The Evolution of Fuzzy Rules as Strategies in Two-Player Games

Abstract

This paper describes simulations using fuzzy that show how Nash equilibrium behavior can be achieved by boundedly rational agents in two-player games with infinite strategy spaces. That is, we show how agents using simple rules of thumb can achieve near-equilibrium outcomes without any overt computation of the equilibrium. This is accomplished by using a genetic algorithm to approximate repeated play. Two games of differing complexities, both with analytic solutions, are examined: a repeated linear-demand Cournot game and a contestable rent game. When fuzzy used only the most recent information, the games we examined converged to outcomes similar to their respective Cournot-Nash equilibrium outcomes. When fuzzy remembered play from the more distant past, we found that the games converged more slowly, if at all. How to model players' strategies has been an important issue in the study of repeated games. The problem has been reasonably well addressed for the case in which strategy choices in the stage game are finite because the history at any point in time consists only of a series of choices from a finite set of stage game strategies. For example, Miller (1996) and Linster (1992, 1994) used finite automata to encode strategies in the repeated prisoner's dilemma. These finite automata, or Moore machines, can capture the idea of bounded rationality, but they can only be used when strategy choices are finite in number. The purpose of this paper is to illustrate the use of fuzzy strategies in two familiar two-player games with continuous strategy spaces. We chose games for which analytic solutions are available so as to enable comparisons between equilibrium outcomes of our fuzzy models and Cournot-Nash, competitive, and collusive outcomes. These simulations based on fuzzy strategies can be used in models for which analytic solutions do not exist or are computationally very difficult to obtain. A rule described by a small number of parameters can express only limited rationality within the context of a complex model, but these might be good to describe human behavior in a variety of situations. Consider as an example the task of running an automobile's heating and cooling system prior to the advent of climate control. A heater temperature/air conditioner lever position is continuously and minutely variable. No doubt a complicated optimal nonlinear response function could be specified in which temperature readings are taken and minute adjustments are made to the lever position. Instead, we hypothesize that as few as three simple might be enough to