
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 twoplayer 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 rulebased games. This paper provides a theoretical foundation for games based on fuzzy rules by developing a static normalform 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 settheoretic 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 gametheoretic 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 constraintsa 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 decisionmaking 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 gametheoretic 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 zerosum games called onesum 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 tradeoff 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.

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,
[[PSI].sub.A](x) = {1 if x [member of] A {0 otherwise.
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...
Games in fuzzy environments.
Author:  Aristidou, Michael 
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COPYRIGHT GALE, Cengage Learning. All rights reserved.
COPYRIGHT GALE, Cengage Learning. All rights reserved.