Games in fuzzy environments.

1. Introduction

In a recent issue of this journal, an interesting paper by West and Linster (2003) used fuzzy rules to show that Nash equilibrium behavior can be achieved by boundedly rational agents in two-player games with infinite strategy spaces. These rules are based on the notion of triangular numbers from fuzzy set theory and are posited as "rules of thumb" type behaviors. Updating based on these rules is utilized in the genetic algorithm developed for the simulations in the repeated game. Their most interesting find is that for fuzzy rules using only the most recent histories, play converges to the analytical Nash equilibria of the games considered in the paper. However there is yet no theoretical foundation for such fuzzy rule-based games. This paper provides a theoretical foundation for games based on fuzzy rules by developing a static normal-form fuzzy game in which both payoffs and strategies of players are modeled as fuzzy sets.

The behavior of players in a game depends on the structure of the game being played. This involves the decisions they face and the information they have when making decisions, how their decisions determine the outcome, as well as the preferences they have over the outcomes. The structure also incorporates the possibility of repetition, the implementation of any correlating devices, and alternative forms of communication. Any imprecision regarding the structure of the game has consequences for the outcome. Yet, in the real world, decision making often takes place in an environment in which the objectives, the constraints, and the outcomes faced by the players are not known in a precise manner. Ambiguities can exist if the components of the game are specified with some vagueness or when the players have their own subjective perception of the game.

Psychological games analyzed by Geanakoplous, Pearce, and Staccehetti (1989) and the model of fairness developed by Rabin (1993) are two examples in which the players have their own interpretation of the game. The psychological game is defined on an underlying material game (the standard game that one normally assumes the agents are playing) in which beliefs about reciprocal behavior by the other players generate additional (psychological) payoffs. Chen, Friedman, and Thisse (1997) have a model of boundedly rational behavior in which the players have a latent subconscious utility function and are not precisely aware of the actual utility associated with each outcome. Over time they learn the true nature of their utility, and play converges to the Nash equilibrium.

In this paper we develop a descriptive theory to analyze games with such characteristics using a fuzzy set-theoretic toolkit. We assume that the components of the game involve subjective perception on the part of the players. The model builds on the work of Bellman and Zadeh (1970), who analyze decision making in a fuzzy environment, and extends it to a game-theoretic setting. A fuzzy set differs from a classical set (referred to as a crisp set hereafter) in that the characteristic function can take any value in the interval [0,1]. In this manner it replaces the binary (Aristotelian) logic framework of set theory and incorporates "fuzziness" by appealing to multivalued logic. For instance, a person who is 6 feet tall can have a high membership value (in the characteristic function sense) in the set of "tall people" and a low membership value in the set of "short people." (1) Providing general tools to model such subjective perceptions is one of the main advantages of fuzzy set theory because dual membership instances of this type cannot arise in the context of crisp sets. The underlying motive behind much of fuzzy set theory is that by introducing imprecision of this sort in a formal manner into crisp set theory, we can analyze complex and realistic versions of problems involving information processing and decision making.

In the conventional approach to decision making, a decision process is represented by (i) a set of alternatives, (ii) a set of constraints restricting choices among the different alternatives, (2) and (iii) a performance function that associates with each alternative the gain (or loss) resulting from the choice of that alternative. When decision making occurs in a fuzzy environment characterized by ambiguity and vagueness, Bellman and Zadeh (1970) argue that a different and perhaps more natural conceptual framework suggests itself. They argue that it is not always appropriate to equate imprecision with randomness and provide a distinction between randomness and fuzziness. (3)

Randomness deals with uncertainty concerning nonmembership or membership of an object in a nonfuzzy set. Fuzziness, on the other hand, is concerned with grades of membership in a set, which may take intermediate values between 0 and 1. A fuzzy goal of an agent is a statement like "my payoff should be approximately 50," and a fuzzy constraint may be expressed as "the outcome should lie in the medium range." This is similar to the rules of thumb followed by the players in West and Linster (2003): If the other player produces x I will produce y. Here x and y can even be a range of numbers or linguistic descriptors such as "medium" or "high." The most important feature of this framework is its symmetry with respect to goals and constraints--a symmetry that erases the differences between them and makes it possible to relate in a particularly simple way the concept of decision making to those of the goals and constraints of a decision process.

Our model is similar in spirit to the Bellman and Zadeh (1970) approach and models the standard game as a set of constraints and goals that can then be solved like a decision-making problem while taking the other player's actions into account. The fuzzy extension of the standard game in our framework will have fuzzy payoffs, which represent the goals of the players. We will define a fuzzy extension of the strategies of both players effectively limiting the choices of both players. The equilibrium concept will be identical to the Nash equilibrium, except that it will now be defined on the fuzzy extension of the game. This is unlike the formulation of fuzzy noncooperative games in Butnariu (1978, 1979) and Billot (1992), where the payoff functions are completely absent because they are subsumed into abstract beliefs. The solution in their model imposes very high information requirements on the definition of a game, making the equilibrium unappealing. Moreover, it is also rather cumbersome to translate their model into standard game-theoretic terms. Our formulation is easier to interpret and is closer to the standard model of noncooperative games.

This paper adds to the literature in several ways. We provide an alternative way to model noncooperative games that is more appropriate in situations where there might be a highly subjective component to the game, thereby providing a foundation to the work of West and Linster (2003). We prove the existence of equilibrium in a fuzzy game. We identify conditions to guarantee a minimum level of payoffs in games involving such subjective elements. We also show the existence of equilibrium in the fuzzy version of zero-sum games called one-sum games. A simple duopoly application is presented before suggesting directions for further work. It is also worth mentioning that, given the descriptive nature of the formulation, there is a trade-off in terms of its predictive abilities. Finally, the paper also provides a review of the existing (albeit small) literature on noncooperative fuzzy games.

The next section describes some of the basic concepts of fuzzy set theory. Section 3 provides a review of the existing work on noncooperative fuzzy games. Section 4 presents the model along with a few results. The final section has some concluding remarks.

