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

TL;DR

This paper applies statistical mechanics techniques to analyze the properties and typical features of Nash equilibria in large, random bimatrix games, revealing that most equilibria have nearly equal payoffs and specific strategic characteristics.

Contribution

It introduces a novel application of statistical mechanics to quantify the number and properties of Nash equilibria in large random games, including payoff distributions and strategy fractions.

Findings

Most equilibria have nearly equal payoffs.

The number of equilibria is exponentially large.

Payoff and strategy distributions depend on payoff matrix correlation.

Abstract

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.