Fuzzy Utility and Equilibria

1774 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 34, NO. 4, AUGUST 2004

Fuzzy Utility and EquilibriaPhilippe De Wilde, Senior Member, IEEE

Abstract—A decision maker is frequently confronted with fuzzyconstraints, fuzzy utility maximization, and fuzziness about thestate of competitors. In this paper we present a framework forfuzzy decision-making, using techniques from fuzzy logic, gametheory, and micro-economics. In the first part, we study the ratio-nality of fuzzy choice. We introduce fuzzy constraints, and showthat this can easily be combined with maximizing a fuzzy utility.The second part of the paper analyzes games with uncertaintyabout the state of the competitors. We implement fuzzy Cournotadjustment, define equilibria, and study their stability. Finally, weshow how a play progresses where the players have uncertaintyabout the state of the other players, and about their utility. Fora likely procedure of utility maximization, the equilibria are thesame as for the game without utility maximization.

Index Terms—Decision making, fuzzy abstract economies, fuzzyCournot equilibrium, fuzzy utility, qualitative reasoning.

I. FUZZY CHOICE AND INTELLIGENT AGENTS

E LECTRONIC commerce allows buyers and sellers touse increasingly complicated decision procedures. Large

amounts of information are available, and computers canprocess this to advise the economic agent in the choice ofan alternative. The information gathered, e.g., over the web,is often inconsistent. If it is to be used in decision making,the decision-making process will have to be able to deal withuncertainty. Humans deal with uncertainty in a natural way,via generalization. Automatic procedures use either probabilitytheory or fuzzy logic. We prefer fuzzy logic because it dealswith linguistic variables in a more intuitive way than probabilitytheory. The linguistic variables are classes that have evolvedover time, in a certain application domain, to be effective ingeneralization. Linguistic variables are the method adoptedby humans to indicate choice and preference, except by thoseeducated in game theory or decision analysis. There is nodoubt that the latter two fields provide rational techniques forutility maximization when there are numerical models thatcharacterize the noise precisely. This paper takes linguisticvariables for granted, a fundamental assumption of fuzzy logic,and models preferences and equilibria in this context.

Many intelligent agents aim to capture the preferences of theirhuman owner. If decisions are to be made by a computer agentin e-commerce, it is essential that the computer agent agreeswith the human owner. Should we study psychology before im-plementing a shop-bot? This would create problems, as a psy-

Manuscript received September 26, 2003; revised April 12, 2004. This workwas presented in part at the 2002 IEEE International Conference on Systems,Man and Cybernetics, Hammamet, Tunisia, 2002. This paper was recommendedby Associate Editor A. Gomez Skarmeta.

The author is with the Intelligent and Interactive Systems Group, Departmentof Electrical Engineering, Imperial College London, London SW7 2BT, U.K.(e-mail: [email protected]).

Digital Object Identifier 10.1109/TSMCB.2004.829775

chological analysis of economical behavior returns results thatare difficult to implement in a computer algorithm. Most econ-omists hold that their theory, micro-economics based on gametheory, gives an accurate description of human economical be-havior. They even have applied the micro-economic paradigmto areas such as social interactions, and irrational behavior inhouseholds and firms [1]. For the management of resources, acore economic activity, the micro-economic approach is pre-vailing. This is what e-commerce is mostly about: buying andselling quantifiable resources. If we can allow the quantities tobe fuzzy, e-commerce and e-management of resources will beeven more widely applied than it is now.

E-commerce and e-management of resources can operate au-tomatically using intelligent software agents. To achieve this,we need to re-formulate micro-economy so that it can deal withfuzzy choice and preferences. The Orlovsky choice function isoften used as the basis for fuzzy choice [2], [3]. We will startfrom an entirely different starting point, immediately taking intoaccount prices of resources that affect the choice among alter-natives. Another approach, ranking based on pairwise compar-isons, is described in [4]. Choice among attributes that have mul-tiple attributes is reviewed in [5]. The attributes of our alterna-tives will be the prices of goods in the consumption bundle. Thiswill allow us to have more specific procedures for ranking thanin [4], [5].

Once a basic concept, such as the Orlovsky choice function, isproposed and adopted, scientists usually start refining and gen-eralizing it. This happened to fuzzy choice functions, just as ithappened to Nash equilibrium, expert systems, etc. Much of thecurrent theory about fuzzy choice has become so abstract thatit is impossible to implement in an e-commerce agent. The re-fined theory of choice can certainly be used to model particularuser’s decisions very accurately, but this matching of theory anduser requires extensive human intervention. If the e-commerceagent has to implement fuzzy choice automatically for a largeclass of users, we have to turn back, and use a more intuitivetheory. Kulshreshtha and Shekar [6] have recently attempted topresent an intuitive perspective on fuzzy preference. It becomesclear from this paper that there is an array of possible choicefunctions, with no clear criteria as to which ones to prefer. Thereare even some intuitive contradictions. The authors point out theneed to conduct experiments to find the most appropriate fuzzypreference relations for real life situations. We will not conductexperiments, but consider the crisp theory of preference closerto the application (resource management), before fuzzifying it.

II. TWO WEAK AXIOMS OF FUZZY REVEALED PREFERENCE

We now explain the theory of choice, based on [7], and itsfuzzification, based on [8]. In the Section III, we will propose aradically different fuzzification.