2. Mathematical Preliminaries

The seminal formulation of the concepts of fuzzy sets is attributed to Zadeh (1965), who generalized the idea of a classical set by extending the range of its characteristic function. (4) Informally, a fuzzy set is a class of objects for which there is no sharp boundary between those objects that belong to the class and those that do not. Here we provide some definitions that are pertinent to our work.

Let X denote a universe of discourse. We distinguish between crisp or traditional and fuzzy subsets of X.

DEFINITION 1. The characteristic function [[PSI].sub.A] of a crisp set A maps the elements of X to the elements of the set {0,1}, i.e., [[PSI].sub.A]: X [right arrow] {0,1}. For each x [member of] X,</p>

<pre> [[PSI].sub.A](x) = {1 if x [member of] A {0 otherwise. </pre> <p>To go from this definition to a fuzzy set we need to expand the set {0,1} to the set [0,1] with 0 and 1 representing the lowest and highest grades of membership (or degree of belongingness), respectively. We introduce two additional definitions about functions before moving to fuzzy sets.

DEFINITION 2. The function f is quasiconcave if and only if, for all x, x' [member of] X and all [lambda] [member of] [0,1] we have

if f(x) [greater than or equal to] f (x') then f((1 - [lambda])x + [lambda] x') [greater than or equal to] f (x').

DEFINITION 3. The function f defined on the convex set X is strictly quasiconcave if and only if for all x, x' [member of] X with x [not equal to] x' and all [lambda] [member of] (0,1) we have

if f(x) [greater than or equal to] f(x') then f((1 - [lambda])x + [lambda] x') > f(x').

In other words, a function is quasiconcave if and only if the line segment joining the points on two level curves lies nowhere below the level curve corresponding to the lower value of the function. A function is strictly quasiconcave if every point, except the endpoints, on any line segment joining points on two level curves yields a higher value for the function than does any point on the level curve corresponding to the lower value of the function. We now introduce the basic mathematics of fuzzy sets.

DEFINITION 4. The membership function [[micro].sub.A] of a fuzzy set A maps the elements of X to the elements of the set [0,1]; i.e., [[micro].sub.A]: X [right arrow] [0,1]. For x [member of] X, [[micro].sub.A] (x) is called the degree of grade of membership.

Membership functions have also been used as belief functions and can be viewed as nonadditive probabilities. For a discussion of these issues see Klir and Yuan (1995) and Billot (1992). The fuzzy set A itself is defined as the graph of [[micro].sub.A]:

a = {(x,y) [member of] X x [0, 1]:y = [[micro].sub.A](x)}.

The sole purpose of this definition is to have something at hand that is literally a set. Ali the properties of fuzzy sets are defined in terms of their membership functions. For example, a fuzzy set A is called normal when [sup.sub.x] [[micro].sub.A] (x) = 1. TO emphasize that, indeed, all the properties of fuzzy sets are actually attributes of their membership functions, suppose that X is a nonempty subset of a Euclidean space. Then A is called convex, if [[micro].sub.A] is quasiconcave. (5) We highlight further some other important definition terms of the membership function.

DEFINITION 5. The fuzzy set B is a subset of the fuzzy set A if and only if

[[micro].sub.B] (x) [less than or equal to] [[micro].sub.A] (x) for all x [member of] X.

DEFINITION 6. The complement of a fuzzy set A is fuzzy set CA with the membership function

[[micro].sub.CA](x) = 1 - [[micro].sub.A](x), x [member of] X.

Elements of X for which [[micro].sub.A] (x) = [[micro].sub.CA] (x) are sometimes referred to using the misleading term "equilibrium points." We now define the basic set-theoretic notions of union and intersection. Let A and B be two fuzzy sets.

DEFINITION 7. The membership function [[micro].sub.F] (x) of the intersection F = A [intersection] B is defined pointwise by [[micro].sub.F] (x) = min{[[micro].sub.A] (x), [[micro].sub.B] (x)}, x [member of] X. Similarly, for the union operation, the membership function of D = A [union] B, we have [[micro].sub.F] (x) = max{[[micro].sub.A] (x), [[micro].sub.B] (x)}, x [member of] X, also defined pointwise.

These definitions of union and intersection in the context of fuzzy sets are from Zadeh (1965). Although alternative formulations of the union and intersection property exist, Bellman and Giertz (1973) prove that this is the most consistent way of defining these operations. They also provide an axiomatic discussion of other standard set-theoretic operations in the context of fuzzy sets. The upper contour sets of a fuzzy set are called [alpha]-cuts and are introduced next.

DEFINITION 8. Let [alpha] [member of] [0,1 ]. The crisp set [A.sub.[alpha]] of elements of X that belong to the fuzzy set A at least to the degree [alpha] is called the [alpha]-cut of the fuzzy set A:

[A.sub.[alpha]]={x [member of] X: [[micro].sub.A](x)[greater than or equal to] [alpha]}.

Moreover, we define the strict ([alpha]-cut [A.sup.*.sub.[alpha]] of A as the crisp set

[A.sup.*.sub.[alpha]]= {x [member of] X: [[micro].sub.A](x)[greater than or equal to] [alpha]}.

In particular, [A.sub.0] = X and [A.sup.*.sub.1] = [empty set]. Also, [A.sup.*.sub.0] is called the support of A or [[micro].sub.A].

We next define the notion of a fuzzified function and the extension principle for a fuzzy function. We say that a crisp function f: X [right arrow] Y is fuzzified when it is extended to act on fuzzy sets defined on X and Y. The fuzzified function for which the same symbol f is used commonly has the formf: F(X) [right arrow] F(Y), where F(X) denotes the fuzzy power set of X (the set of all fuzzy subsets of X). The principle for fuzzifying crisp functions (or crisp relations) is called the extension principle.

DEFINITION 9. The Extension Principle: Any given function f: X [right arrow] Y induces two functions

f: F(X) [right arrow] F(Y) [f.sup.-1]: F(X) [right arrow] F(Y)

defined as [f(A)](y) = [sup.x:y=f(x)] A(x) for all A [member of] F(X), and [[f.sup.-1](B)](x) = B[f (x)] for B [member of] F(Y), respectively.

