Evolution of Beliefs and the Nash Equilibrium of Normal Form Games

In models of learning, it is recognized that the path of play displays some conspicuous patterns when players use simple rules in assessing their opponents' behavior. If the players themselves become aware of such patterns, they may want to utilize them in an attempt to better assess their opponents' behavior. This paper formulates a simple two-person model of learning that allows such pattern recognition and discusses its implications. In particular, it focuses on the convergence of players' beliefs to a mixed Nash equilibrium of a game. Consider the learning process of two myopic players who repeatedly play the game of matching pennies as depicted in Fig. 1. Suppose that both players use fictitious play to assess their opponents' behavior. Namely, each player maximizes the current-period payoff with the assumption that the opponent's next mixed action is given by the empirical frequency of the opponent's past pure action choices. article no. 0098