1083-4419/04$20.00 © 2004 IEEE

DE WILDE: FUZZY UTILITY AND EQUILIBRIA 1775

The set of alternatives is called . A preference relationassigns the number 0 or 1 to two alternatives

and , where means that alternative is at least as goodas , i.e., weakly dominates . The strict preference relationis defined by but not . On the same set ofalternatives , the standard fuzzy preference relation

indicates the degree to which is at least as good as, a number between 0 and 1. It is clear that the crisp preference

relation is the limit of the fuzzy preference relation , wherethe degree can only take on values 0 or 1.

Often, more than one alternative is acceptable. It is not pos-sible to implement this via a function; hence the concept ofchoice structure was introduced. A choice structure is denotedby ( , ). Here is a set of nonempty subsets of . An ele-ment is called a budget set. It consists of a number ofalternatives. is a choice rule that assigns a nonempty set ofchosen elements for every budget set . It repre-sents the choice made by the agent of one or more alternativesfrom the set . These alternatives are acceptable alterna-tives to the agent. A fuzzy choice rule assigns a nonempty fuzzyset of chosen elements for every budget set .

A fuzzy subset is defined in the standard way as

(1)

where indicates the membership function of a set . This re-duces to the crisp notion of subset, when the membership func-tions can only take on the values 0 or 1. For this reason the fuzzychoice function is a simple extension of the crisp one, and wewill denote both by ( , ).

A choice structure induces a preference relation called therevealed preference relation defined as follows:

(2)

where we read as “ is revealed as least as good as”, meaning that both alternatives have to be in a budget set,

with the preferred one also in the choice set. Note that the otheralternative can also be in the choice set. It is again possible todefine the strict preference relation

(3)

The revealed fuzzy preference relation is defined as follows(following [8], but with our notation)

(4)

This definition reduces to (2) for membership functions that cantake on only the values 0 and 1, but it is only one amongst manypossibilities. It is even possible to use linguistic variables forthe values of , as in [9]. The linguistic variables have to bedefined by membership functions, that can be related to .

Differences in the literature, and counter-intuitive definitionsstart when one tries to define fuzzy strict preference relations,the fuzzy equivalent of . A fuzzy strict preference relationcan be defined as

(5)

This makes the relation anti-symmetric, as is requiredfor a strict order. In [8], the proposal for a fuzzy strict preferencerelation is

(6)

For this relation , we have to find a definition based onfuzzy choice rules. Remark that in (4) and (6), the maximum istaken over all budget sets containing and , butis independent of the degree to which is in the fuzzy choiceset . It is the function that will be used in the formula-tion of one of the most fundamental axioms of in the theory ofchoice, the weak axiom of revealed preference [10].

The weak axiom of revealed crisp preference states the fol-lowing.

Axiom 1—Crisp Weak Axiom: If for some with ,we have , then for any with , ,

we must also have .This means that if there are two budget sets both containing

two alternatives, and one alternative is chosen in the first set,and the second alternative is chosen in the second set, then thefist alternative must also be chosen in the second set. This isan intuitive way of requiring consistency. In terms of revealedpreferences, this means that, within a given choice structure, if

is revealed at least as good as , then cannot be revealedto be strictly better than . Many authors have been dissatisfiedwith this axiom, and have investigated alternatives. Aizerman[11] studied postulates of Sen and Samuelson in the context ofoptimization. Aleskerov [12] allows for an error in the rational-izability of choice by a utility function. We will study an al-ternative to this via fuzzy utility. Pattanaik and Xu [13], [14]rank opportunity sets, and relate this to freedom of choice, andavenue that we do not pursue in this paper. Bandyopadhyay etal. [15], [16] allow the consumer’s choice to be stochastic, andderive a stochastic counterpart of the weak axiom of revealedpreference. This nonfuzzy approach is promising, and one canlook forward to seeing an application in game theory. We agreewith Barrett, Pattanaik and Salles that “In real life, exact choicesare induced by fuzzy preferences” [17], [18].

The weak axiom of revealed fuzzy preference exists in mul-tiple forms ([8], [17], our notation); an example follows.

Axiom 2—Fuzzy Weak Axiom, Version 1: For all alternativesand , implies .Axiom 3—Fuzzy Weak Axiom, Version 2: For all alternativesand , and for all , implies

.These two fuzzy axioms are not intuitive, and are not general-

izations of the crisp axiom. Rather than searching for other, pref-erence-based choice functions, as suggested in [17], we fuzzifyat the level of budget sets in the next section. Rather than exactchoices induced by fuzzy preferences, we will have choices in-duced by fuzzy budget sets.

III. WEAK PROPERTY FOR REVEALED FUZZY PREFERENCE FOR

BUDGET SETS

Assume an agent manages resources, and has quantitiesof them. The resources have prices .

These prices can also be virtual, for example when networked


agents manage tasks, or when the resources are intangible.However abstract the resources may be, we feel that for agentsin an e-commerce context, they can always be quantified andpriced. As we study multiagent systems with many agents,the market, not a single agent, will determine the price. Theresource vector is denoted by , and the price vector by .The agent has a level of wealth . This can again be virtual orreal, and expresses the buying power or power to recruit otheragents, that an agent has.

