trajectory

whose initial point in 119877119871is mapped into 119877

119872 The shape of

the functions in the case of Figure 6(b) is similar to the oneshown in Figure 5

4 Conclusion

In this paper we have analyzed a dynamical version of theBraess paradox with nonimpulsive commuters who changeroad proportionally to the cost differenceWe were interestedto prove that with nonimpulsive commuters there exists aunique equilibrium both in the original network with twochoices and in the ternary onewhen a single link is addedWehave shown that both models are quite robust We were ableto provide an analytical proof of the global attractivity of theunique fixed point in the one-dimensional case when onlytwo roads are available andweusednumerical techniques andsimulations to give evidence in the two-dimensional case (iewhen a new road is added) as a rigorous proof can be doneonly for particular border cases

A limitation of our contribution is considering homo-geneous populations in spite of the fact that the evidencesupports heterogeneity Nevertheless heterogeneity of com-mutersrsquo behavior cannot be easily classified yet even when

conducting experiments common individual patterns arejust identified and classified On the other hand whenobserving the aggregate behavior inmany cases the literatureprovides evidence in support of the paradox outcome thatis the convergence to the Nash equilibrium The impulsiveagents in [30] react faster than the proportional agentspresented in this paper yet as a population they fail toachieve the Nash equilibria in certain cases while in thispaper the aggregate behavior does not We proposed thisaggregate behavior because it replies the convergence to theNash equilibria and at the same time it may be the resultof modeling some observed individual behaviors such asimitation and free riding

Taking into account the limitation of homogeneous pop-ulation analysis a natural extension of our contribution isconsidering heterogeneous populations with both impulsiveand nonimpulsive commuters We have reasons to believethat in such a case some properties of the different behaviorsare inherited and some are lost Therefore the results weprovide in this paper match those for impulsive populationsand are paramount to extend the analysis to more realisticcases In factmdashas the empirical results showmdashheterogeneouspopulations dynamics provide a better fit of what has beenobserved when considering human participants interaction

Furthermore it will be interesting to consider general-izations to networks with richer architectures Finally weobserve that since with the class of behavior we considerin this paper that dynamics depends on the shape of thecost functions it will be interesting to extend the analysis tononlinear cost functions

12 Discrete Dynamics in Nature and Society

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

V Avrutin is supported by the European Community withinthe scope of the project ldquoMultiple-discontinuity inducedbifurcations in theory and applicationsrdquo (Marie Curie Actionof the 7th Framework Programme Contract Agreement noPIEF-GA-2011-300281) U Merlone has worked under theauspices of COST Action IS1104 ldquoThe EU in the new complexgeography of economic systems models tools and policyevaluationrdquo

References

[1] R Arnott and K Small ldquoThe economics of traffic congestionrdquoAmerican Scientist vol 82 no 5 pp 446ndash455 1994

[2] T Bass ldquoRoad to ruinrdquo Discover pp 56ndash61 1992[3] B Calvert and G Keady ldquoBraessrsquos paradox and power-law non-

linearities in networksrdquoThe Journal of the AustralianMathemat-ical Society B Applied Mathematics vol 35 no 1 pp 1ndash22 1993

[4] J E Cohen andC Jeffries ldquoCongestion resulting from increasedcapacity in single-server queueing networksrdquo IEEEACMTrans-actions on Networking vol 5 no 2 pp 305ndash310 1997

[5] J E Cohen and F P Kelly ldquoA paradox of congestion in a queuingnetworkrdquo Journal of Applied Probability vol 27 no 3 pp 730ndash734 1990

[6] S Dafermos and A Nagurney ldquoOn some traffic equilibriumtheory paradoxesrdquo Transportation Research B Methodologicalvol 18 no 2 pp 101ndash110 1984

[7] C Fisk ldquoMore paradoxes in the equilibrium assignment prob-lemrdquo Transportation Research B Methodological vol 13 no 4pp 305ndash309 1979

[8] W H K Lam ldquoEffects of road pricing on system performancerdquoTraffic Engineering amp Control vol 29 no 12 pp 631ndash635 1988

