Complex Behaviours in binary Choice Models with Global or Local Social Influence

Complex Behaviours in binary Choice Models

with Global or Local Social Influence

Denis Phan (1,2) - Stephane Pajot

(1) CREM CNRS Université de Rennes I

(2) ICI Université de Bretagne Occidentale

AE2006 : A Symposium in Agent-based Computational Methods in Finance, Game Theory and their applications, Aalborg, Denmark,

September 14-15.

This work is part of the project 'ELICCIR' supported by the joint program "Complex Systems in Human and Social Sciences“of the French Ministry of Research and of the CNRS. M.B.G., J-P. N. and D.P. are CNRS members

AE 2006 - [email protected] 214-15 September 2006

Related literature - two kind of heterogeneities

(1) Consumption with externality Social dependence & Bandwagon effect (Veblen, Leibenstein 1950) Telecommunication and network effect

(Rabeneau, Sthal 1974, Curien Gensollen 1987, Rohlf 2002) Becker (1974): “Restaurant Pricing and Other Examples of Social Influences on Price”

Granovetter, Soong (1986) ”Threshold effects of Interpersonal effects in consumer demand” Partial equilibrium approach # Random (walrassian) economies: Föllmer (1974), Horst

(2) Discrete choice with social influence Schelling (1973, 1978), Granovetter (1978), Galam, Gefen, Shapir (1982) Becker (1974), Glaeser, Sacerdote, Scheinkman (1996) Glaeser, Scheinkman (2002)

(3) BDD versus GNP model BDD: (Durlauf 1997, Blume, Durlauf 2001, 2003, Brock Durlauf 2001a, 2001b…) GNP: (Nadal et al. 2003, Gordon et al. 2005; Nadal et al. 2005, Phan, Semeshenko 2006)

BBD: Random Utility Model (Thurstone 1927, Luce 1959) + Quantal Choice Analysis (Luce, Supes 1965, McFadden 1974) : idiosyncratic heterogeneity concerns the random term > “Classic Ising Model with annealed disorder”each alternative random term is i.i.d. double exponential (extreme value type I) distributed > probabilistic choices: the join distribution of choice is logistic

GNP: The heterogeneity concerns the fixed idiosyncratic willingness to pay (IWA) > “Quenched Random Field Ising Model” > deterministic maximization


Outline and motivation

•Plurality of equilibria is a generic propriety of discrete choice models with heterogeneous, idiosyncratic preferences and social influence in case of sufficiently strong social influence, for a large class of mono-modal pdf (Gordon, Nadal, Phan, Semeshenko, 2006)

Part I : Discrete choice with social influence :Agent’s choices and collective outcomeUniqueness vs Multiplicity > hence, coordination plm.

Part II : Multi-Agent Simulation analysis :Finite Size effects; Global or Local interactionsAvalanches et Intermediate equilibrium positions ( due to the irregular discrete distribution of IWA among Agents)

Multiplicity of path-dependant equilibrium positions due to local structures (clusters) Sethna’s hysteresis

I - Discrete choice with social influence :

Agent’s choices and collective outcome.

Uniqueness vs Multiplicity (coordination)


the général framework of the GNP modelheterogeneous (fixed) IWA and social influence

•A population of N agents Each agent i has to make a binary choice : to adopt /buy (i1) or not (i=0)

•Each customer's willingness to adopt/ pay (IWP) is the sum of two terms:

•An idiosyncratic term Hi randomly distributed in the population H : mean value of the distribution

Si = i : deviation with respect to the mean, of pdf f(Si)

•A social influence term: a weighted sum of the choices of other agents

i

ik kki

1J

weight given by agent i to the choices of his neighbours

number of « neighbours » of i « neighbourhood » of i

choice of neighbour k (0 or 1)

i

i ik kki

1H J


Simplifying hypothesis for analytical purpose

• strategic complementarity: making the same choice as the others is advantageous

•homogeneous social influence (Jik=J) :

•global neighbourhood and large N :

• insensitive to fluctuations : single agents cannot influence individually the collective term J

•Agent choice to maximize:

•Where P is an exogenous cost (Price)

ik

ki

J1

Ji

weight of neighbours' choices

fraction of i’s neighbours that adopt

N

1kk

N

)ik(1k

ki N1

1N1

= fraction of buyers

Jik > 0

i i i i iW ( ) S ( ) (H P J )


Idiosyncratic Willingness to pay distribution: hypothesesGordon, Nadal, Phan, Semeshenko (2006) Discrete Choices under Social Influence: Generic

Properties

H1. Modality: f is unimodal, that is it has a unique maximum.H2. Smoothness: f is non zero, continuous, and at least piecewise twice

continuously differentiable inside its support, ]xm, xM[ , where xm and xM may be finite or equal to ( in the latter case f is strictly monotonically decreasing towards zero as x)

H3 Boundedness: the maximum of f, fB (that may be reached at xm or xM if

these numbers are finite), is finite:prototypical cases

A - Unbounded supports: The support of the distribution is the real axis;

• Typical exemple: the logit distribution (we do not assume that the pdf is symmetric)

•Supplementary hypothesis for extreme values:H 4-5 the pdf has a finite mean value and a finite variance

B- Compact supports: the support of the distribution is some interval[xm, xM] with xm and xM finite; the pdf is continuous on [xm, xM] and continuously derivable on ]xm, xM[ with a unique maximum on [xm, xM].


-10 -5 5 10

0.2

0.4

0.6

0.8

1

For a given P, four categories of agents

Ex 1 : logistic pdfNormalization by standard deviationp = P / ; h = H / ; j = J /

22

2e e3.

e e

1 F s

1F s s h p j.

