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The myth of the Folk Theorem

Author

Listed:
  • Borgs, Christian
  • Chayes, Jennifer
  • Immorlica, Nicole
  • Kalai, Adam Tauman
  • Mirrokni, Vahab
  • Papadimitriou, Christos

Abstract

The Folk Theorem for repeated games suggests that finding Nash equilibria in repeated games should be easier than in one-shot games. In contrast, we show that the problem of finding any Nash equilibrium for a three-player infinitely-repeated game is as hard as it is in two-player one-shot games. More specifically, for any two-player game, we give a simple construction of a three-player game whose Nash equilibria (even under repetition) correspond to those of the one-shot two-player game. Combined with recent computational hardness results for one-shot two-player normal-form games ([Daskalakis et al., 2006], [Chen et al., 2006] and [Chen et al., 2007]), this gives our main result: the problem of finding an (epsilon) Nash equilibrium in a given n×n×n game (even when all payoffs are in {-1,0,1}) is PPAD-hard (under randomized reductions).

Suggested Citation

  • Borgs, Christian & Chayes, Jennifer & Immorlica, Nicole & Kalai, Adam Tauman & Mirrokni, Vahab & Papadimitriou, Christos, 2010. "The myth of the Folk Theorem," Games and Economic Behavior, Elsevier, vol. 70(1), pages 34-43, September.
  • Handle: RePEc:eee:gamebe:v:70:y:2010:i:1:p:34-43
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    References listed on IDEAS

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    1. Gilboa, Itzhak & Zemel, Eitan, 1989. "Nash and correlated equilibria: Some complexity considerations," Games and Economic Behavior, Elsevier, vol. 1(1), pages 80-93, March.
    2. Kalai, Ehud & Lehrer, Ehud, 1993. "Rational Learning Leads to Nash Equilibrium," Econometrica, Econometric Society, vol. 61(5), pages 1019-1045, September.
    3. Drew Fudenberg & Eric Maskin, 2008. "The Folk Theorem In Repeated Games With Discounting Or With Incomplete Information," World Scientific Book Chapters, in: Drew Fudenberg & David K Levine (ed.), A Long-Run Collaboration On Long-Run Games, chapter 11, pages 209-230, World Scientific Publishing Co. Pte. Ltd..
    4. Shane M. Greenstein (ed.), 2006. "Computing," Books, Edward Elgar Publishing, number 3171.
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    Cited by:

    1. Halpern, Joseph Y. & Pass, Rafael & Seeman, Lior, 2019. "The truth behind the myth of the Folk theorem," Games and Economic Behavior, Elsevier, vol. 117(C), pages 479-498.
    2. 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).
    3. 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.
    4. Dvijotham, Krishnamurthy & Rabani, Yuval & Schulman, Leonard J., 2022. "Convergence of incentive-driven dynamics in Fisher markets," Games and Economic Behavior, Elsevier, vol. 134(C), pages 361-375.

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