Finding, counting, or determining the existence of Nash equilibria, where players must play optimally given each others' actions, are known to be computational intractable problems. We ask whether weakening optimality to the requirement that each player merely avoid worst responses -- arguably the weakest meaningful rationality criterion -- yields tractable solution concepts. We show that it does not: any solution concept with this minimal guarantee is ``as intractable'' as pure Nash equilibrium. In general games, determining the existence of no-worst-response action profiles is NP-complete, finding one is NP-hard, and counting them is #P-complete. In potential games, where existence is guaranteed, the search problem is PLS-complete. Computational intractability therefore stems not only from the requirement of optimality, but also from the requirement of a minimal rationality guarantee for each player. Moreover, relaxing the latter requirement gives rise to a tractability trade-off between the strength of individual rationality guarantees and the fraction of players satisfying them.

翻译：在博弈论中，寻找、计数或判定纳什均衡（即每位参与者必须在给定其他参与者行动下选择最优策略）的存在性，已被证明是计算上不可处理的问题。我们探讨的问题是：若将最优性要求弱化为每位参与者仅需避免最差响应——这可以说是最弱的有意义理性准则——是否会产生可处理的解概念。我们证明事实并非如此：任何具有此最低保证的解概念都与纯策略纳什均衡“同样不可处理”。在一般博弈中，判定无最差响应行动组合的存在性是NP完全的，寻找这样的组合是NP难的，而计数此类组合则是#P完全的。在势博弈中，虽然存在性得以保证，但搜索问题却是PLS完全的。因此，计算不可处理性不仅源于最优性要求，也源于对每位参与者的最低理性保证要求。此外，放宽后一要求会在个体理性保证的强度与满足该保证的参与者比例之间产生一种可处理性的权衡。