Detailed expositions of different aspects of fuzzy set theory and their numerous applications can be found in a number of excellent textbooks (see for instance Zimmerman 1991; Klir and Yuan 1995).

3. Review of the Existing Literature

In this section we review alternative approaches to modeling noncooperative fuzzy games. The most prominent theoretical work in this area is the formulation of noncooperative fuzzy games by Butnariu (1978, 1979) and Billot (1992). In the Butnariu-Billot formulation, players have the usual strategies and beliefs about what strategies the other players will choose in the game. These beliefs are described by fuzzy sets over the strategy space of the other players. Players in such a fuzzy game will choose strategies that maximize the membership value of their beliefs about the other players and, while doing this, try to minimize the restrictions they impose on others. They do not explicitly pursue an objective function. However, the equilibrium concept requires very restrictive assumptions, i.e., full information about beliefs, making the formulation quite uninteresting. A detailed description of this model is given by Haller and Sarangi (2000). The interested reader may refer to this paper for a critique and reformulation of the Butnariu-Billot model.

In this section we discuss other approaches to noncooperative fuzzy games as well as some applications. (6) The two other theoretical approaches in the literature only provide techniques to analyze zero-sum games. Campos (1989) uses linear programming to model matrix games, and Billot (1992) uses lexicographic fuzzy preferences to identify equilibria in a normal form game. We also discuss two applications of fuzzy sets to industrial organization. The first, by Greenhut, Greenhut, and Mansur 0995), is an application to modeling a quantity setting oligopoly, and the second application, from Goodhue (1998), analyzes collusion through a fuzzy trigger strategy. We start with the two theoretical approaches followed by the applications.

The Linear Programming Approach

Campos (1989) introduces a number of different types of linear programming (LP) models to solve zero-sum fuzzy normal form games. In his formulation, each player's strategy set is a crisp set, but players have imprecise knowledge about the payoffs. A zero-sum two-person fuzzy game is represented by G = ([S.sub.1], [S.sub.2], [??]), where [S.sub.1] and [S.sub.2] denote the pure strategy sets of the two players. Assume that player 1 is the row player and use i for her strategies, and player 2 is the column player, and hence his strategies will be referred to by j. We assume that player 1 has m strategies and player 2 has n strategies. Then [??] = ([??]) is an m x n matrix of fuzzy numbers, i.e., numbers that lie in the [0,1] interval. The fuzzy numbers are defined by their membership functions as follows:

[[micro].sub.ij] = [??] [right arrow] [0,1], i [member of] [S.sub.1], j [member of] [S.sub.2].

This membership function captures the information that player 1 has about her payoffs and also the information about player 2's payoffs associated with the ith strategy and jth strategy choices by the two players, respectively.

Campos (1989) argues that payoffs need to be represented by fuzzy numbers because in many real-world situations players may not be aware of their exact payoffs. In standard game-theoretic terms, the above operation using the membership function just normalizes the payoffs of each player to the interval [0,1]. However, because the players have imprecise knowledge of their own payoffs, Campos (1989) allows for "soft constraints"; i.e., each player is willing to permit some flexibility in satisfying the constraints. Hence, we can write down player l's problem as

Max v

s.t. [summation over (i)] [??] [greater than or equal to] v, j [member of] [S.sub.2]

[s.sub.i][greater than or equal to] 0, i [member of] [S.sub.1], [SIGMA (i)] [s.sub.i] = 1,

where [greater than or equal to] represents the fuzzy constraint, v represents the security level for player 1, and [s.sub.i] [member of] [S.sub.1]. (7) Notice that the problem now involves double fuzziness because the payoff functions are represented by membership functions and the constraint is also fuzzy. The LP problem in the above form is intractable and needs to be modified further. For this we define [u.sub.i] = [s.sub.i]/v, and thus v = [SIGMA][s.sub.i]/[SIGMA][u.sub.i] = 1/[SIGMA][u.sub.i]. Using this we can restate the original LP problem in terms of its dual as Min[SIGMA][u.sub.i] subject to [[SIGMA].sub.i]][??][u.sub.i] [greater than or equal to] > 1, j [member of] [S.sub.2], whose solution is easier to obtain. The resolution of a fuzzy constraint of the type shown above relies on a technique introduced by Adamo (1980). The fuzzy constraint is now replaced by a convex constraint given by [[SIGMA].sub.i]][??][u.sub.i] [??] 1 - [??](1 - [alpha]), j [member of] [S.sub.2], is a fuzzy number that expresses the maximum violation that player 1 will permit in the accomplishment of his constraint and [??] is the relation the decision maker chooses for ranking the fuzzy numbers. (8)

Fuzzy set theory provides numerous ways of ranking fuzzy numbers. Campos (1989) considers five different ways of ranking fuzzy numbers and for each case rewrites the constraints using fuzzy triangular numbers. Two of these are based on the work of Yager (1981) and involve the use of a ranking function or index that maps the fuzzy numbers onto [??]. A third approach involves the use of [alpha]-cuts and is based on the work of Adamo (1980). The last two approaches rank fuzzy numbers using possibility theory. This stems from the work of Dubois and Prade (1983). Finally, the five different parametric LP models obtained through this transformation process are solved using conventional LP techniques to identify their fuzzy solutions. This exercise is performed with different numerical examples.

The primary limitation of this approach is that it is suited only for linear programming-type formulations, thereby limiting it to zero-sum games. This places great restriction on the payoffs and also the constraints. Moreover, the paper demonstrates its results primarily through specific numerical examples and thus is devoid of generalizations.