The resources that an agent can afford are usually calculatedfrom the Walrasian [7] budget set . As the de-cision making by software agents in e-commerce is exclusivelygoverned by such constraints, we feel that the Walrasian budgetset is the right concept to fuzzify, not the choice function.

We will denote by the degree to which an alternativebelongs to the budget set. The fuzziness arises because

the inequality constraint may only hold to a certaindegree. It is the leniency that your banker shows you, or the de-gree to which a company is willing to break its rules to satisfycustomers. This flexibility is necessary to break deadlocks. Hu-mans show it, and economic software agents have to have it asa feature.

The function depends on the alternatives in a special way.The flexibility has to be a function of the difference ,because this is the budget surplus or budget deficit. This differ-ence shows how much capability the agent still has ,or whether it has exceeded its wealth. The degree will alsodepend directly on the wealth, for this allows us to take into ac-count such facts as that a small deficit is irrelevant if the wealthis large, etc. On the other hand, should not depend directlyon or . This is because is only concerned with wealth, notwith prices per unit, or units of resources, the latter two beingof a different dimension from . Moreover, the prices are set bythe market, but the tolerance for to exceed depends onlyon the individual decision maker.

Two particularly useful functions for are

(7)

and

(8)We will call such membership functions fuzzy budget constraintsor resource constraints interchangeably in the sequel.

Function (7), illustrated in Fig. 1, indicates that all alterna-tives that are in the crisp Walrasian budget set are in the fuzzybudget set to a degree one. This degree then decreases linearlyto 0 until exceeds . The quantity indicatesthe flexibility of the wealth limit . Function (8), illustrated inFig. 2, decreases gradually from near 1 to near 0 as increases,taking the value 1/2 when . The slope indicateshow hard the budget constraint is, with a hard constraint im-plemented via a large .

Walras’ law, for all , can be exactly satisfied by (7),when , but will never be exactly satisfied for (8), where

Fig. 1. A piecewise-linear fuzzy budget set for two goods in quantities x andx , and with prices 3 and 5, respectively. The wealth limit equals 4 and can beexceeded by 1.

Fig. 2. A smooth fuzzy budget set for two goods in quantities x and x , withprices 3 and 5, respectively. The wealth limit 4 can be exceeded to an arbitraryamount.

can only approach 1 asymptotically. Any linear transforma-tion of prices or resource amounts will preserve the mem-bership functions (7) and (8) in the same form. For the samereasons, any hyperplane in the -dimensional -space will cutthe surfaces (7) and (8) according to a piecewise linear or a tanhfunction respectively.

The Walrasian demand correspondence is the amountof goods at prices that can be consumed given wealth .Normally this is a point within the budget set, or on the edge ofthe budget set if Walras’ law is fulfilled. If the budgetset is a fuzzy set , we define the demand correspondence asa fuzzy set with

(9)

Many different amounts of goods can be consumed, each to adifferent degree. If , then cannot beconsumed.

The Walrasian demand correspondence is homoge-neous of degree zero if for any , and


. In the fuzzy version, homogeneity of degree zero be-comes a property of the membership function of the fuzzybudget set. It can easily be seen that (7) and (8) will be a homo-geneous of degree zero if , respectively, are constants,this means that the flexibility on the budget constraint is inde-pendent of the wealth.

The Walrasian demand correspondence is different froma price-wealth situation . The price-wealth situationis simply the actual consumption of goods at prices andwealth , all crisp numbers. There is nothing fuzzy about aprice-wealth situation. The demand correspondence, however,is fuzzy because of the fuzzy budget set. It is the originality ofour approach that we only introduce fuzziness via fuzzy budgetconstraints, and not via fuzzy choice relations.

Now that the resource constraints are formulated as a fuzzybudget set , we are able to formulate a much more intuitivefuzzy weak axiom. The crisp weak axiom of revealed preferencefor a Walrasian demand function is as follows.

Axiom 4—Crisp Weak Axiom: For any and , ifand , then .

It can be shown [7] that this is equivalent to axiom 1. Thefuzzy version can now be obtained in an intuitive way, if themembership function is introduced.

Property 1—Fuzzy Weak Property: For any two fuzzy budgetsets and , andany two price-wealth situations , the fuzzyweak property holds to a degree

(10)

An illustration is given in Fig. 3.It can be shown that if the budget sets are crisp, the fuzzy

weak property always holds to a degree 1, and hence is alwaystrue. So instead of a fuzzification of the crisp weak axiom, aswith axioms 2 and 3 at the end of Section II, we have foundnot an axiom, but a property. This property holds to a certaindegree. If the membership function of the budget set is “flat”,then the ridge in Fig. 3 will not be very pronounced, indicatingindifference in the choice of consumption bundle . Onthe other hand, if drops sharply at , the ridge will beclearly defined.

The fuzzy budget set and the fuzzy weak property have givenus an intuitive approach to fuzzy constraints, without the neces-sity to define fuzzy preference relations. In the next section, wewill show how the fuzzy budget set can be combined with utilitymaximization.

