We study the computational complexity of the problem of computing local min-max equilibria of games with a nonconvex-nonconcave utility function $f$. From the work of Daskalakis, Skoulakis, and Zampetakis [DSZ21], this problem was known to be hard in the restrictive case in which players are required to play strategies that are jointly constrained, leaving open the question of its complexity under more natural constraints. In this paper, we settle the question and show that the problem is PPAD-hard even under product constraints and, in particular, over the hypercube.

翻译：我们研究了计算具有非凸非凹效用函数 $f$ 的博弈的局部最小-最大均衡问题的计算复杂性。根据 Daskalakis、Skoulakis 和 Zampetakis [DSZ21] 的研究，已知在参与者被要求采用联合约束策略的限制性情况下，该问题是困难的，这使其在更自然约束下的复杂性成为一个开放问题。在本文中，我们解决了这个问题，并证明即使在乘积约束下，特别是超立方体上，该问题也是 PPAD 困难的。