A Fuzzy Game with Lexicographic Preferences 

Billot (1992) develops an alternative model of fuzzy games using fuzzy lexicographic preorderings. This rudimentary formulation is applicable only to zero-sum games. Unlike the model originally developed by Butnariu (1978, 1979), this is an ordinal game and differs from the standard game-theoretic formulation only by allowing for fuzzy lexicographic preferences. He introduces an axiom called the Axiom of Local Nondiscrimination according to which a player is assumed to be indifferent between two very close options. However, such indifference is not unique, and its intensity is allowed to take values between 0 and 1. This degree of intensity is captured using a membership function. Next it is shown that under the above axiom, a fuzzy lexicographic preorder can be represented by a continuous utility function defined on a connected referential set X [subset or equal to] [??].

A normal form game is defined as G = [([[SIGMA].sub.i]], [P.sub.i]).sup.n.sub.i=[zeta]] where [[summation of].sub.i] is the strategy space, which is assumed to be a real convex set. An individual strategy is denoted by [[alpha].sub.i] [member of] [[summation of].sub.i], and the strategies of all the other players by [[alpha].sub.-1] [member of] [[SIGMA].sub.-i] = [X.sub.j[not equal to]i] [[summation of].sub.j]. Here [P.sub.i] denotes the payoff of function, [P.sub.i]: [X.sub.j[not equal to]i] [[SIGMA].sub.j] [right arrow] [??] and is assumed to be continuous. Next he introduces a transformation function that orders the strategies lexicographically based on the payoffs they yield. Recall that under the axiom, a fuzzy lexicographic preorder can be represented by a continuous utility function. Because this utility function can now be defined on the set of strategies, Billot calls this a strategic utility function. He then proves an existence result for two-person zero sum games under fairly simple conditions. It is shown that if the axiom is satisfied, the strategy space is compact and convex, and the payoff function is continuous, then an equilibrium will exist. Further, it is shown that for inessential games where the payoffs and the strategies satisfy the conditions listed above, the equilibrium set derived using fuzzy lexicographic preferences contains the usual set of Nash equilibria.

In terms of technicalities Billiot's work is interesting because it proves an existence result, and the equilibria derived using lexicographic preferences contain the usual Nash equilibria. The drawbacks lie in the fact that it is necessary to assume lexicographic preferences and that results are applicable mainly to zero-sum games.

A Fuzzy Approach to Oligopolistic Competition

Greenhut, Greenhut, and Mansur (1995) use fuzzy set theory to model oligopolistic competition. Their objective is to characterize the problem of a real-world oligopolistic market from the perspective of the decision maker in a firm. A firm i may be ranked as a strong or a weak rival by firm j depending on the degree of its inclusion in the oligopoly. For example, in the soft drink industry Coke and Pepsi are the dominant firms, but smaller rivals also exist, and each of the two leading firms may be interested in taking the actions of these smaller rivals into account. The degree of inclusion of these small firms in the oligopoly then quantifies the importance that ought to be given to the actions of the smaller rivals. It is argued that quantification of real world setting in this manner will be of great help to these decision makers.

Greenhut, Greenhut, and Mansur (1995) claim that an oligopoly can be described as competition among a few firms producing similar products. They use the three different fuzzy sets to model the vague (italicized) linguistic terms in the definition of an oligopoly above. (9) Each fuzzy descriptor captures the degree to which a particular firm belongs to the oligopolistic market when compared with a representative firm [F.sub.i] whose membership in the oligopoly is of degree one. The first category is similar products and is used to model the notion that firms do not produce exactly identical products. The membership function expresses how a particular firm's product compares to the product of the representative firm. The fuzzy set [S.sup.*] contains the membership value of each firm in the industry vis-a-vis product similarity.

The next aspect of oligopolies that is modeled in the paper is the degree of interdependence between firms. This is denoted by the fuzzy set 1", which is the fuzzy set of firms whose membership grades represent the degree of perceived interdependence between a firm and the firm [F.sub.i], quantifying the degree of strategic rivalry between firms.

The third category mentioned in their formulation is the notion of a few firms. The fuzzy set [F.sup.*] denotes the fuzzy membership of firms in the industry where a degree of membership is assigned to the discrete numbers belonging to N. The authors regard the number of firms in the industry to be inexact by appealing to the possibility of free entry and exit, and the fact that geographic boundaries between competing firms are not well defined.

The oligopoly itself is denoted by the fuzzy set [O.sup.*], which is a combination of [S.sup.*], [I.sup.*], and [F.sup.*]. It now expresses the fact that an oligopoly involves competition among a few interdependent firms producing similar products. The degree of membership of any particular firm in [O.sup.*] is obtained by applying Zadeh's Extension Principle. The authors illustrate their point by means of a numerical example. Using numerical examples they also show how fuzzy set theory can be used to compute a fuzzy Herfindahl Index.

Although the approach suggested by Greenhut, Greenhut, and Mansur (1995) is interesting, it does not provide satisfactory answers to basic oligopoly questions. The membership grades used in their examples are completely subjective and arguably arbitrary. The authors argue that though they use subjective membership functions, accepting the possibility of a fuzzy model will allow us to develop more realistic oligopoly models in conjunction with econometric techniques, which may be used to obtain membership functions. Further, although it may be hard to quibble with the idea of using a fuzzy set to model product homogeneity, the last two fuzzy categories used in defining an oligopoly, namely the notion of interdependent firms and a few firms, is clearly debatable. Interdependence among firms is usually not a subjective issue, and the notion of a few firms based on geography is more appropriate for retail stores. In a certain sense, the paper also fails to deliver in that it does not suggest how to solve a quantity-setting or a price-setting game between firms after computing the degree of inclusion of each firm in the oligopoly. This clearly remains an open research question, and West and Linster (2003) make a beginning in this direction. Despite these shortcomings, the paper has practical value because it can help a decision maker in formulating action plans. The paper concludes on a more philosophical note, claiming that fuzzy modeling opens up a host of possibilities despite its subjective elements.

