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Monotone Imitation Dynamics in Large Populations

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  • Imhof, Lorens
  • Fudenberg, Drew

Abstract

We analyze a class of imitation dynamics with mutations for games with any finite number of actions, and give conditions for the selection of a unique equilibrium as the mutation rate becomes small and the population becomes large. Our results cover the multiple-action extensions of the aspiration-and-imitation process of Binmore and Samuelson [Muddling through: noisy equilibrium selection, J. Econ. Theory 74 (1997) 235–265] and the related processes proposed by Benaı¨m and Weibull [Deterministic approximation of stochastic evolution in games, Econometrica 71 (2003) 873–903] and Traulsen et al. [Coevolutionary dynamics: from finite to infinite populations, Phys. Rev. Lett. 95 (2005) 238701], as well as the frequency-dependent Moran process studied by Fudenberg et al. [Evolutionary game dynamics in finite populations with strong selection and weak mutation, Theoretical Population Biol. 70 (2006) 352–363]. We illustrate our results by considering the effect of the number of periods of repetition on the selected equilibrium in repeated play of the prisoner's dilemma when players are restricted to a small set of simple strategies.

Suggested Citation

  • Imhof, Lorens & Fudenberg, Drew, 2008. "Monotone Imitation Dynamics in Large Populations," Scholarly Articles 3196338, Harvard University Department of Economics.
    • Handle: RePEc:hrv:faseco:3196338
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    References listed on IDEAS

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    1. Fudenberg Drew & Kreps David M., 1993. "Learning Mixed Equilibria," Games and Economic Behavior, Elsevier, vol. 5(3), pages 320-367, July.
    2. Fudenberg, Drew & Levine, David, 1998. "Learning in games," European Economic Review, Elsevier, vol. 42(3-5), pages 631-639, May.
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    9. Drew Fudenberg & David K. Levine, 1998. "The Theory of Learning in Games," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262061945, December.
    10. P. Young, 1999. "The Evolution of Conventions," Levine's Working Paper Archive 485, David K. Levine.
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    13. Martin A. Nowak & Akira Sasaki & Christine Taylor & Drew Fudenberg, 2004. "Emergence of cooperation and evolutionary stability in finite populations," Nature, Nature, vol. 428(6983), pages 646-650, April.
    14. Fudenberg, Drew & Imhof, Lorens A., 2006. "Imitation processes with small mutations," Journal of Economic Theory, Elsevier, vol. 131(1), pages 251-262, November.
    15. Binmore, Ken & Samuelson, Larry, 1997. "Muddling Through: Noisy Equilibrium Selection," Journal of Economic Theory, Elsevier, vol. 74(2), pages 235-265, June.
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    Cited by:

    1. Veller, Carl & Hayward, Laura K., 2016. "Finite-population evolution with rare mutations in asymmetric games," Journal of Economic Theory, Elsevier, vol. 162(C), pages 93-113.
    2. Michel Benaïm & Jörgen Weibull, 2009. "Mean-field approximation of stochastic population processes in games," Working Papers hal-00435515, HAL.
    3. Sandholm, William H., 2012. "Stochastic imitative game dynamics with committed agents," Journal of Economic Theory, Elsevier, vol. 147(5), pages 2056-2071.
    4. Izquierdo, Luis R. & Izquierdo, Segismundo S. & Sandholm, William H., 2019. "An introduction to ABED: Agent-based simulation of evolutionary game dynamics," Games and Economic Behavior, Elsevier, vol. 118(C), pages 434-462.
    5. Darong DAI & Kunrong SHEN, 2014. "Stochastic evolutionary cartel formation," Theoretical and Applied Economics, Asociatia Generala a Economistilor din Romania / Editura Economica, vol. 0(1(590)), pages 7-26, January.
    6. Ellison, Glenn & Fudenberg, Drew & Imhof, Lorens A., 2009. "Random matching in adaptive dynamics," Games and Economic Behavior, Elsevier, vol. 66(1), pages 98-114, May.
    7. repec:agr:journl:v:1(590):y:2014:i:1(590):p:7-26 is not listed on IDEAS
    8. Sandholm, William H. & Staudigl, Mathias, 2016. "Large Deviations and Stochastic Stability in the Small Noise Double Limit, I: Theory," Center for Mathematical Economics Working Papers 505, Center for Mathematical Economics, Bielefeld University.
    9. Jakub Bielawski & {L}ukasz Cholewa & Fryderyk Falniowski, 2024. "The emergence of chaos in population game dynamics induced by comparisons," Papers 2412.06037, arXiv.org, revised Apr 2025.
    10. Chatterjee, Krishnendu & Reiter, Johannes G. & Nowak, Martin A., 2012. "Evolutionary dynamics of biological auctions," Theoretical Population Biology, Elsevier, vol. 81(1), pages 69-80.
    11. Man, Priscilla T.Y., 2012. "Efficiency and stochastic stability in normal form games," Games and Economic Behavior, Elsevier, vol. 76(1), pages 272-284.

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