[9] J D Murchland ldquoBraessrsquos paradox of traffic flowrdquo Transporta-tion Research vol 4 no 4 pp 391ndash394 1970

[10] R Steinberg and W I Zangwill ldquoThe prevalence o f BraessrsquoparadoxrdquoTransportation Science vol 17 no 3 pp 301ndash318 1983

[11] M Frank ldquoThe Braess paradoxrdquo Mathematical Programmingvol 20 no 1 pp 283ndash302 1981

[12] T Roughgarden Selfish Routing and the Price of Anarchy MITpress 2005

[13] B B Fu C X Zhao and S B Li ldquoThe analysis of Braessrsquo paradoxand robustness based on dynamic traffic assignment modelsrdquoDiscrete Dynamics in Nature and Society vol 2013 Article ID796842 8 pages 2013

[14] C Fisk and S Pallottino ldquoEmpirical evidence for equilibriumparadoxes with implications for optimal planning strategiesrdquoTransportation Research Part A General vol 15 no 3 pp 245ndash248 1981

[15] H Youn M T Gastner and H Jeong ldquoPrice of anarchy intransportation networks efficiency and optimality controlrdquoPhysical Review Letters vol 101 no 12 Article ID 128701 2008

[16] K Aoki Y Ohtsubo A Rapoport and T Saijo ldquoEffects of priorinvestment and personal responsibility in a simple network

gamerdquo Current Research in Social Psychology vol 13 no 2 pp10ndash21 2007

[17] J Morgan H Orzen and M Sefton ldquoNetwork architectureand traffic flows experiments on the Pigou-Knight-Downs andBraess paradoxesrdquoGames and Economic Behavior vol 66 no 1pp 348ndash372 2009

[18] A Rapoport V Mak and R Zwick ldquoNavigating congested net-works with variable demand experimental evidencerdquo Journal ofEconomic Psychology vol 27 no 5 pp 648ndash666 2006

[19] A Rapoport T Kugler S Dugar and E J Gisches ldquoBraessparadox in the laboratory an experimental study of route choicein traffic networks with asymmetric costsrdquo inDecisionModelingand Behavior in Uncertain and Complex Environment T KuglerJ C Smith T Connolly and Y J Son Eds pp 309ndash337Springer New York NY USA 2008

[20] A Rapoport T Kugler S Dugar and E J Gisches ldquoChoice ofroutes in congested traffic networks experimental tests of theBraess paradoxrdquo Games and Economic Behavior vol 65 no 2pp 538ndash571 2009

[21] E J Gisches and A Rapoport ldquoDegrading network capacitymay improve performance private versus public monitoring inthe Braess paradoxrdquoTheory and Decision vol 73 no 2 pp 267ndash293 2012

[22] A Dal Forno and U Merlone ldquoReplicating human interactionin Braess paradoxrdquo in Proceedings of the Simulation Conference(WSC rsquo13) pp 1754ndash1765 IEEE 2013

[23] M Ausloos H Dawid and U Merlone ldquoSpatial interactionsin agent-based modelingrdquo in Complexity and GeographicalEconomics Topics and Tools P Commendatore S S Kayamand I Kubin Eds Springer Berlin Germany 2014

[24] G-I Bischi and U Merlone ldquoGlobal dynamics in binary choicemodels with social influencerdquo The Journal of MathematicalSociology vol 33 no 4 pp 277ndash302 2009

[25] G I Bischi L Gardini and U Merlone ldquoImpulsivity in binarychoices and the emergence of periodicityrdquoDiscrete Dynamics inNature and Society vol 2009 Article ID 407913 22 pages 2009

[26] JH PattonM S Stanford andE S Barratt ldquoFactor structure ofthe Barratt Impulsiveness Scalerdquo Journal of Clinical Psychologyvol 51 no 6 pp 768ndash774 1995

[27] G I Bischi L Gardini and U Merlone ldquoPeriodic cycles andbifurcation curves for one-dimensional maps with two discon-tinuitiesrdquo Journal of Dynamical Systems GeometricTheories vol7 no 2 pp 101ndash123 2009