Fuzzy Trigger Strategies

Goodhue (1998) applies fuzzy set theory to model collusive behavior. She examines the Green and Porter (1984) model by assuming that firms can use fuzzy trigger strategies. Prices are expressed as fuzzy sets. There are a finite number (I) of fuzzy sets denoted by [P.sub.i] describing the level of prices in linguistic terms. For example, "low prices" denotes one such set. The degree of membership of a price in any particular set captures the extent to which it possesses as the properties associated with that set. Uncertainty that firms face regarding the realization of demand is also modeled as a fuzzy set. One example of such a set is the set that expresses the fact that "demand is low." There are J such sets, each denoted by [D.sub.j]. The chance of cheating in this model is defined on these two sets, which is made possible through the application of the Extension Principle. Interestingly, she finds that the fuzzy trigger pricing game reverses the standard cyclic price war prediction. Collusion-sustaining price wars are most likely to occur during times of high demand. The fuzzy model also predicts that markets with relatively volatile prices are more likely to undergo collusion-sustaining price wars. The paper has some practical merits because it provides a decision maker with a set of tools that will help understand price wars and possible collusion. A drawback is the fact that it is a model of fuzzy trigger strategies and not fuzzy games themselves.

On the Relationship to the Existing Literature

Our goal in this paper is to provide an alternative way of modeling fuzzy games by extending the decision theory framework of Bellman and Zadeh (1970). This is done with the objective of providing a theoretical basis to the work of West and Linster (2003). The generality of our model is an improvement over the linear programming-type formulations, and we do provide results for zero-sum games without requiring restrictions on preferences. We believe that our model goes beyond the Greenhut, Greenhut, and Mansur (1995) framework because it does provide an equilibrium concept. Although Goodhue's (1998) formulation is practical, it is quite specific, being designed to provide a fuzzy extension of Green and Porter's (1984) work. Our model is more general and therefore does not provide predictions as precise as Goodhue's (1998) and should be viewed as a first step toward developing more general fuzzy game-theoretic models.

4. The Model

The model developed here uses the Bellman and Zadeh (1970) approach to fuzzy decision making. We start with a standard normal-form game and then proceed to fuzzify it. Let G = (N, S, II) be the triple that defines a standard normal form game where N= {1,2} is the set of players in the game. For each player i we denote her strategies by [S.sub.i] and a particular strategy chosen from this set by [s.sub.i]. A strategy profile is denoted by ([s.sub.1], [s.sub.2]) = s [member of] S where S = [S.sub.1] x [S.sub.2]. Each player's payoff function is denoted by [II.sub.i]. S[right arrow]R.

Because the environment in which the game is being played is complex, the game that is actually played may vary from the standard game described above. The players create their own fuzzy version of the game, which can be attributed to different factors like their own subjective perception of the game, presence of vaguely defined concepts, or even boundedly rational behavior. Recall, for instance, the idea of the subconscious utility function explored by Chen, Friedman, and Thisse (1997) where the players have only a vague notion of their actual utility function. Bacharach's (1993) Variable Frame Theory is also similar in the sense that different games are associated with different variable universes and lead to a different focal point in each associated game. Similarly in the environment of West and Linster (2003) the Coumot game is computationally complex. See also the reason cited by Campos (1989) as to why a fuzzy game might be appropriate in many situations. Thus, a fuzzy game provides a way of understanding the player's modus operandi in complex game-theoretic conditions.

We will now define a fuzzy version of this game. For each player i the constraint set is given by

[[mu].sub.i]: [S.sub.1] [right arrow] [0, 1].

Note that [[mu].sub.i] represents a player's perceived or fuzzy constraints. The constraints vary in their degree of feasibility, and only some of the constraints might be considered completely feasible, that is, have a membership value of one. This acts as a constraint on his choice of strategies, It stems from his own understanding of the game as well from his beliefs about the other player. It might capture, for instance, player i's belief about the other player's type or about her rationality. It can also be used to eliminate dominated strategies. We will call this a perception constraint, which in this simple case is assumed to be entirely static and nonadaptable.

We next define for each player a nonempty goal function

[[gamma].sub.i] : S [right arrow] [0, 1].

This represents each player's aspiration level, i.e., the player's fuzzy goal function. Although this function is defined over the action space, it could also be defined over the payoff space by considering a mapping from the action space to the payoff space, which is then mapped onto the unit interval. (10) This fuzzy membership function could be used to capture some alternatives to utility maximization or like altruistic behavior or satisficing. The usual normal form game is now replaced by a modified game in a fuzzy environment, which we will call a "fuzzy game." This fuzzy game is formally expressed by the tripe [G.sup.f] = (N, [mu], [gamma]).

Discussion 1

On the interpretation of [mu] and [gamma]

We now provide a brief discussion of a fuzzy game and its constituent elements. The aim of this paper has been to develop a fuzzy version of a standard game when the players are faced with a complex environment as in the implementation of the Cournot game in the West and Linster (2003) formulation. The perception constraint la can be viewed as an expression of the player's rationality. Sometimes it may be possible to specify all the available strategies in an abstract way, but the resulting set may be too large to be evaluated exhaustively. For example, if the problem has a combinatoric explosion of possibilities or a continuum of strategies, then players might not evaluate all the possible strategies because they do not consider all of them feasible. Moreover, it can also be used to capture a player's boundedly rational behavior or a player's beliefs about other players. For example, a set of closed and consistent beliefs can be used to define a curb set (Basu and Weibull, 1991), restricting choices only to a particular subset of available strategies. Iterated elimination of strictly or weakly dominated strategies can be captured by assigning successively lower values to dominated strategies. For example, this would accommodate Basu's (1994) explanation of why subjects will not play the Nash equilibrium in a Traveler's Dilemma. More interestingly, it could also explain behavior in the Traveler's Dilemma as the punishment-reward structure varies as in Capra et al. (1999). Other refinement criteria could also be captured in a similar way. Note also that [[mu].sub.i] must be nonempty; otherwise the player does not think that there are any feasible strategies to choose from. (11)

The goal function [gamma] is a way of describing the player's perception of his objectives. Note that the goal function need not alter the payoff rankings, and this is the most likely outcome in a game. However, it could be used to model some alternatives to utility maximization or combine different objectives. It could also be used to model fairness of the type suggested by Rabin (1993) because the goal function reorders the payoff function. In Rabin's formulation, players get more or less utility in addition to what they get from the payoff function of the original game, depending on whether they feel their opponent is being nice or mean to them. Similarly, this methodology of reordering payoffs can also be extended to other psychological games (Geanakoplos, Pearce, and Stacchetti 1989). Thus, the two membership functions defined above are quite general and can embody a whole range of possibilities. They can be used to explore sophistication in the players' reasoning, and the goal function can capture elements of psychological games and formulations of the sort suggested by behavioral game theory.