IV. FUZZY-UTILITY MAXIMIZATION UNDER FUZZY RESOURCE

CONSTRAINTS

Once constraints on resources are laid down in a budget set,the next step is to maximize utility, given a budget set. Fuzzyconstraints go together with fuzzy objectives. The latter can bemodeled by a fuzzy utility function. Fuzzy utility functions havebeen introduced via fuzzy random variables [19], in a desire tofind a treatment compatible with Bayesian statistics. Anotherapproach is to use fuzzy numbers in an ordinary utility function[20].

Fig. 3. Two resources, and two budget sets defined by x + x = 1 andx =2+2x = 1. Expression (8) with �(w) = 1 is chosen for the membershipfunctions � of the two fuzzy budget sets. The degree to which the fuzzy weakproperty holds,� (x ; x ), is plotted, together with a projection of the contourlines. The ridge in � coincides with the bisector of x +x = 1 and x =2+2x = 1.

Fuzzy graphs [21] are one of the most intuitive ways to quan-tify linguistic uncertainty, if the membership functions of thevariables that are being related in the graph, are well known.Let the universe of discourse be quantities of the resources.Instead of the numerical values , of the previous sec-tion, the decision maker uses linguistic variables with member-ship functions , , . For example,could be the membership function of a “small” quantity of re-source 2, and the membership function for a “large” quantityof resource 2. The membership functions are functions of realnumbers which we will denote by . The membershipfunctions of the utility (“small utility”, “large utility” etc.) aredenoted by , .

A fuzzy utility can now be defined as a fuzzy graph

(11)

An example is given in Fig. 4. As the resource vector is alsodenoted (and the price vector ), note that depends onand , the latter variable indicating the utility.

If the utility is subject to budget constraints, the member-ship function will be the minimum of the utility and budgetmembership functions, . The utility maximizationproblem consists in choosing the resource allocation or con-sumption bundle that maximizes this minimum

(12)


Fig. 4. Fuzzy utility of a single resource (L = 1). A small amount of resource1 has membership function � , a large amount � , low utility � , high utility� . The x variable is the amount of resource, the x variable quantifies theutility. In most practical applications there will be multiple resources.

This fuzzy resource-constrained utility maximizationproblem is computationally intensive as formulated in (12),because of the succession of maximization and minimizations.Fortunately it is possible to significantly simplify (12). Thishinges on the observation that, for arbitrary numbers , , ,

, ,we have

(13)

The easiest way to prove this is by investigating the 32 differentpossibilities for ranking the variables , , , , . It is pos-sible to generalize (13) to

(14)

Property (14) is important for fuzzy graphs with constraints.In words, it says that when a fuzzy graph is constrained [theminimum with in (14)], this is equivalent to constraining thefuzzy relations (“rectangles”) that make up the fuzzy graph. Wecan now use (14) to simplify (12)

(15)

where the last minimum is taken over membership func-tions: for the resources, for the utility, andfor the budget constraint. This is illustrated for in Fig. 5.

The hyperbolic tangent budget constraint (8) can also be ap-plied for an equivalent price and wealth situation to generateFig. 6.

We can see that the peaks in the fuzzy graphs that representedthe statement “a large amount of resource x has a high utility”become significantly smaller when a budget constraint is ap-plied. Because of this, “a large amount of resource x havinga high utility” belongs to the constrained fuzzy utility set to alesser degree than it does to the unconstrained fuzzy utility set.This is an intuitive result if we consider that all consumers op-erate in a market environment in which commodities have asso-ciated prices and economic agents have personal wealth levels.

Fig. 5. Fuzzy utility of a single resource (L = 1), constrained by the fuzzy

budget set of Fig. 2 and (7). The resource vector xy , here a single variable xy

is that value of x where the fuzzy constrained utility is maximal. Comparethis with Fig. 4: because of the minimization procedure, the second peak hasdisappeared, the remaining peak is lower and has a different slope.

Fig. 6. How fuzzy utility is affected by fuzzy budget constraints.

If we assume that a given consumer makes “rational” deci-sions, we should expect that she would attempt to maximize herutility in some sense. For the unconstrained case with just oneresource, when a consumer has infinite wealth, we should ex-pect to see her choose a large amount of this resource in orderto derive a high utility. However, using the more realistic modelin which the consumer has an associated wealth, we find thatselecting a large amount of the resource realizes a high utilitythat belongs to the constrained fuzzy utility set to a lesser de-gree. In this way, we can view this degree as the likelihood thata high utility will be achieved. For the constrained case it is thenseen, as expected, that choosing a small amount of the resourcerealizes a greater likelihood of a small utility being achieved.Following this theme, we should expect that as the price of theresource increases or the wealth of the consumer decreases, thelikelihood of achieving a high utility becomes smaller. The fol-


Fig. 7. A fuzzy utility reduced by fuzzy budget constraints.

Fig. 8. The same price-wealth combination as in Fig. 7, but for a larger �.

lowing plots show how the constrained fuzzy utility varies fordifferent values of price, wealth, and flexibility of the wealthlimit. Gaussian membership functions are used to represent thelinguistic terms and the budget constraint is given by the hyper-bolic tangent function. The flexibility in (8) and Figs. 7 and8 is inversely proportional to the leniency that is available to theconsumer; a large value represents a harder constraint with lessflexibility.

