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Learning in Perturbed Asymmetric Games

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Abstract

We investigate the stability of mixed strategy equilibria in 2 person (bimatrix) games under perturbed best response dynamics. A mixed equilibrium is asymptotically stable under all such dynamics if and only if the game is linearly equivalent to a zero sum game. In this case, the mixed equilibrium is also globally asymptotically stable. Global convergence to the set of perturbed equilibria is shown also for (rescaled) partnership games (also know as games of identical interest). Some applications of these results to stochastic learning models are given.

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  • Josef Hofbauer & Ed Hopkins, 2000. "Learning in Perturbed Asymmetric Games," ESE Discussion Papers 53, Edinburgh School of Economics, University of Edinburgh.
  • Handle: RePEc:edn:esedps:53
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    Cited by:

    1. Joseph Abdou & Nikolaos Pnevmatikos & Marco Scarsini, 2014. "Uniformity and games decomposition," Documents de travail du Centre d'Economie de la Sorbonne 14084, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    2. Jim Engle-Warnick & Ed Hopkins, 2006. "A Simple Test of Learning Theory," Levine's Bibliography 321307000000000724, UCLA Department of Economics.
    3. Fudenberg, Drew & Takahashi, Satoru, 2011. "Heterogeneous beliefs and local information in stochastic fictitious play," Games and Economic Behavior, Elsevier, vol. 71(1), pages 100-120, January.
    4. Benndorf, Volker & Martínez-Martínez, Ismael, 2017. "Perturbed best response dynamics in a hawk–dove game," Economics Letters, Elsevier, vol. 153(C), pages 61-64.
    5. Kets, W., 2007. "The Minority Game : An Economics Perspective," Discussion Paper 2007-53, Tilburg University, Center for Economic Research.
    6. Andriy Zapechelnyuk, 2009. "Limit Behavior of No-regret Dynamics," Discussion Papers 21, Kyiv School of Economics.
    7. Benaïm, Michel & Hofbauer, Josef & Hopkins, Ed, 2009. "Learning in games with unstable equilibria," Journal of Economic Theory, Elsevier, vol. 144(4), pages 1694-1709, July.
    8. Hofbauer, Josef & Sandholm, William H., 2007. "Evolution in games with randomly disturbed payoffs," Journal of Economic Theory, Elsevier, vol. 132(1), pages 47-69, January.
    9. Mengel, Friederike, 2012. "Learning across games," Games and Economic Behavior, Elsevier, vol. 74(2), pages 601-619.
    10. repec:eee:gamebe:v:107:y:2018:i:c:p:109-122 is not listed on IDEAS
    11. Xing Gao & Weijun Zhong & Shue Mei, 2013. "Stochastic Evolutionary Game Dynamics and Their Selection Mechanisms," Computational Economics, Springer;Society for Computational Economics, vol. 41(2), pages 233-247, February.
    12. Häfner, Samuel, 2018. "Stable biased sampling," Games and Economic Behavior, Elsevier, vol. 107(C), pages 109-122.
    13. Hoffmann, Eric, 2016. "On the learning and stability of mixed strategy Nash equilibria in games of strategic substitutes," Journal of Economic Behavior & Organization, Elsevier, vol. 130(C), pages 349-362.
    14. Sandholm, William H., 2007. "Evolution in Bayesian games II: Stability of purified equilibria," Journal of Economic Theory, Elsevier, vol. 136(1), pages 641-667, September.
    15. Kets, W., 2008. "Networks and learning in game theory," Other publications TiSEM 7713fce1-3131-498c-8c6f-3, Tilburg University, School of Economics and Management.
    16. Josef Hofbauer & William H. Sandholm, 2001. "Evolution and Learning in Games with Randomly Disturbed Payoffs," Vienna Economics Papers 0205, University of Vienna, Department of Economics.
    17. Hopkins, Ed & Posch, Martin, 2005. "Attainability of boundary points under reinforcement learning," Games and Economic Behavior, Elsevier, vol. 53(1), pages 110-125, October.
    18. Perkins, S. & Leslie, D.S., 2014. "Stochastic fictitious play with continuous action sets," Journal of Economic Theory, Elsevier, vol. 152(C), pages 179-213.
    19. Kets, W. & Voorneveld, M., 2007. "Congestion, Equilibrium and Learning : The Minority Game," Discussion Paper 2007-61, Tilburg University, Center for Economic Research.
    20. Golman, Russell, 2012. "Homogeneity bias in models of discrete choice with bounded rationality," Journal of Economic Behavior & Organization, Elsevier, vol. 82(1), pages 1-11.
    21. Dai, Darong, 2012. "Learning Nash Equilibria," MPRA Paper 40040, University Library of Munich, Germany.
    22. Leslie, David S. & Collins, E.J., 2006. "Generalised weakened fictitious play," Games and Economic Behavior, Elsevier, vol. 56(2), pages 285-298, August.
    23. Ratul, Lahkar, 2011. "The dynamic instability of dispersed price equilibria," Journal of Economic Theory, Elsevier, vol. 146(5), pages 1796-1827, September.
    24. Russell, Golman, 2011. "Quantal response equilibria with heterogeneous agents," Journal of Economic Theory, Elsevier, vol. 146(5), pages 2013-2028, September.

    More about this item

    Keywords

    games; learning; best response dynamics; stochastic fictitious play; mixed strategy equilibria; zero sum games;

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • D83 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Search; Learning; Information and Knowledge; Communication; Belief; Unawareness

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