Using the two notions developed above, we can now determine the player's decision set, which, in the words of Bellman and Zadeh (1970, pg. B 149), is the "... confluence of goals and constraints," defined by [[delta].sub.i]: S [right arrow] [0,1] with [[delta].sub.i](s) = min {[[mu].sub.i] ([s.sub.i]), [[Gamma].sub.i] ([s.sub.i])}. As can be easily deduced, [[delta].sub.i] is basically the intersection of the set of goals and the constraints facing a player. (12) We now provide an explanation for how to interpret [delta] and when it is useful.

Discussion 2

On the interpretation of [delta]

The player's decision set [delta] can be interpreted as follows: For the particular strategy choice [s.sub.2] [member of] [S.sub.2] of player 2, [[delta].sub.i] (., [s.sub.2]) represents player 1 's response to this strategy using the Bellman and Zadeh approach. This means that player 1 must formulate [[delta].sub.i] and [[mu].sub.i] accordingly, that is, player 1 must follow the above rule when computing his goal function. Essentially it says that for any strategy profile each player first requires that his constraints and goals be satisfied simultaneously. Moreover, at this juncture the player chooses the one that is the easiest to satisfy. Thus, the decision set expresses the degree of compatibility between a player's perceptions and his goals. This facilitates decision making on the part of the player.

It should be obvious that this approach does not have any advantage in "simple games," by which we mean games whose structure is so transparent that the components would remain unchanged even in a fuzzy environment. Because it imposes a symmetry between the goals and the constraints, it would be useful in games that involve a large number of strategies and require sophistication in the reasoning process, or in games with multiple equilibria. It is perhaps more of a heuristic way of looking at a game when computing the equilibrium might be difficult, a claim echoed by West and Linster (2003). By putting the strategies and payoffs on a common platform and ensuring that they are simultaneously satisfied, one can argue that it makes it easier to solve such a game. Indeed, as West and Linster (2003) argue, there is no obvious way to model the actions of players in a repeated game, and one solution might be to use fuzzy rules for action choice.

In the standard game each strategy profile leads to an outcome over which players have references. In a sense our fuzzy version of a game allows players to rank their strategy choices based on their model of the other player's behavior. Players can also have preferences over the outcomes that capture their own objectives, including such notions as fairness, as suggested by Rabin (1993). Finally, players attempt to choose the strategy that maximizes their payoff given their beliefs about the other players. This makes [delta] the confluence of goals and constraints, the natural variable to maximize. In other words, a fuzzy version of a standard game is a simple way to model boundedly rational behavior and, as suggested above, may be viewed as a heuristic way of looking at some games. In such games choosing to maximize the payoff function may lead to a suboptimal situation because it would fail to take into account the rationality of the other player.

We now proceed to define a Nash equilibrium in the fuzzy version of the game, after which we identify conditions for its existence.

DEFINITION 10. A strategy tuple ([s.sup.*.sub.1], [s.sup.*.sub.2]) is a Nash equilibrium in [G.sup.f] if, for all i [member of] N, we have

[[delta].sub.i](s.sup.*) [greater than or equal to] [[delta].sub.i]([s'.sub.i], [s.sup.*.sub.-i]) for all [s'.sub.i] [member of] [S.sub.i].

Under certain standard conditions on the membership functions, we now show that such equilibrium will always exist for the fuzzy extension [G.sup.f] of the game defined above. We will first assume the following.

ASSUMPTION: For all i [member of] N we assume that [S.sub.i] is compact and convex and the payoffs are continuous.

THEOREM 1. For a game [G.sup.f]=(N,[mu],[gamma]), if [[delta].sub.i] is nonempty, continuous, and strictly quasiconcave in a player's own strategies, then [G.sup.f] has at least one Nash equilibrium.

PROOF. For each player i [member of] N, define the best response function for i as [r.sub.i]: S [right arrow] [S.sub.i] as follows

for all s [member of] S, [r.sub.i](s) = arg max [[delta].sub.i]([t.sub.i], [s.sub.-i]).

From the above conditions it is obvious that such an [r.sub.i] (s) must exist and is unique. We also define the best response function r: S [right arrow] S if for all s [member of] S, r(s) = [r.sub.1](s), [r.sub.2](s)]. Because [S.sub.i] is compact and convex for all i, it follows that S is compact and convex. Now through contradiction we will show that r is continuous by showing that [r.sub.i] (s) is continuous for all i.

Suppose not. Then there exists s [member of] S, and a sequence {[s.sup.n]} in S such that [s.sup.n] [right arrow] s, but [r.sub.i]([s.sup.n]) does not converge to [r.sub.i](s). This and the compactness of S implies that there is a subsequence that converges to [t.sub.i] [not equal to] [r.sub.i] (s). Without loss of generality, suppose that {[s.sup.n]} itself converges to [t.sub.i]. Because [[delta].sub.I][[s.sup.n] \ [r.sub.i] ([s.sup.n])] [greater than or equal to] [[delta].sub.I][[s.sup.n] \ [r.sub.i] (s)] for all n, it follows from the continuity of [[delta].sub.i] that [[delta].sub.i]([t.sub.i], [s.sub.-i]) [greater than or equal to] [[delta].sub.i][r.sub.i](s), [s.sub.-I]. This is a contradiction because [r.sub.i](s) is the unique best response of player i to s.

