The Computational Intractability of Not Worst Responding

TL;DR

This paper demonstrates that even when relaxing the rationality criterion to avoid worst responses, finding such profiles remains computationally intractable across various game types, highlighting fundamental complexity barriers.

Contribution

It proves that minimal rationality guarantees do not simplify computational complexity, establishing intractability results for no-worst-response profiles in general and potential games.

Findings

Determining existence is NP-complete in general games.

Finding a no-worst-response profile is NP-hard.

Counting such profiles is #P-complete.

Abstract

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.