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How long to equilibrium? The communication complexity of uncoupled equilibrium procedures

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  • Hart, Sergiu
  • Mansour, Yishay

Abstract

We study the question of how long it takes players to reach a Nash equilibrium in uncoupled setups, where each player initially knows only his own payoff function. We derive lower bounds on the communication complexity of reaching a Nash equilibrium, i.e., on the number of bits that need to be transmitted, and thus also on the required number of steps. Specifically, we show lower bounds that are exponential in the number of players in each one of the following cases: (1) reaching a pure Nash equilibrium; (2) reaching a pure Nash equilibrium in a Bayesian setting; and (3) reaching a mixed Nash equilibrium. We then show that, in contrast, the communication complexity of reaching a correlated equilibrium is polynomial in the number of players.

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  • Hart, Sergiu & Mansour, Yishay, 2010. "How long to equilibrium? The communication complexity of uncoupled equilibrium procedures," Games and Economic Behavior, Elsevier, vol. 69(1), pages 107-126, May.
    • Handle: RePEc:eee:gamebe:v:69:y:2010:i:1:p:107-126
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    2. Tom Johnston & Michael Savery & Alex Scott & Bassel Tarbush, 2023. "Game Connectivity and Adaptive Dynamics," Papers 2309.10609, arXiv.org, revised Nov 2023.
    3. Babichenko, Yakov & Rubinstein, Aviad, 2022. "Communication complexity of approximate Nash equilibria," Games and Economic Behavior, Elsevier, vol. 134(C), pages 376-398.
    4. Pradelski, Bary S.R. & Young, H. Peyton, 2012. "Learning efficient Nash equilibria in distributed systems," Games and Economic Behavior, Elsevier, vol. 75(2), pages 882-897.
    5. 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.
    6. Jiang, Albert Xin & Leyton-Brown, Kevin, 2015. "Polynomial-time computation of exact correlated equilibrium in compact games," Games and Economic Behavior, Elsevier, vol. 91(C), pages 347-359.
    7. Fabrizio Germano & Peio Zuazo-Garin, 2017. "Bounded rationality and correlated equilibria," International Journal of Game Theory, Springer;Game Theory Society, vol. 46(3), pages 595-629, August.
    8. Cheung, Yun Kuen & Cole, Richard & Devanur, Nikhil R., 2020. "Tatonnement beyond gross substitutes? Gradient descent to the rescue," Games and Economic Behavior, Elsevier, vol. 123(C), pages 295-326.
    9. Yakov Babichenko, 2014. "How long to Pareto efficiency?," International Journal of Game Theory, Springer;Game Theory Society, vol. 43(1), pages 13-24, February.
    10. Babichenko, Yakov, 2013. "Best-reply dynamics in large binary-choice anonymous games," Games and Economic Behavior, Elsevier, vol. 81(C), pages 130-144.
    11. Babichenko, Yakov, 2012. "Completely uncoupled dynamics and Nash equilibria," Games and Economic Behavior, Elsevier, vol. 76(1), pages 1-14.
    12. Itai Arieli & H Peyton Young, 2011. "Stochastic Learning Dynamics and Speed of Convergence in Population Games," Economics Series Working Papers 570, University of Oxford, Department of Economics.
    13. Yakov Babichenko, 2018. "Fast Convergence of Best-Reply Dynamics in Aggregative Games," Mathematics of Operations Research, INFORMS, vol. 43(1), pages 333-346, February.
    14. Meir, Reshef & Kalai, Gil & Tennenholtz, Moshe, 2018. "Bidding games and efficient allocations," Games and Economic Behavior, Elsevier, vol. 112(C), pages 166-193.
    15. Jonathan Newton, 2018. "Evolutionary Game Theory: A Renaissance," Games, MDPI, vol. 9(2), pages 1-67, May.
    16. Babaioff, Moshe & Blumrosen, Liad & Schapira, Michael, 2013. "The communication burden of payment determination," Games and Economic Behavior, Elsevier, vol. 77(1), pages 153-167.
    17. Yakov Babichenko, 2012. "Best-Reply Dynamics in Large Anonymous Games," Discussion Paper Series dp600, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.
    18. Amir Danak & Shie Mannor, 2012. "Approximately optimal bidding policies for repeated first-price auctions," Annals of Operations Research, Springer, vol. 196(1), pages 189-199, July.

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    More about this item

    Keywords

    Uncoupled dynamics Nash equilibrium Communication complexity Correlated equilibrium Speed of convergence;

    JEL classification:

    • C78 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Bargaining Theory; Matching Theory
    • C70 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - General
    • C79 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Other

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