V. FUZZY COURNOT ADJUSTMENT

Fuzzy Cournot adjustment is a model with players whosestate is a single number. We will limit ourselves here to thecase of two players, as the generalization to multiple players

Fig. 9. Player 2 starting the Cournot adjustment, using its decision functiond to adjust its state x . Player 1 then plays, etc.

is straightforward. One player observes the state of the otherplayer and adjusts her own state according to her decision func-tion. The other player then does the same, and so on, until anequilibrium is reached. The adjustment process can also diverge,and reach a limit cycle or a chaotic attractor. Cournot adjustmentis a dynamical system.

Denote the state of player at time by . The time is dis-crete, . A point on the trajectory of the Cournot ad-justment at time is denoted by . The initial state of thesystem is . The decision function of player is denotedby . It is a function of the state of the other player. If player 1moves first, the points on the trajectory are

(16)

When player 2 moves first, the trajectory is

(17)

This is illustrated in Fig. 9.It is clear that the trajectory only depends on the state of the

player who does not move first. When 1 moves first, for ex-ample, the trajectory only depends on . This affects the na-ture of the attraction basins. When 1 moves first, the attractionbasins are horizontal strips. For players, we find

Property 2: The attraction basins of an -player Cournot ad-justment starting with player are

(18)

where , .An example is shown in Fig. 10.The existence of an attractor depends on the slope of the de-

cision functions of the players. When the decision functions


Fig. 10. A Cournot adjustment with eight attractors. Player 2 starts, hence theattraction basins are vertical strips. The attractors are indicated by diamonds,and the intervals ]x ; x [ of the attraction basins ]x ; x [�] �1;+1[ by segments of black dots approximately at the height (x -value) ofthe attractors. When the segments are at x = 0, there are no attractors in thatvertical strip.

are linear, with slope and , respectively, theCournot adjustment converges when [22]

(19)

A condition that holds locally will also hold globally for lineardecision functions. There is one attractor or none. In the lattercase, the Cournot adjustment diverges. The condition (19) forthe existence of an attractor will still be valid sufficiently closeto the equilibrium in the case of continuous nonlinear decisionfunctions. What we have observed is that, for the same distancebetween the initial point and the attractor, the more moves areneeded to reach the attractor, the more likely condition (19) isviolated, despite the existence of an attractor. Further away froman attractor, the linearization does not hold any more. The readershould compare Figs. 9 and 11 for an example.

In the fuzzy Cournot adjustment, the player does not knowexactly what the state of the other player is. This incompleteinformation can arise for two reasons. The player may not beable to observe the state of the other player very well, or shemay observe the state perfectly but only make decisions basedon a coarse discretization of what she observes. We will rep-resent the incomplete information by a linguistic variable. Thedifferent values this variable can take on are characterized bymembership functions. We will assume that these membershipfunctions do not overlap, although the case of overlapping mem-bership functions can easily be dealt with, at the expense of amore complicated notation. Every players’ decision function,perfectly know to herself, now takes on discrete values, depen-dent on the value of the linguistic variable of the state of theother player. The value of the decision function stays in the do-main of the membership functions. This can be seen in Fig. 12.

Fig. 11. A Cournot adjustment trajectory, with circles indicating the pointswhere condition (19) is violated, despite the existence of an attractor. Thelinearization of the decision functions would not lead to the same attractor inthese points.

Fig. 12. A fuzzy Cournot adjustment. The decision functions d (x ) andd (x ) are step functions. The trajectory is the continuous black line. Thetrajectory leads to the intersection of two segments, and the attractor is thecartesian product of these two segments.

If we denote by the function that is 1 in the in-terval , and 0 elsewhere, and similarly for values of

, the decision functions of the players can be denoted by

(20)where and , respectively, are the number of distinct valuesfor the linguistic variables of the state of player 1 and 2, re-spectively, and and are numbers. For example, if player1 only bases her decisions on whether the state of player 2 islarge, medium, or small, then , and her decision function

will be a step function consisting of three steps.The attractors are not points anymore, but rectangles. If, for

a certain and , the lines andintersect, then all points in the rectangle


Fig. 13. A fuzzy Cournot adjustment. The whole Cartesian product of the twosegments intersecting at the diamond is an attractor.

either belong to an attractor or do not. If, for example, numbersin are denoted by “large”, and numbers inas “small”, then (“large”, “small”) is a fuzzy equilibrium of thefuzzy Cournot adjustment. This means that player 1 keeps herstate at “large”, and player 2 keeps her state at “small”, and thisis consistent with the decision functions of both players. Thisis illustrated in Fig. 12, where the black line of the trajectoryleads to the intersection of two segments, and the attractor is theCartesian product of these two segments.

Not all intersecting segments are attractors. In Fig. 13 theattractors are indicated by a diamond. The whole Cartesianproduct of the two segments intersecting at the diamond is anattractor. In the crisp case, Fig. 13 would show an alternationof stable attractors (diamonds) and unstable points (segmentintersections without diamonds). In the fuzzy case, the samegraph shows a different dynamical behavior. As can be seenin Fig. 12, after an initial move, the trajectory follows thesegments of the piecewise linear decision functions. When itterminates in an attractor, its graph stops at the intersection oftwo segments, but the state of the system can continue to varyin the rectangle formed by the cartesian product of the twointersecting segments. The pair of linguistic variables, however,does not change anymore.

