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Stochastic Learning Dynamics and Speed of Convergence in Population Games

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  • Itai Arieli
  • H Peyton Young

Abstract

Consider a finite, normal form game G in which each player position is occupied by a population of N individuals, and the payoff to any given individual is the expected payoff from playing against a group drawn at random from the other positions. Assume that individuals adjust their behavior asynchronously via a stochastic better reply dynamic. We show that when G is weakly acyclic, convergence occurs with probability one, but the expected waiting time to come close to Nash equilibrium can grow exponentially in N. Unlike previous results in the literature our results show that Nash convergence can be exponentially slow even in games with very simple payoff structures. We then show that the introduction of aggregate shocks to players' information and/or payoffs can greatly accelerate the learning process. In fact, if G is weakly acyclic and the payoffs are generic, the expected waiting time to come e-close to Nash equilibrium is bounded by a function that is polynomial e-1, exponential in the number of strategies, and independent of the population size N.

Suggested Citation

  • 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.
  • Handle: RePEc:oxf:wpaper:570
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    File URL: http://www.economics.ox.ac.uk/materials/papers/5259/young5702.pdf
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    References listed on IDEAS

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    1. Sandholm, William H. & Hofbauer, Josef, 2011. "Survival of dominated strategies under evolutionary dynamics," Theoretical Economics, Econometric Society, vol. 6(3), September.
    2. Hart, Sergiu & Mas-Colell, Andreu, 2006. "Stochastic uncoupled dynamics and Nash equilibrium," Games and Economic Behavior, Elsevier, vol. 57(2), pages 286-303, November.
    3. Foster, Dean P. & Young, H. Peyton, 2003. "Learning, hypothesis testing, and Nash equilibrium," Games and Economic Behavior, Elsevier, vol. 45(1), pages 73-96, October.
    4. Foster, Dean P. & Young, H. 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. 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.
    7. Germano, Fabrizio & Lugosi, Gabor, 2007. "Global Nash convergence of Foster and Young's regret testing," Games and Economic Behavior, Elsevier, vol. 60(1), pages 135-154, July.
    8. Sergiu Hart & Andreu Mas-Colell, 2003. "Uncoupled Dynamics Do Not Lead to Nash Equilibrium," American Economic Review, American Economic Association, vol. 93(5), pages 1830-1836, December.
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    Cited by:

    1. Babichenko, Yakov, 2013. "Best-reply dynamics in large binary-choice anonymous games," Games and Economic Behavior, Elsevier, vol. 81(C), pages 130-144.

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