The value of random zero-sum games
Romain Cosson, Laurent Massoulié
TL;DR
This work analyzes the value $v(M)$ of a two-player zero-sum game defined by a random payoff matrix, focusing on Gaussian and orthogonal models. Using Gaussian comparison inequalities and convex-geometry tools, it derives tight tail bounds and scaling laws for $v(M)$: in the square Gaussian case, $\sigma(v(M))=\Theta(1/n)$, while in a near-square rectangular regime with $m=n+\lambda\sqrt{n}$ the mean scales as $\Theta(\lambda/n)$ with fluctuations $O(1/n)$; for random orthogonal matrices, $\mathrm{sd}(v) = O(n^{-3/2})$. The results bridge probability, convex geometry, and theoretical computer science, offering a probabilistic lens on minimax values and potential phase-transition phenomena as matrix aspect ratio changes. The work also sketches connections to statistical physics via ground-state interpretations and outlines open directions for discrete distributions, algorithmic implications, and the structure of optimal strategies. Overall, the paper provides new tail bounds, clarifies scaling regimes, and proposes a geometric framework for analyzing random games with broad potential impact in CS and physics.
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 $v(M) = \min_{x\inΔ_n}\max_{y\in Δ_m}x^T M y$. 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 $m = n+λ\sqrt{n}$, showing that the expected value of the game is of order $\fracλ{n}$, 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.