Because r is continuous and S is compact and convex, we know by Brouwer's fixed-point theorem that there exists [s.sup.*] [member of] S, such that r([s.sup.*]) = [s.sup.*]. Thus, for all i [member of] N, [[delta].sub.i]([s.sup.*]) [greater than or equal to] [[delta].sub.i]([s.sub.i], [s.sup.*.sub.-i] for all [s.sub.i] [member of] [s.sub.i]. So [s.sup.*] is a Nash equilibrium. QED.

The fuzzy set theoretic formulation here allows us to compare the tension between the player's aspirations and constraints by assigning numerical values to strategies and payoffs in the interval [0,1]. We believe that this is in fact the most appealing feature of the version of fuzzy games developed in our paper.

We now investigate an issue that arises quite naturally in such a game. Assume that a player has a given goal function. We will now identify conditions on his strategies that will enable him to ensure a certain fixed level of payoff [[gamma].sub.0]. In other words, this is similar to the concept of security level in a zero-sum game. For this purpose we assume that the players adopt a cautious approach and follow a maximin type of reasoning. For each player define the following number

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This number defines the maximum payoff a cautious player can ensure for herself. Note also that a low [C.sub.i] implies that there is a big gap between a player's aspirations and her feasible choices. Next we also define the [alpha]-cut of the set [S.sub.i] as [S.sup.[[gamma].sub.0.].sub.i] = {[s.sub.i] [member of] [[mu].sub.i] ([s.sub.i]) [greater than or equal to] [[gamma].sub.0]}. Also let [S.sub.i]([[gamma].sub.0]) = {[s.sub.i] [member of][S.sub.i]: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [[gamma].sub.i](s) [greater than or equal to] [[gamma].sub.0]} [subset or equal to] [S.sub.i].

PROPOSITION 1. If [S.sub.i]([[gamma].sub.0]) [not equal to] [PHI] and [S.sub.i]([[gamma].sub.0]) [intersection] [S.sup.[[gamma].sub.0].sub.i] [not equal to] [PHI], [C.sub.i] [greater than or equal to] [[gamma].sub.0].

PROOF. If [S.sub.i]([[gamma].sub.0]) [not equal to] [PHI], then it is possible that [C.sub.i] < [[gamma].sub.0], and hence the player cannot always guarantee the desired payoff. If [S.sub.i]([[gamma].sub.0]) [not equal to] [PHI] and if [[mu].sub.i([s.sub.i]) [greater than or equal to] [[gamma].sub.0] for at least one [s.sub.i] [member of] [S.sub.i] ([[gamma].sub.0]), then from the definition of [C.sub.i] it is easy to check that [C.sub.i] [greater than or equal to] [[gamma].sub.0] will always be true. QED.

The proposition illustrates for a given goal function what restrictions on the constraint set will ensure a prespecified payoff like [[gamma].sub.0]. Note that this situation, however, need not be an equilibrium, unless we consider a one-sum game. (13) This is stated in the next corollary.

COROLLARY. Let [G.sup.f] be a one-sum game, and [[delta].sub.i] be nonempty, continuous, and strictly quasiconcave in a player's own strategies. If [C.sub.i] [greater than or equal to] [[gamma].sub.0], then it is also a Nash equilibrium of [G.sup.f].

PROOF. We know that an equilibrium exists because the conditions required for Theorem 1 are satisfied. Then from the definition of [C.sub.i] where players use a maximin type of reasoning, it follows that it is the equilibrium of the fuzzy one-sum game. QED.

Preplay communication has some interesting implications for this model. Suppose the players can communicate before playing the game. This would clearly affect their perception set or the set of feasible choices that each player has. With preplay communication player i will have a better notion of the strategies that player j will choose. Denote this by [S'.sub.j] [subset] [S.sub.j]. This affects the minimum payoff a player can ensure for herself, i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

provided [S'.sub.j] is a strict subset of the set of original strategies, [C'.sub.i] [greater than or equal to] [C.sub.i]. Hence, exchange of information between the two players has potentially interesting possibilities in this context. In a standard game, preplay communication does not actually alter the set of feasible choices--it facilitates coordination. In a fuzzy game, on the other hand, preplay communication actually allows for better coordination by reducing the feasible choices because it reduces the set of available choices.

Duopoly Example

In what follows we set up a basic duopoly model and discuss the implication of making it a fuzzy game. We consider a single-period homogeneous product Cournot duopoly. The inverse demand function in this market is given by the standard linear formulation

p = a - bQ, Q = [q.sub.1] + [q.sub.2] with a, b > 0.

We also assume that both firms have identical constant marginal cost functions given by C([q.sub.i]) : c[q.sub.i], i = 1, 2. We can now write the profit function as

[[PI].sub.i]([q.sub.1], [q.sub.2]) = (a - bQ - C)[q.sub.i], i = 1,2.

In the fuzzy version of this game, the constraint set is assumed to be a crisp set. Thus, each firm considers all its strategies equally feasible, i.e., [[mu].sub.i]([q.sub.i]) = 1 for all [q.sub.i] and for i = 1,2. In order to make things simple, we assume that the strategy set is compact and defined by [q.sub.i] [member of] [O, a/b]. The goal function, however, is fuzzy, and each firm believes that the collusive outcome is the best possible outcome. Hence, the membership function is single peaked such that

[[gamma].sub.i](a - c / 4b, a - c / 4b)= 1 for i = 1,2.

An example of a membership function with this property is

[[gamma].sub.i]([q.sub.i],[q.sub.2]) = exp(-([q.sub.1] - a-c / 4b) - ([q.sub.2] - a-c / 4b)).

Recall that West and Linster (2003) consider fuzzy rules of the type where player i reasons as follows: If player j chooses to produce an amount x, I will produce an amount y. In a sense, the above membership function captures such fuzzy rules because in this function player i considers deviations from the optimal value for each player. A number of variations of this membership function can also be constructed using absolute values. Further, we have chosen to have crisp strategies and fuzzy goals, but it will be equally possible to define the game in terms of fuzzy strategies. For instance replace [[gamma].sub.i](.) with [[mu].sub.i](.) and define [[gamma].sub.i](.) as follows. Let [[gamma].sub.i]([q.sub.1], [q.sub.2]) = 1 for [q.sub.1] [member of] [(a - c) / 4b, (a - c)/2b] and [[gamma.sub.i] ([q.sub.1], [q.sub.2]) = 0 elsewhere. Then it can be verified that we will get the results obtained in this model.

