Anticipatory Learning in Two-Person Games

Abstract

A learning process for 2-person games in normal form is introduced. The game is assumed to be played repeatedly by two large populations, one for player 1 and one for player 2. Every individual plays against changing opponents in the other population. Mixed strategies are adapted to experience. The process evolves in discrete time. All individuals in the same population play the same mixed strategy. The mixed strategies played in one period are publicly known in the next period. The payoff matrices of both players are publicly known. In a preliminary version of the model, the individuals increase and decrease probabilities of pure strategies directly in response to payoffs against last period’s observed opponent strategy. In this model, the stationary points are the equilibrium points, but genuinely mixed equilibrium points fail to be locally stable. On the basis of the preliminary model an anticipatory learning process is defined, where the individuals first anticipate the opponent strategies according to the preliminary model and then react to these anticipated strategies in the same way as to the observed strategies in the preliminary model. This means that primary learning effects on the other side are anticipated, but not the secondary effects due to anticipations in the opponent population. Local stability of the anticipatory learning process is investigated for regular games, i.e., for games where all equilibrium points are regular. Astability criterion is derived which is necessary and sufficient for sufficiently small adjustment speeds. This criterion requires that the eigenvalues of a matrix derived from both payoff matrices are negative. It is shown that the stability criterion is satisfied for 2x 2-games without pure strategy equilibrium points, for zero-sum games and for games where one player’s payoff matrix is the unit matrix and the other player's payoff matrix is negative definite. Moreover, the addition of constants to rows or columns of payoff matrices does not change stability. The stability criterion is related to an additive decomposition of payoffs reminiscent of a two way analysis of variance. Payoffs are decomposed into row effects, column effects and interaction effects. Intuitively, the stability criterion requires a preponderance of negative covariance between the interaction effects in both players’ payoffs. The anticipatory learning process assumes that the effects of anticipations on the other side remain unanticipated. At least for completely mixed equilibrium points the stability criterion remains unchanged, if anticipations of anticipation effects are introduced.