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.

翻译：我们研究定义在随机矩阵 $M\in \mathbb{R}^{n\times m}$ 上的双人零和博弈的价值，其定义为 $v(M) = \min_{x\inΔ_n}\max_{y\in Δ_m}x^T M y$。在 $n=m$ 且 $M$ 的条目为独立同分布标准高斯变量的设定下，我们证明了 $v(M)$ 的标准差量级为 $\frac{1}{n}$。这证实了可追溯至20世纪80年代的一项实验猜想。我们还研究了 $M$ 为矩形高斯矩阵（其中 $m = n+λ\sqrt{n}$）的情形，表明博弈的期望价值量级为 $\fracλ{n}$，以及 $M$ 为随机正交矩阵的情形。我们的技术基于概率论论证和凸几何。我们认为，对随机博弈的研究可为理论计算机科学中的各类问题提供新的见解。