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Discounted Repeated Games Having Computable Strategies with No Computable Best Response under Subgame-Perfect Equilibria

Discounted Repeated Games Having Computable Strategies with No Computable Best Response under... A classic result in computational game theory states that there are infinitely repeated games where one player has a computable strategy that has a best response, but no computable best response. For games with discounted payoff, the result is known to hold for a specific class of games—essentially generalizations of Prisoner’s Dilemma—but until now, no necessary and sufficient condition is known. To be of any value, the computable strategy having no computable best response must be part of a subgame-perfect equilibrium, as otherwise a rational, self-interested player would not play the strategy.We give the first necessary and sufficient conditions for a two-player repeated game \( G \) to have such a computable strategy with no computable best response for all discount factors above some threshold. The conditions involve existence of a Nash equilibrium of the repeated game whose discounted payoffs satisfy certain conditions involving the min–max payoffs of the underlying stage game. We show that it is decidable in polynomial time in the size of the payoff matrix of \( G \) whether it satisfies these conditions. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png ACM Transactions on Economics and Computation (TEAC) Association for Computing Machinery

Discounted Repeated Games Having Computable Strategies with No Computable Best Response under Subgame-Perfect Equilibria

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References (43)

Publisher
Association for Computing Machinery
Copyright
Copyright © 2022 Association for Computing Machinery.
ISSN
2167-8375
eISSN
2167-8383
DOI
10.1145/3505585
Publisher site
See Article on Publisher Site

Abstract

A classic result in computational game theory states that there are infinitely repeated games where one player has a computable strategy that has a best response, but no computable best response. For games with discounted payoff, the result is known to hold for a specific class of games—essentially generalizations of Prisoner’s Dilemma—but until now, no necessary and sufficient condition is known. To be of any value, the computable strategy having no computable best response must be part of a subgame-perfect equilibrium, as otherwise a rational, self-interested player would not play the strategy.We give the first necessary and sufficient conditions for a two-player repeated game \( G \) to have such a computable strategy with no computable best response for all discount factors above some threshold. The conditions involve existence of a Nash equilibrium of the repeated game whose discounted payoffs satisfy certain conditions involving the min–max payoffs of the underlying stage game. We show that it is decidable in polynomial time in the size of the payoff matrix of \( G \) whether it satisfies these conditions.

Journal

ACM Transactions on Economics and Computation (TEAC)Association for Computing Machinery

Published: Apr 8, 2022

Keywords: Repeated games

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