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An Evolutionary Interpretation Of Mixed-Strategy Equilibria

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  • Joerg Oechssler

    (Columbia University)

Abstract

One of the more convincing interpretations of mixed strategy equilibria describes a mixed equilibrium as a steady state in a large population in which all players use pure strategies but the population as a whole mimics a mixed strategy. To be complete, however, this interpretation requires a good story about how the population arrives at the appropriate distribution over pure strategies. In this paper I attempt to give an explanation based on an evolutionary, stochastic learning process. Convergence properties of these processes have been studied extensively but almost exclusively for the case of convergence to pure Nash equilibria. Here I study the conditions under which an evolutionary process converges to population mixed-strategy equilibria. I find that not all mixed equilibria can be justified as the result of the evolutionary learning process even if the equilibrium is unique. For symmetric 2x2 and 3x3 games I give necessary and sufficient conditions for convergence and for n*n games I give a sufficient condition. For cases in which the conditions are not satisfied counterexamples are given, in which the process enters a limit cycle.

Suggested Citation

  • Joerg Oechssler, 1994. "An Evolutionary Interpretation Of Mixed-Strategy Equilibria," Game Theory and Information 9404001, University Library of Munich, Germany.
  • Handle: RePEc:wpa:wuwpga:9404001
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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. Ellison, Glenn, 1993. "Learning, Local Interaction, and Coordination," Econometrica, Econometric Society, vol. 61(5), pages 1047-1071, September.
    3. Jordan J. S., 1993. "Three Problems in Learning Mixed-Strategy Nash Equilibria," Games and Economic Behavior, Elsevier, vol. 5(3), pages 368-386, July.
    4. Robson, Arthur J. & Vega-Redondo, Fernando, 1996. "Efficient Equilibrium Selection in Evolutionary Games with Random Matching," Journal of Economic Theory, Elsevier, vol. 70(1), pages 65-92, July.
    5. Ritzberger, Klaus & Weibull, Jorgen W, 1995. "Evolutionary Selection in Normal-Form Games," Econometrica, Econometric Society, vol. 63(6), pages 1371-1399, November.
    6. Rosenthal, R W, 1979. "Sequences of Games with Varying Opponents," Econometrica, Econometric Society, vol. 47(6), pages 1353-1366, November.
    7. J&00F6;rg Oechssler, 1999. "Competition Among Conventions," Computational and Mathematical Organization Theory, Springer, vol. 5(1), pages 31-44, May.
    8. Kandori Michihiro & Rob Rafael, 1995. "Evolution of Equilibria in the Long Run: A General Theory and Applications," Journal of Economic Theory, Elsevier, vol. 65(2), pages 383-414, April.
    9. Young, H Peyton, 1993. "The Evolution of Conventions," Econometrica, Econometric Society, vol. 61(1), pages 57-84, January.
    10. Noldeke Georg & Samuelson Larry, 1993. "An Evolutionary Analysis of Backward and Forward Induction," Games and Economic Behavior, Elsevier, vol. 5(3), pages 425-454, July.
    11. Kandori, Michihiro & Mailath, George J & Rob, Rafael, 1993. "Learning, Mutation, and Long Run Equilibria in Games," Econometrica, Econometric Society, vol. 61(1), pages 29-56, January.
    12. Bergin, James & Lipman, Barton L, 1996. "Evolution with State-Dependent Mutations," Econometrica, Econometric Society, vol. 64(4), pages 943-956, July.
    13. Samuelson, L., 1991. "How to Tremble if you Must," Working papers 9122, Wisconsin Madison - Social Systems.
    14. Martin J. Osborne & Ariel Rubinstein, 1994. "A Course in Game Theory," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262650401.
    15. Oechssler, Jorg, 1997. "Decentralization and the coordination problem," Journal of Economic Behavior & Organization, Elsevier, vol. 32(1), pages 119-135, January.
    16. Ariel Rubinstein, 1988. "Comments on the interpretation of game theory (Now published in Econometrica, 59 (1991), pp.909-924.)," STICERD - Theoretical Economics Paper Series 181, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
    17. Crawford, Vincent P., 1985. "Learning behavior and mixed-strategy Nash equilibria," Journal of Economic Behavior & Organization, Elsevier, vol. 6(1), pages 69-78, March.
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    Cited by:

    1. Marie-Laure Cabon-Dhersin & Nathalie Etchart-Vincent, 2013. "Wording and gender effects in a Game of Chicken. An explorative experimental study," Working Papers hal-00796708, HAL.
    2. Carlos Alós-Ferrer, 2003. "Finite Population Dynamics and Mixed Equilibria," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 5(03), pages 263-290.
    3. Marie-Laure Cabon-Dhersin & Nathalie Etchart-Vincent, 2011. "Cooperation: the power of a single word? Some experimental evidence on wording and gender effects in a Game of Chicken," Working Papers hal-00741973, HAL.
    4. DeMichelis, Stefano & Germano, Fabrizio, 2000. "On the Indices of Zeros of Nash Fields," Journal of Economic Theory, Elsevier, vol. 94(2), pages 192-217, October.
    5. Oechssler, Jorg, 1997. "Decentralization and the coordination problem," Journal of Economic Behavior & Organization, Elsevier, vol. 32(1), pages 119-135, January.
    6. Fei Shi, 2015. "Long-run technology choice with endogenous local capacity," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 59(2), pages 377-399, June.
    7. Jörg Oechssler & Karl H Schlag, 1997. "Loss of Commitment? An Evolutionary Analysis of Bagwell’s Example," Levine's Working Paper Archive 598, David K. Levine.
    8. Carlos Alós-Ferrer & Georg Kirchsteiger & Markus Walzl, 2010. "On the Evolution of Market Institutions: The Platform Design Paradox," Economic Journal, Royal Economic Society, vol. 120(543), pages 215-243, March.
    9. Demichelis, Stefano & Germano, Fabrizio, 2002. "On (un)knots and dynamics in games," Games and Economic Behavior, Elsevier, vol. 41(1), pages 46-60, October.
    10. Jorg Oechssler & Karl Schlag, 1997. "An Evolutionary Analysis of Bagwell's Example," Game Theory and Information 9704001, University Library of Munich, Germany, revised 11 Apr 1997.
    11. Carlos Alós-Ferrer & Nick Netzer, 2015. "Robust stochastic stability," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 58(1), pages 31-57, January.
    12. Thomas Norman, 2010. "Cycles versus equilibrium in evolutionary games," Theory and Decision, Springer, vol. 69(2), pages 167-182, August.
    13. Wilfred Amaldoss & Robert J. Meyer & Jagmohan S. Raju & Amnon Rapoport, 2000. "Collaborating to Compete," Marketing Science, INFORMS, vol. 19(2), pages 105-126, November.

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    More about this item

    JEL classification:

    • C7 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory
    • D8 - Microeconomics - - Information, Knowledge, and Uncertainty

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