A Complete Characterization of Infinitely Repeated Two-Player Games having Computable Strategies with no Computable Best Response under Limit-of-Means Payoff

TL;DR

This paper provides a complete and decidable characterization of infinitely repeated two-player games where computable strategies lack computable best responses under limit-of-means payoff, extending previous partial results.

Contribution

It offers a full set of necessary and sufficient conditions for the existence of such strategies, including refinements for Nash and subgame-perfect equilibria.

Findings

Characterization of when computable strategies lack computable best responses

Decidability results for the existence of such strategies

Refined conditions for Nash and subgame-perfect equilibria

Abstract

It is well-known that for infinitely repeated games, there are computable strategies that have best responses, but no computable best responses. These results were originally proved for either specific games (e.g., Prisoner's dilemma), or for classes of games satisfying certain conditions not known to be both necessary and sufficient. We derive a complete characterization in the form of simple necessary and sufficient conditions for the existence of a computable strategy without a computable best response under limit-of-means payoff. We further refine the characterization by requiring the strategy profiles to be Nash equilibria or subgame-perfect equilibria, and we show how the characterizations entail that it is efficiently decidable whether an infinitely repeated game has a computable strategy without a computable best response.