The value of random zero-sum games

2026-01-12

AI Summary

This study investigates the statistical properties of the value in random two-player zero-sum games, focusing on payoff matrices drawn from Gaussian and orthogonal ensembles. By integrating tools from probability theory and convex geometry, the authors rigorously establish that when the payoff matrix is an $n \times n$ standard Gaussian matrix, the standard deviation of the game value scales as $O(1/n)$, thereby confirming a conjecture based on empirical observations from the 1980s. The analysis is further extended to rectangular matrices, revealing that for dimensions $n \times (n + \lambda\sqrt{n})$, the expected game value is $O(\lambda/n)$. This work represents the first theoretical treatment of random orthogonal matrices and non-square settings in this context, significantly advancing the understanding of the asymptotic behavior of values in random zero-sum games.

Technology Category

• Game Theory and Economic Paradigms: Behavioral Game Theory
• Reasoning under Uncertainty: Stochastic Optimization
• Multiagent Systems: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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.

Problem

• random zero-sum games
• game value
• Gaussian matrix
• orthogonal matrix
• probabilistic analysis

Innovation

• random zero-sum games
• Gaussian random matrices
• value concentration
• convex geometry
• probabilistic methods