Matrix Games, Mixed Strategies, and Statistical Mechanics

TL;DR

This paper applies statistical mechanics techniques to analyze large random matrix games, deriving analytical formulas for optimal strategies, game value, and strategy distribution, including the fraction of non-contributing pure strategies.

Contribution

It introduces a novel approach using statistical mechanics to study the properties of optimal mixed strategies in large random matrix games, providing analytical insights.

Findings

Derived formulas for game value and strategy distribution.

Calculated the fraction of pure strategies not used in optimal strategies.

Validated analytical results with numerical simulations.

Abstract

Matrix games constitute a fundamental problem of game theory and describe a situation of two players with completely conflicting interests. We show how methods from statistical mechanics can be used to investigate the statistical properties of optimal mixed strategies of large matrix games with random payoff matrices and derive analytical expressions for the value of the game and the distribution of strategy strengths. In particular the fraction of pure strategies not contributing to the optimal mixed strategy of a player is calculated. Both independently distributed as well as correlated elements of the payoff matrix are considered and the results compared with numerical simulations.