The fuzzy Cournot adjustment can be seen as a discretizationor quantization of the continuous Cournot adjustment. The na-ture of the equilibria is totally different, however, because thecontinuous Cournot adjustment always has points as equilibria.When the granularity of the fuzzy decision functions decreases,the Cartesian product equilibria will have decreasing volume.When the granularity is coarse, the fuzzy problem can have adifferent number of equilibria than the continuous one. An ex-ample is given in Fig. 14.

The number of equilibria in any Cournot adjustment, fuzzy ornot, is less than the number of intersections of the decision func-tions and . This number of intersections is impossible topredict for arbitrary nonlinear decision functions. The step func-tions in the fuzzy Cournot adjustment are nonlinear, hence it isnot possible to give an upper bound for the number of equilibria.

Fig. 14. A fuzzy Cournot adjustment with decision functions d and d thatare discretizations of continuous decision functions.

Fig. 15. Cournot adjustment where the two decision functions have beendisturbed by noise that is uniformly distributed within the bands indicated inthe figure. The trajectory is longer than it would be without the noise, and itcomes close to several equilibria, but does not get attracted to them.

A necessary condition for existence of an equilibrium can bederived from 16. This equation shows that, if player 1 movesfirst, the dynamics are an iteration of the mapping .An equilibrium will exist when this mapping has a fixed point.Banach’s fixed-point theorem [23] gives conditions for this. Thediscrete step functions in the fuzzy Cournot adjustment do notpose a problem for applying Banach’s fixed point theorem, sothat this theorem gives an existence condition for a fuzzy equi-librium.

The fuzzy Cournot decision functions form regions aroundpoint equilibria of deterministic, continuous Cournot adjust-ment. This allows the problem to get out of equilibria withlow stability in a way similar to the addition of noise towiden attraction regions and fuse attraction basins. This is atechnique that has been used in neural networks and simulatedannealing. Fig. 15 shows this for noise bands around thedecision functions.


When there are players, their decision functions are-dimensional stepfunctions. For player , the decision function

is

(21)

where is the number of linguistic variables that are neededby the other players to describe the state of player , the arethe weights that are now assigned not to segments, as in the caseof two players, but to -dimensional hypercubes. A fuzzyequilibrium in this player game is a -dimensional hypercube

where a trajectory ends.

VI. EQUILIBRIA IN ABSTRACT FUZZY ECONOMIES

Fuzzy equilibria have been defined in the context of abstractfuzzy economies, also called generalized fuzzy games [24]. Werepeat their definition here, but slightly simplified and with anadapted notation, so that a comparison with our definition (21)will be possible. Player has a set of alternatives . Thereare players. Denote by the closure of a crisp set, and by

the alpha-cut of a fuzzy set. Every player has a preferencecorrespondence , which is a fuzzy set

defined on . This preference correspondence asso-ciates a fuzzy set on the players’ alternatives with all crisp statesof the other players. Every player has a strategy correspondence

. Every player also has a preference func-tion , which associates a number for everystate of the other players, and a strategy function

.A fuzzy equilibrium of an abstract fuzzy economy is a state

of all players such that

(22)

(23)

Is this the same as the fuzzy equilibrium we have definedfor fuzzy Cournot adjustment, a -dimensional hypercube

, where a trajectory ends? The condition (22)means that, as the game is played, i.e. the strategy correspon-dence is iterated, the equilibrium is a result of this iteration.This is the reason that the existence of such equilibria canbe derived from fixed-point theorems. We do have iterationin our fuzzy Cournot adjustment, and the intersection of thesegments lies in the closure of the attraction basin. Whereour fuzzy equilibrium definition differs from (22) is that wemodel uncertainty about the states of the other players, and(22), using a strategy correspondence, models uncertainty inthe decision of the player. Our model can deal both with aplayer being uncertain about the state of other players, andwith a player who is indifferent about some states of the otherplayers, and expresses her indifference by describing the stateof the other players with linguistic variables. An abstract fuzzyeconomy only deals with the latter.

Another major difference is of course that our fuzzy equilibriaare regions in state space, where the abstract economy equilibriaare points in state space. We feel that when a consensus arisesamong players with fuzzy preferences, the ensuing equilibriumshould be fuzzy, not crisp.

Note that the alpha-cut to the level in (22) merelyserves to indicate that the membership function of the fuzzy set

has to exceed a certain threshold in the strategy correspon-dence.

Condition (23) implies that in the equilibrium, the strategycorrespondence cannot be further iterated while improving thepreferences. In our framework, this would mean a fuzzy Cournotadjustment, while maximizing the utility. Condition (23) canalso be modified to include extra constraints [24].

VII. FUZZY EQUILIBRIA AND FUZZY UTILITY MAXIMIZATION

In this final section, we bring together the notions of fuzzyequilibria, developed in Section V on fuzzy Cournot adjustment,and fuzzy utility maximization, developed in Section IV. Thereare no resource constraints in this part.