1 exp s

(1) i = 0

sm(1)= p h j >0

(2) i = 1

s(0)= p h < 0

(3) i = 1 for = e

s(e ) = p h j. e = 0

i i i i iw ( ) s ( ) (h p j )

(4) i = 0 for = e


0 0.2 0.4 0.6 0.8 1

0.25

0.5

0.75

1

1.25

1.5

1.75

2

dP ( ) H ,J D

0 0.2 0.4 0.6 0.8 1

0.25

0.5

0.75

1

1.25

1.5

1.75

2

dP ( ) H ,J D

0 0.25 0.5 0.75 10

2

4

6

-2

0

2

4

0 0.25 0.5 0.75 1

Function: D(,J ) f or 0 J 50 0.25 0.5 0.75 1

0

2

4

6

-2

0

2

4

0 0.25 0.5 0.75 1

Function: D(,J ) f or 0 J 5

Uniqueness or multiplicity of equilibrium(s)

•Relationship with the “classical” downward-slopping demand curve (Becker 1974)

JB

J = 3

J = 1

for J {0,1,2,3}

s j

h p

s:

s:

(h s ) p j 0

1- F(-s ) s ( )

D( ) ( ) j

(1)

(1 bis)

(2)

with :

average ex - ante surplus

marginal customer :

Uniqueness :Moderate Social

InfluenceAssumption

Glaeser, Scheinkman, (2002)

Triangular distribution :

Phan, Semeshenko(2006)


Unbounded distributions: the Logistic case

-3 -2 -1 0 1 2 3

0,0

0,1

0,2

0,3

0,4

0,5

2cosh2f

2 4 6 8 10j

-3

-2

-1

0

1

2

hp

under blue low , above red high

single

solution

great

small

Two solutions

customers phase diagramwith Logistic distribution

(Nadal et al. 2005)

h-p

h < p

h > p

jjB

fraction of buyers

0 0.25 0.5 0.75 1

02.5

57.5

10

0

1

2

3

4

5

0 0.25 0.5 0.75 1

02.5

57.5

10

Inverse demand

Nadal J.P., Phan D., Gordon M.B., Vannimenus J. (2005) Quantitative

FinanceGordon M.B. Nadal J.P., Phan D.,

Semeshenko V (2006) Discrete Choicesunder Social Influence: Generic

Properties


Bounded Distributions: triangular cases

0 1 2 3 4 5 6

-4

-2

0

2

=1

0<<1

=0

b

h-p

j

b

-2b

jB 2b 3b

-b/20<<1

customers phase diagram :

distributions:

-2 0 2 4

0.0

0.2

0.4

2/(3b)

h+2bh

xh-b

f(hi)

Asymmetric triangular

²

Symmetric triangular on [-a,+a] (Phan, Semeshenko 2006)

0 1 2 3 4 5 6

-4

-3

-2

-1

0

1

2

3

JJ

1

JB

HC

01

0

0

01

,

0,1

1

a2 max

min

a

a

0 < < 1

= 0 = 0

= 1

= 1

= 1

_ a

2

+

a J

Dmax(, j)

Dmin(, j)

= 0+= 1

+, , +=1 = 0

+

- a

a

hp

jj*jB

fraction of buyers

coexistence of 2 solutions

-2 -1 0 1 20.25

0.3

0.35

0.4

0.45

0.5

symmetric triangular

II – Multi-Agent Simulation analysis :Finite Size effects

& Global or Local interactions


Direct versus indirect adoption,chain effect and avalanche process

Indirect effect of prices: « chain » or « dominoes »

effectVariation in price

( P1 P2 )

Change of agent i

Change of agent k

k t i 1 2

k t i 2 2

H S ( P ) P

H S ( P ) P

i t i 1 2H S ( P ) P

Variation in price

( P1 P2 )

Change of agent i

Change of agent j

Direct effect of prices


Chain effect, avalanches and hysteresis

0

10

20

30

40

50

60

70

80

90

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33

Chronology and sizes of induced adoptions in the avalanche when decrease from

1.2408 to 1.2407

0

200

400

600

800

1000

1200

1400

1 1,1 1,2 1,3 1,4 1,5

First order transition (strong connectivity)

i i i kk

V H J . P

P = H + JP = H

Homogeneous

population: Hi = H i

0

200

400

600

800

1000

1200

1400

1 1,1 1,2 1,3 1,4 1,5

= 5 = 20


Global or Local interactions + FSE

0%

20%

40%

60%

80%

100%

1,9 1,95 2 2,05 2,1

C 0%

20%

40%

60%

80%

100%

0 1 2 3 4

C

Full connectivityOne dimensional, periodic two nearest neighbours

Symmetric Triangular pdf

(Phan, Semeshenko 2006)


Finite Size effects

Symmetric Triangular distribution (Phan, Semeshenko 2006)

Theoretical equilibrium plots with J = 0, 1, JB 2.5, J*, 4

and upstream branch with J= 3, J*, 3.8, 4

0%

20%

40%

60%

80%

100%

1 1,25 1,5 1,75 2

C

•Simulated


Sethna’s inner hystersis

(neighbourhood = 8, H = 1, J = 0.5, = 10)

Sub trajectory : [1,18-1,29]

0

200

400

600

800

1000

1200

1400

1,1 1,15 1,2 1,25 1,3 1,35 1,4

AB

Complex Behaviours in binary Choice Models with Global or Local Social Influence

Documents

Transcript of Complex Behaviours in binary Choice Models with Global or Local Social Influence