Entropy and typical properties of Nash equilibria in two-player games

– We use techniques from the statistical mechanics of disordered systems to analyse the properties of Nash equilibria of bimatrix games with large random payoff matrices. By means of an annealed bound, we calculate their number and analyse the properties of typical Nash equilibria, which are exponentially dominant in number. We find that a randomly chosen equilibrium realizes almost always equal payoffs to either player. This value and the fraction of strategies played at an equilibrium point are calculated as a function of the correlation between the two payoff matrices. The picture is complemented by the calculation of the properties of Nash equilibria in pure strategies. Game theory seeks to model problems of strategic decision-making arising in economics, sociology, or international relations. In the generic set-up, a number of players chooses between different strategies, the combination of which determines the outcome of the game specified by the payoff to each player. Contrary to the situation in ordinary optimization problems, these payoffs are in general different for different players leading to a competitive situation where each player tries to maximize his individual payoff. One of the cornerstones of modern economics and game theory is therefore the concept of a Nash equilibrium [1], see also [2], which describes a situation where no player can unilaterally improve his payoff by changing his individual strategy given the other players all stick to their strategies. However, this concept is thought to suffer from the serious drawback that in a typical game-theoretical situation there is a large number of Nash equilibria with different characteristics but no means of telling which one will be chosen by the players, as would be required of a predictive theory. (∗) E-mail: [email protected] (∗∗) E-mail: [email protected] (∗∗∗)Unité Mixte de Recherche du Centre National de la Recherche Scientifique et de l’Ecole Normale Supérieure.