PROPOSITION 2. The fuzzy duopoly game as defined above has a Cournot-Nash equilibrium.

PROOF. It is easily verified that all the requirements of Theorem 1 are satisfied. Note that using [[mu].sub.i]([q.sub.1], [q.sub.2]) and [[gamma].sub.i]([q.sub.1],[q.sub.2]) for i =1, 2, we can define [[delta].sub.i] as the minimum of these two functions for each player. Because the constraint set is a crisp set, the confluence of the goals and constraints will just be the goal function, i.e., [[delta].sub.i]([q.sub.1], [q.sub.2]) = [[delta].sub.i]([q.sub.1],[q.sub.2]). Hence [[delta].sub.i] is nonempty, continuous, and concave, reaching a maximum at ((a - c)/4b, (a - c)/4b). Hence, an equilibrium exists. QED.

Because the two firms are symmetric in all respects, (Q/4, Q/4) where (a - c)/4b = Q/4 is indeed an equilibrium. Hence, we see that the collusive outcome can easily be supported as an equilibrium in the fuzzy game. Note that unlike the operational version of West and Linster (2003), our theoretical model does not require a finite and discrete number of possible choices for both players because there is no computational complexity issue in our model. Moreover, we do not require linearity of the response function--for our purpose continuity alone is sufficient. It is also obvious that different types of membership functions can be used to support other situations such as the Cournot-Nash outcome as an equilibrium of the fuzzy sets. This illustrates the importance of the beliefs that firms have about each other and the role played by their own goals in strategic interaction. As a final word of caution it is worth mentioning that because the membership functions are quite subjective, one can argue that this is also a weakness of the approach. (14)

5. Conclusion

The aim of this paper has been to provide a theoretical foundation for the West and Linster (2003)-type formulations. In games played in complex environments, the players very often develop their own understanding of the game and may use reasoning processes that make the game simpler to solve. We argue that such a heuristic process can be described as a fuzzy game.

This paper develops a static model of a fuzzy game by extending the fuzzy decision theory of Bellman and Zadeh (1970). The model developed is simple and exploratory in nature. We identify conditions that guarantee the existence of equilibrium as well as how to attain a certain minimum payoff in the game. Further, a very simple example demonstrates that fuzzy modeling allows us to sustain collusive behavior. It differs from the earlier work on fuzzy games by being closest to the standard gametheoretic framework. Although it seems to have a realistic flavor, future research should consider more sophisticated applications and compare the results of this approach with those obtained under the standard game-theoretic formulation. We believe the most promising applications would involve cheap talk and the modeling of curb sets using a fuzzy environment. Other interesting issues would be to link the constraints faced by a player or his perception set to different degrees of bounded rationality. This would allow us to investigate equilibrium selection and refinements from a different perspective.

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Michael Aristidou, Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA. and Sudipta Sarangi, Department of Economics, Louisiana State University, Baton Rouge, LA 70803, USA; E-mail [e-mail]; corresponding author.

We would like to thank the Editor and two anonymous referees for their comments and suggestions. All remaining errors ale ours.

Received January 2004; accepted February 2005.

(1) Note that the notions of "tallness" and "shortness" are themselves context related.

(2) Note that the sets of alternatives could be a restricted set. However, these are restrictions that are a part of the definition of the problem, whereas constraints on the choice set arise from the player's perception of the decision-making situation.

(3) There are a number of alternative ways to solve fuzzy decision problems. A particularly interesting approach is that of Li and Yen (1995), which relies on linguistic variables. Semantics are used to create a descriptor frame, which redefines the decision-making problem.

(4) The roots of Zadeh's work on fuzzy sets can be traced back to work on multi valued logic by the philosopher Max Black (1937).

(5) This does not require that the graph of [[mu].sub.A] itself be convex. Take, in particular, a crisp set A. Then A [subset or equal to] X is convex if and only if its characteristic function [[PSI].sub.A] is quasiconcave. The latter does not imply, however, that the graph of [[PSI].sub.A] is convex.

(6) The West and Linster (2003) paper, although relevant, has not been included in this review because its salient features are discussed elsewhere in the paper.

(7) Player 2's problem is a standard minimization problem and for the sake of brevity is not shown here. The interested reader may refer to Friedman (1990) for more on zero-sum games.

(8) The only requirement for this relation is that it must preserve the ranking of the fuzzy numbers under linear transformations.

(9) The presence of these imprecise linguistic terms is cited as the main reason for using fuzzy techniques instead of relying on probabilistic methods.

(10) Thus, [[gamma].sub.i] may be construed as the composition of two mappings: [II.sub.i]: [S.sub.1] x [S.sub.2] [right arrow] R and [[GAMMA].sub.i] R [right arrow] [0,1].

(11) This has some interesting implications. If [[mu].sub.i]: is indeed empty, the player does not believe that any strategies are feasible options and hence is unwilling to participate in the game. In order to avoid issues of this sort, we assume that the constraint set is a normal fuzzy set.

(12) The membership function [[mu].sub.F] (x) F = A [intersection] B is defined pointwise by [[mu].sub.F] (x) {[[mu].sub.A] (x) [mu].sub.B] (x)} x [member of] X.

(13) Note that there is no fuzzy zero-sum game because [[delta].sub.i] [member of] [0,1] always. The analogue of the zero-sum game is the complementary one-sum game, where [[delta].sub.i](s) = 1 - [[delta].sub.j](s). In other words player j's decision set is the complement of player i's decision set. This is stated in the next corollary.

(14) In general, it could be argued that fuzzy techniques provide a way of analyzing problems that decision makers can use according to their need in conjunction with their subjective beliefs about the problem. Hence, the wide popularity of this approach in a number of engineering problems.