[28] G-I Bischi and U Merlone ldquoBinary choices in small and largegroups a unifiedmodelrdquo Physica A vol 389 no 4 pp 843ndash8532010

[29] A Dal Forno L Gardini and U Merlone ldquoTernary choicesin repeated games and border collision bifurcationsrdquo ChaosSolitons amp Fractals vol 45 no 3 pp 294ndash305 2012

[30] A Dal Forno and U Merlone ldquoBorder-collision bifurcationsin a model of Braess paradoxrdquo Mathematics and Computers inSimulation vol 87 pp 1ndash18 2013

[31] A Dal Forno V Giorgino and U Merlone ldquoAgent-based mod-eling with GTmethodology An examplerdquoWorking Paper 2014

[32] L Gardini U Merlone and F Tramontana ldquoInertia in binarychoices continuity breaking and big-bang bifurcation pointsrdquoJournal of Economic Behavior amp Organization vol 80 no 1 pp153ndash167 2011

[33] W H Sandholm Population Games and Evolutionary Dynam-ics The MIT Press Cambridge Mass USA 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Discrete Dynamics in Nature and Society

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

V Avrutin is supported by the European Community withinthe scope of the project ldquoMultiple-discontinuity inducedbifurcations in theory and applicationsrdquo (Marie Curie Actionof the 7th Framework Programme Contract Agreement noPIEF-GA-2011-300281) U Merlone has worked under theauspices of COST Action IS1104 ldquoThe EU in the new complexgeography of economic systems models tools and policyevaluationrdquo

References

[1] R Arnott and K Small ldquoThe economics of traffic congestionrdquoAmerican Scientist vol 82 no 5 pp 446ndash455 1994

[2] T Bass ldquoRoad to ruinrdquo Discover pp 56ndash61 1992[3] B Calvert and G Keady ldquoBraessrsquos paradox and power-law non-

linearities in networksrdquoThe Journal of the AustralianMathemat-ical Society B Applied Mathematics vol 35 no 1 pp 1ndash22 1993

[4] J E Cohen andC Jeffries ldquoCongestion resulting from increasedcapacity in single-server queueing networksrdquo IEEEACMTrans-actions on Networking vol 5 no 2 pp 305ndash310 1997

[5] J E Cohen and F P Kelly ldquoA paradox of congestion in a queuingnetworkrdquo Journal of Applied Probability vol 27 no 3 pp 730ndash734 1990

[6] S Dafermos and A Nagurney ldquoOn some traffic equilibriumtheory paradoxesrdquo Transportation Research B Methodologicalvol 18 no 2 pp 101ndash110 1984

[7] C Fisk ldquoMore paradoxes in the equilibrium assignment prob-lemrdquo Transportation Research B Methodological vol 13 no 4pp 305ndash309 1979

[8] W H K Lam ldquoEffects of road pricing on system performancerdquoTraffic Engineering amp Control vol 29 no 12 pp 631ndash635 1988

[9] J D Murchland ldquoBraessrsquos paradox of traffic flowrdquo Transporta-tion Research vol 4 no 4 pp 391ndash394 1970

[10] R Steinberg and W I Zangwill ldquoThe prevalence o f BraessrsquoparadoxrdquoTransportation Science vol 17 no 3 pp 301ndash318 1983

[11] M Frank ldquoThe Braess paradoxrdquo Mathematical Programmingvol 20 no 1 pp 283ndash302 1981

[12] T Roughgarden Selfish Routing and the Price of Anarchy MITpress 2005

[13] B B Fu C X Zhao and S B Li ldquoThe analysis of Braessrsquo paradoxand robustness based on dynamic traffic assignment modelsrdquoDiscrete Dynamics in Nature and Society vol 2013 Article ID796842 8 pages 2013

[14] C Fisk and S Pallottino ldquoEmpirical evidence for equilibriumparadoxes with implications for optimal planning strategiesrdquoTransportation Research Part A General vol 15 no 3 pp 245ndash248 1981

