The Computational Intractability of Not Worst Responding

2026-02-11

AI Summary

This study investigates whether relaxing the rationality requirement in games—from optimal best responses to merely avoiding worst responses—reduces the computational complexity of equilibrium computation. Leveraging computational complexity theory, including analyses of NP, #P, and PLS completeness, the work establishes lower bounds on problem difficulty in both general and potential games. The findings reveal that even under this weakened rationality notion, deciding the existence of such equilibria remains NP-complete, finding one is NP-hard, and counting them is #P-complete; in potential games, the search problem is PLS-complete. These results demonstrate that computational intractability stems not only from the assumption of optimality but also from the minimal rationality constraint imposed on all players, uncovering a computability trade-off between the strength of individual rationality and the proportion of players required to satisfy it.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Multiagent Systems: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Social Networks and Social Media: Computational social science
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

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.

Problem

• Computational Intractability
• Nash Equilibrium
• Worst Response
• Rationality
• Complexity

Innovation

• computational intractability
• Nash equilibrium
• no-worst-response
• complexity theory
• bounded rationality