As in Cournot adjustment, let us again assume two players1 and 2 whose state is a number and , respectively. Thisstate can be a number specifying the amount of a resource (e.g.,money) that the player possesses. If the game is a board game(e.g., chess), the state can be a number describing the positionof the player’s pieces on the board. In a game such as chess, thenumber of different states of a player’s pieces is of course enor-mous. Naturally, players have developed linguistic variables thatdescribe a class of states—for example, “the queen dominatesthe center”, “the way is open for the bishop”, “the left side isopen”, “the queen is protected by the knight”, etc. These lin-guistic variables can be specified in terms of membership func-tions on the state space. We observe that players of intermediatelevel and above play the game in terms of these linguistic vari-ables, incorporating the state space only for observing the rulesof the game.

Evaluation functions cannot be constructed in terms of thestate space for many games [25]. It is easier to assign utilities tolinguistic variables describing regions of state space.

In Section IV, the were numerical amounts of resource, andwere the membership functions of linguistic variables of

amounts of resource. In Section V, was the state of player, which can be a resource amount, a board position, etc. The

reader should bear in mind that can represent a state in agame, and are membership functions of linguistic vari-ables describing either a quantity of resource (e.g., “small”),or the degree to which state belongs to a linguistic variable(e.g., “the left side is open”). From now on we use to indicatethe state of player in the game. Players are indexed by , andmembership functions by . This is consistent with the utilitymaximization formalism in Section IV, because every player hasone resource, .

As there are two players, they possess resources in amountsand . They each have different fuzzy utilities defined as

fuzzy graphs as in (11)

(24)


Fig. 16. The fuzzy utility of the state x of player 1. A small amount ofresource 1 has membership function � , a large amount � , low utility � ,high utility � . The x variable quantifies the utility. A similar graph existsfor player 2.

and

(25)

where the utility of player 1 depends on the state , andthe utility linguistic variable . Here, is a numericalvariable used to define the utility membership function. It is nota state of a player, just as was not a resource in (11). Foran example, see Fig. 16.

As the players are playing a game using Cournot adjustment,they still have their decision functions and . Theplayers now want to use these, while at the same time max-imizing their utility. The decision function acts as the budgetconstraint in (12), but it has a crisp value. Let us concentrate onplayer 1, and drop the time dependencies to simplify the nota-tion. This player would move its state to , but also has tomaximize her utility .The state has to be biased toward the state where theutility is maximal.

There are an infinity of ways to do this, but we propose thefollowing. The decision function can only take on a dis-crete number of values, see for example Fig. 14. We want to lookfor maximal utility “around” the value that the player1 would normally move to. Call the value that is justsmaller than , and the value just larger. We willdefuzzify the utility between these two values, using the argmaxoperator, and then take the average of this value and forthe player to move to. Hence, is replaced by

(26)

The maximization over is unconstrained, we look at theextreme values of the utility. The meaning of (26) is that aplayer may change her decision in order to maximize utility, butnever by so much that the derivative of the decision function ischanged.

Fig. 17. Fuzzy chess with fuzzy utility maximization. The variable x

represents the position of player 1 on the board, and x that of player 2.Membership functions are defined (but not shown) on the intervals denoted“(dominate) center”, “(attack with) queen”, and “(defend) pawns”. Themembership functions for “queen” can be different for the two players.Membership functions can overlap. Players also have utility functions for theirrespective choices. These utility functions are fuzzy graphs defined in (24)and (25). The trajectory reaches the dashed rectangle (the fuzzy equilibrium),and stays in that rectangle. The trajectory ends when the play ends. This isdetermined by the rules of the play.

The game can now proceed as described in Section V. The tra-jectories look similar as those in Fig. 13, but their corner pointslie off the decision function graph. They always lie within the in-tervals for player 1 and similarly for player2. Now remember from Section V that the attractors are rect-angles. The trajectory with utility maximization (26) will endwithin the same rectangle as the trajectory without utility max-imization because of the restriction on in the utility maxi-mization. Whereas the latter will end on the intersection of thetwo decision functions and , the former will usually end offthe intersection, but within the rectangle. We can conclude thatfuzzy Cournot adjustment with utility maximization (26) has thesame fuzzy equilibria as without.

Let us consider a simple example of a chess game playerby two (very) fuzzy players. Player 1 has only two linguisticvariables to characterize the position of her pieces: dominatethe center, with membership function , and attack withthe queen, with membership function . She attacheshigh utility to dominating the center, and low utility

to attacking with the queen. Player 2 also has onlytwo characterizations of the position of her pieces: defendthe pawns , and attack with the queen . Sheplaces high utility on defending the pawns, and lowutility on attacking with the queen . The decisionfunction of player 1 is so that she dominates the center whenthe other player attacks with the queen, and attacks with thequeen when the other player defends her pawns. Player 2’sdecision function is so that she always defends her pawns. Thisgame has an equilibrium where player 1 attacks with the queen,and player 2 defends her pawns. Any chess player worth hersalt will know where in the state space this play will end. Thisis illustrated in Fig. 17.


Applications of our fuzzy Cournot adjustment are whereverCournot adjustment is used and where players have limited in-formation about the moves of other players, and/or are uncertainabout their own utility assignments. Cournot adjustment is oftenused for resource allocation and price setting in a micro-eco-nomic context. The automatic diary slot allocation described in[26] can be reformulated using fuzzy Cournot adjustment, forexample. The same can be done for the network of traders in[27].

