We study equilibrium finding in polymatrix games under differential privacy constraints. Prior work in this area fails to achieve both high-accuracy equilibria and a low privacy budget. To better understand the fundamental limitations of differential privacy in games, we show hardness results establishing that no algorithm can simultaneously obtain high accuracy and a vanishing privacy budget as the number of players tends to infinity. This impossibility holds in two regimes: (i) We seek to establish equilibrium approximation guarantees in terms of Euclidean \emph{distance} to the equilibrium set, and (ii) The adversary has access to all communication channels. We then consider the more realistic setting in which the adversary can access only a bounded number of channels and propose a new distributed algorithm that: recovers strategies with simultaneously vanishing \emph{Nash gap} (in expected utility, also referred to as \emph{exploitability}) and \emph{privacy budget} as the number of players increases. Our approach leverages structural properties of polymatrix games. To our knowledge, this is the first paper that can achieve this in equilibrium computation. Finally, we also provide numerical results to justify our algorithm.

翻译：本文研究在差分隐私约束下多矩阵博弈中的均衡求解问题。现有工作未能同时实现高精度均衡与低隐私预算。为深入理解博弈中差分隐私的基本限制，我们证明了当玩家数量趋于无穷时，不存在能同时获得高精度与零隐私预算的算法。这一不可能性体现在两个场景中：(i) 以欧几里得距离度量到均衡集的近似精度；(ii) 对手可访问所有通信信道。随后，我们考虑了更现实的场景——对手仅能访问有限数量的信道，并提出一种新型分布式算法：随着玩家数量增加，该算法能同步实现纳什间隙（期望效用层面，亦称可剥削性）与隐私预算的收敛。该方法利用了多矩阵博弈的结构性质。据我们所知，这是首篇在均衡计算中实现该目标的论文。最后，我们通过数值实验验证了算法的有效性。