Games in Fuzzy Environments

Abstract

1. IntroductionIn a recent issue of this journal, an interesting paper by West and Linster (2003) used fuzzy to show that Nash equilibrium behavior can be achieved by boundedly rational agents in two-player games with infinite strategy spaces. These 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 is utilized in the genetic algorithm developed for the simulations in the repeated game. Their most interesting find is that for fuzzy 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 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 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. …