Fuzzy equilibria occur in many board games, e.g., chess. Inaddition to the game with very broad strategy classes introducedabove, we show in Table I an actual game.1 The notation ofthe moves is standard, but before every move we have listed alinguistic variable that the player who is about to move, has usedto make the move. This variable is a fuzzy description of theboard position. It indicates to which degree each player thinksthat she or her opponent dominates the play. The actual valuesare listed in the caption to the table. The game ends in the fuzzyequilibrium where both players agree that black has a winningadvantage. Black wins. This equilibrium is reached abruptly, inmove 21. This is no surprise. If one plays the game, which isnot at a high level, one observes that it only becomes clear atthe very end to both players that black has a winning advantage.

VIII. CONCLUSION

In this paper we have presented a framework for fuzzydecision-making that is different from previous attempts. Wehave studied the rationality of fuzzy choice by introducingfuzzy weak axioms of revealed preference. In Section III weintroduced fuzzy budget sets to model constraints on resources.This led to the fuzzy weak property. The degree (10) to whichthis property holds is a quantification of the consistency offuzzy preferences.

The fuzzy budget set provided the right starting point forfuzzy utility maximization in Section IV. We showed in (15) thata fuzzy budget set can be combined with fuzzy utility maximiza-tion without generating too much computational complexity.

Choice and utility maximization are but one aspect of deci-sion-making. In Section V, we introduced fuzzy games wherethe state of each player is a number, and the players have a de-cision function. We defined fuzzy equilibria and their stability.Fuzzy equilibria are regions in state space. The fuzziness in ourgames was generated by uncertainty about the states of the otherplayers, not by uncertainty about what decision to make. Thelatter approach was explained in Section VI.

Finally, we combined the framework in Sections IV and V toshow how a play progresses where the players have uncertaintyabout the state of the other players, and about their utility. Fora likely procedure of utility maximization, (26), the equilibriawere the same as for the game without utility maximization.

A player or an economic agent can now follow a systematiccourse of action when confronted with fuzzy constraints,fuzzy utility maximization, and fuzziness about the state ofcompetitors. We aim to explore further are whether similarity

1Available [Online] at: http://www.chessdoctor.com/chessdownloads/00 011.4.18.htm

TABLE IA CHESS GAME, WITH BEFORE EVERY

MOVE, AN ANNOTATION OF HOW THE PLAYER WHO IS ABOUT TO MOVE

PERCEIVES HER POSITION. NOTATION: =, EQUAL POSITION; +�, WHITE HAS

A WINNING ADVANTAGE;+=, WHITE HAS A CLEAR ADVANTAGE;+==, WHITE

HAS A SLIGHT ADVANTAGE;�+, BLACK HAS A WINNING ADVANTAGE; =+,BLACK HAS A CLEAR ADVANTAGE; ==+, BLACK HAS A SLIGHT ADVANTAGE.THE FINAL FUZZY EQUILIBRIUM IS (�+,�+) IN 21, WHEN THE GAME ENDS

0-1 WITH A WIN FOR BLACK. THERE ARE INTERMEDIATE FUZZY EQUILIBRIA,e.g. (=, =) IN 1, BUT THE GAME MOVES OUT OF THESE, BECAUSE

THE PLAYERS MAKE MOVES. WHICH MOVES ARE MADE DEPENDS ON

HOW THE PLAYER PERCEIVES HER POSITION

in the membership functions employed by the players affectsthe dynamical behavior of the game. We need to deal moreexplicitly with decision functions for overlapping membershipfunctions. We want to show that this framework easily extendsto games with multiple players. We need to find out whetherthere needs to be consistency of utility functions with decisionfunctions. Does this affect the convergence speed? Furthermore,we think that our fuzzy equilibria are fundamentally differentfrom Bayesian equilibria, but more work is needed. Finally,we would like to investigate evolutionary games with fuzzyuncertainties. Maybe membership functions can be replaced bydrawing strategies from suitable probability distributions.

ACKNOWLEDGMENT

The author would like to thank Prof. L. Zadeh for hospitalityat BISC at theUniversity of California Berkeley, Prof. F. Hop-pensteadt for hospitality at the Center for Systems Science andEngineering Research, Arizona State University, and Prof. G.Weisbuch, who hosted the institute Networks, Dynamics, andSocio-economics at the ICTP Trieste, Italy. Part of the work pre-sented here was done at those locations. Thanks to Dr. J. Mestelfor pointing out chess annotation symbols.

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Philippe De Wilde (M’00–SM’03) is a SeniorLecturer with the Intelligent Systems and NetworksGroup, Department of Electrical and ElectronicEngineering, Imperial College London, London,U.K. His research interests include equilibria offuzzy systems, incomplete information in games,learning in fuzzy games, learning in networks,evolving networks, complexity and repair of couplednetworks, communications and scalability inecosystems. He works to discover biological andsociological principles that can improve the design

of decision making and of networks. He was previously a Research Fellow withBritish Telecom in 1994 and a Laureate with the Royal Academy of Sciences,Letters and Fine Arts of Belgium in 1988.

Dr. De Wilde is an Associate Editorfor the IEEE TRANSACTIONS ON SYSTEMS,MAN, AND CYBERNETICS—PART B: CYBERNETICS. He is a member of IEEESystems, Man and Cybernetics Society and the Society for Industrial and Ap-plied Mathematics (SIAM).

Fuzzy Utility and Equilibria

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