The Distribution of Optimal Strategies in Symmetric Zero-sum Games

TL;DR

This paper analyzes the probability distribution of optimal strategies in symmetric zero-sum games with randomly drawn skew-symmetric payoff matrices, revealing how likely certain strategies are under broad distributional assumptions.

Contribution

It characterizes the probability that an optimal strategy randomizes over specific actions in symmetric zero-sum games drawn from general regular distributions.

Findings

Provides formulas for the distribution of optimal strategies.

Applies to a wide class of natural probability distributions.

Enhances understanding of strategic behavior in random symmetric games.

Abstract

Given a skew-symmetric matrix, the corresponding two-player symmetric zero-sum game is defined as follows: one player, the row player, chooses a row and the other player, the column player, chooses a column. The payoff of the row player is given by the corresponding matrix entry, the column player receives the negative of the row player. A randomized strategy is optimal if it guarantees an expected payoff of at least 0 for a player independently of the strategy of the other player. We determine the probability that an optimal strategy randomizes over a given set of actions when the game is drawn from a distribution that satisfies certain regularity conditions. The regularity conditions are quite general and apply to a wide range of natural distributions.