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On the communication complexity of approximate Nash equilibria

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  • Goldberg, Paul W.
  • Pastink, Arnoud

Abstract

We study the problem of computing approximate Nash equilibria of bimatrix games, in a setting where players initially know their own payoffs but not the other player's. In order to find a solution of reasonable quality, some amount of communication is required. We study algorithms where the communication is substantially less than the size of the game. When the communication is polylogarithmic in the number of strategies, we show how to obtain ϵ-approximate Nash equilibrium for ϵ≈0.438, and for well-supported approximate equilibria we obtain ϵ≈0.732. For one-way communication we show that ϵ=12 is the best approximation quality achievable, while for well-supported equilibria, no value of ϵ<1 is achievable. When the players do not communicate at all, ϵ-Nash equilibria can be obtained for ϵ=34; we also provide a corresponding lower bound of slightly more than 12 on the smallest constant ϵ achievable.

Suggested Citation

  • Goldberg, Paul W. & Pastink, Arnoud, 2014. "On the communication complexity of approximate Nash equilibria," Games and Economic Behavior, Elsevier, vol. 85(C), pages 19-31.
    • Handle: RePEc:eee:gamebe:v:85:y:2014:i:c:p:19-31
    DOI: 10.1016/j.geb.2014.01.009
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    References listed on IDEAS

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    1. Paul Goldberg & Rahul Savani & Troels Sørensen & Carmine Ventre, 2013. "On the approximation performance of fictitious play in finite games," International Journal of Game Theory, Springer;Game Theory Society, vol. 42(4), pages 1059-1083, November.
    2. Sergiu Hart & Andreu Mas-Colell, 2013. "Stochastic Uncoupled Dynamics And Nash Equilibrium," World Scientific Book Chapters, in: Simple Adaptive Strategies From Regret-Matching to Uncoupled Dynamics, chapter 8, pages 165-189, World Scientific Publishing Co. Pte. Ltd..
    3. Sergiu Hart & Andreu Mas-Colell, 2013. "A Simple Adaptive Procedure Leading To Correlated Equilibrium," World Scientific Book Chapters, in: Simple Adaptive Strategies From Regret-Matching to Uncoupled Dynamics, chapter 2, pages 17-46, World Scientific Publishing Co. Pte. Ltd..
    4. , P. & , Peyton, 2006. "Regret testing: learning to play Nash equilibrium without knowing you have an opponent," Theoretical Economics, Econometric Society, vol. 1(3), pages 341-367, September.
    5. Sergiu Hart & Yishay Mansour, 2013. "How Long To Equilibrium? The Communication Complexity Of Uncoupled Equilibrium Procedures," World Scientific Book Chapters, in: Simple Adaptive Strategies From Regret-Matching to Uncoupled Dynamics, chapter 10, pages 215-249, World Scientific Publishing Co. Pte. Ltd..
    6. Sergiu Hart & Andreu Mas-Colell, 2013. "Uncoupled Dynamics Do Not Lead To Nash Equilibrium," World Scientific Book Chapters, in: Simple Adaptive Strategies From Regret-Matching to Uncoupled Dynamics, chapter 7, pages 153-163, World Scientific Publishing Co. Pte. Ltd..
    7. Foster, Dean P. & Vohra, Rakesh V., 1997. "Calibrated Learning and Correlated Equilibrium," Games and Economic Behavior, Elsevier, vol. 21(1-2), pages 40-55, October.
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    Cited by:

    1. Babichenko, Yakov & Rubinstein, Aviad, 2022. "Communication complexity of approximate Nash equilibria," Games and Economic Behavior, Elsevier, vol. 134(C), pages 376-398.
    2. J. E. Abdou & E. Safatly & B. Nakhle & A. El Khoury, 2017. "High-Dimensional Nash Equilibria Problems and Tensors Applications," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 19(03), pages 1-25, September.
    3. Cole, Richard & Tao, Yixin, 2021. "On the existence of Pareto Efficient and envy-free allocations," Journal of Economic Theory, Elsevier, vol. 193(C).

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    More about this item

    Keywords

    Two player; Mixed strategy; Approximate equilibria; Efficient algorithms;
    All these keywords.

    JEL classification:

    • C7 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory
    • D83 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Search; Learning; Information and Knowledge; Communication; Belief; Unawareness

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