[15] H Youn M T Gastner and H Jeong ldquoPrice of anarchy intransportation networks efficiency and optimality controlrdquoPhysical Review Letters vol 101 no 12 Article ID 128701 2008

[16] K Aoki Y Ohtsubo A Rapoport and T Saijo ldquoEffects of priorinvestment and personal responsibility in a simple network

gamerdquo Current Research in Social Psychology vol 13 no 2 pp10ndash21 2007

[17] J Morgan H Orzen and M Sefton ldquoNetwork architectureand traffic flows experiments on the Pigou-Knight-Downs andBraess paradoxesrdquoGames and Economic Behavior vol 66 no 1pp 348ndash372 2009

[18] A Rapoport V Mak and R Zwick ldquoNavigating congested net-works with variable demand experimental evidencerdquo Journal ofEconomic Psychology vol 27 no 5 pp 648ndash666 2006

[19] A Rapoport T Kugler S Dugar and E J Gisches ldquoBraessparadox in the laboratory an experimental study of route choicein traffic networks with asymmetric costsrdquo inDecisionModelingand Behavior in Uncertain and Complex Environment T KuglerJ C Smith T Connolly and Y J Son Eds pp 309ndash337Springer New York NY USA 2008

[20] A Rapoport T Kugler S Dugar and E J Gisches ldquoChoice ofroutes in congested traffic networks experimental tests of theBraess paradoxrdquo Games and Economic Behavior vol 65 no 2pp 538ndash571 2009

[21] E J Gisches and A Rapoport ldquoDegrading network capacitymay improve performance private versus public monitoring inthe Braess paradoxrdquoTheory and Decision vol 73 no 2 pp 267ndash293 2012

[22] A Dal Forno and U Merlone ldquoReplicating human interactionin Braess paradoxrdquo in Proceedings of the Simulation Conference(WSC rsquo13) pp 1754ndash1765 IEEE 2013

[23] M Ausloos H Dawid and U Merlone ldquoSpatial interactionsin agent-based modelingrdquo in Complexity and GeographicalEconomics Topics and Tools P Commendatore S S Kayamand I Kubin Eds Springer Berlin Germany 2014

[24] G-I Bischi and U Merlone ldquoGlobal dynamics in binary choicemodels with social influencerdquo The Journal of MathematicalSociology vol 33 no 4 pp 277ndash302 2009

[25] G I Bischi L Gardini and U Merlone ldquoImpulsivity in binarychoices and the emergence of periodicityrdquoDiscrete Dynamics inNature and Society vol 2009 Article ID 407913 22 pages 2009

[26] JH PattonM S Stanford andE S Barratt ldquoFactor structure ofthe Barratt Impulsiveness Scalerdquo Journal of Clinical Psychologyvol 51 no 6 pp 768ndash774 1995

[27] G I Bischi L Gardini and U Merlone ldquoPeriodic cycles andbifurcation curves for one-dimensional maps with two discon-tinuitiesrdquo Journal of Dynamical Systems GeometricTheories vol7 no 2 pp 101ndash123 2009

[28] G-I Bischi and U Merlone ldquoBinary choices in small and largegroups a unifiedmodelrdquo Physica A vol 389 no 4 pp 843ndash8532010

[29] A Dal Forno L Gardini and U Merlone ldquoTernary choicesin repeated games and border collision bifurcationsrdquo ChaosSolitons amp Fractals vol 45 no 3 pp 294ndash305 2012

[30] A Dal Forno and U Merlone ldquoBorder-collision bifurcationsin a model of Braess paradoxrdquo Mathematics and Computers inSimulation vol 87 pp 1ndash18 2013

[31] A Dal Forno V Giorgino and U Merlone ldquoAgent-based mod-eling with GTmethodology An examplerdquoWorking Paper 2014

[32] L Gardini U Merlone and F Tramontana ldquoInertia in binarychoices continuity breaking and big-bang bifurcation pointsrdquoJournal of Economic Behavior amp Organization vol 80 no 1 pp153ndash167 2011

[33] W H Sandholm Population Games and Evolutionary Dynam-ics The MIT Press Cambridge Mass USA 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of