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Non-Computable Strategies and Discounted Repeated Games

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  • William R. Zame

    (UCLA)

Abstract

A number of authors have used formal models of computation to capture the idea of "bounded rationality" in repeated games. Most of this literature has used computability by a finite automaton as the standard. A conceptual difficulty with this standard is that the decision problem is not "closed." That is, for every strategy implementable by an automaton, there is some best response implementable by an automaton, but there may not exist any algorithm for finding such a best response that can be implemented by an automaton. However, such algorithms can always be implemented by a Turing machine, the most powerful formal model of computation. In this paper, we investigate whether the decision problem can be closed by adopting Turing machines as the standard of computability. The answer we offer is negative. Indeed, for a large class of discounted repeated games (including the repeated Prisoner's Dilemma) there exist strategies implementable by a Turing machine for which no best response is implementable by a Turing machine.
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Suggested Citation

  • William R. Zame, 1995. "Non-Computable Strategies and Discounted Repeated Games," UCLA Economics Working Papers 735, UCLA Department of Economics.
    • Handle: RePEc:cla:uclawp:735
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    References listed on IDEAS

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    1. Canning, David, 1992. "Rationality, Computability, and Nash Equilibrium," Econometrica, Econometric Society, vol. 60(4), pages 877-888, July.
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    9. Knoblauch Vicki, 1994. "Computable Strategies for Repeated Prisoner's Dilemma," Games and Economic Behavior, Elsevier, vol. 7(3), pages 381-389, November.
    10. Rubinstein, Ariel, 1986. "Finite automata play the repeated prisoner's dilemma," Journal of Economic Theory, Elsevier, vol. 39(1), pages 83-96, June.
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    Cited by:

    1. Jakub Dargaj & Jakob Grue Simonsen, 2020. "A Complete Characterization of Infinitely Repeated Two-Player Games having Computable Strategies with no Computable Best Response under Limit-of-Means Payoff," Papers 2005.13921, arXiv.org, revised Jun 2020.
    2. Richter, Marcel K. & Wong, Kam-Chau, 1999. "Computable preference and utility," Journal of Mathematical Economics, Elsevier, vol. 32(3), pages 339-354, November.
    3. Conitzer, Vincent & Sandholm, Tuomas, 2008. "New complexity results about Nash equilibria," Games and Economic Behavior, Elsevier, vol. 63(2), pages 621-641, July.
    4. Stephen J. Decanio, 1999. "Estimating The Non‐Environmental Consequences Of Greenhouse Gas Reductions Is Harder Than You Think," Contemporary Economic Policy, Western Economic Association International, vol. 17(3), pages 279-295, July.
    5. Ying-Fang Kao & Ragupathy Venkatachalam, 2021. "Human and Machine Learning," Computational Economics, Springer;Society for Computational Economics, vol. 57(3), pages 889-909, March.
    6. Dargaj, Jakub & Simonsen, Jakob Grue, 2023. "A complete characterization of infinitely repeated two-player games having computable strategies with no computable best response under limit-of-means payoff," Journal of Economic Theory, Elsevier, vol. 213(C).

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    More about this item

    JEL classification:

    • C51 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Construction and Estimation
    • E32 - Macroeconomics and Monetary Economics - - Prices, Business Fluctuations, and Cycles - - - Business Fluctuations; Cycles
    • O11 - Economic Development, Innovation, Technological Change, and Growth - - Economic Development - - - Macroeconomic Analyses of Economic Development
    • O47 - Economic Development, Innovation, Technological Change, and Growth - - Economic Growth and Aggregate Productivity - - - Empirical Studies of Economic Growth; Aggregate Productivity; Cross-Country Output Convergence

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