The value of random zero-sum games

TL;DR

This paper analyzes the value of random zero-sum games with Gaussian and orthogonal matrices, confirming conjectures about their fluctuations and expected values, and explores implications for theoretical computer science.

Contribution

It provides rigorous results on the distribution and expectation of game values for Gaussian and orthogonal random matrices, confirming longstanding conjectures.

Findings

Standard deviation of game value is of order 1/n for Gaussian matrices.

Expected game value is of order λ/n when matrix dimensions are asymmetrically scaled.

Results are derived using probabilistic and convex geometric methods.

Abstract

We study the value of a two-player zero-sum game on a random matrix $M\in \mathbb{R}^{n\times m}$, defined by $v(M) = \min_{x\in\Delta_n}\max_{y\in \Delta_m}x^T M y$. In the setting where $n=m$ and $M$ has i.i.d. standard Gaussian entries, we prove that the standard deviation of $v(M)$ is of order $\frac{1}{n}$. This confirms an experimental conjecture dating back to the 1980s. We also investigate the case where $M$ is a rectangular Gaussian matrix with $m = n+\lambda\sqrt{n}$, showing that the expected value of the game is of order $\frac{\lambda}{n}$, as well as the case where $M$ is a random orthogonal matrix. Our techniques are based on probabilistic arguments and convex geometry. We argue that the study of random games could shed new light on various problems in theoretical computer science.