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An Evolutionary Interpretation of Mixed-Strategy Equilibria

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  • Oechssler, Jorg

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.

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Bibliographic Info

Article provided by Elsevier in its journal Games and Economic Behavior.

Volume (Year): 21 (1997)
Issue (Month): 1-2 (October)
Pages: 203-237

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Handle: RePEc:eee:gamebe:v:21:y:1997:i:1-2:p:203-237

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Web page: http://www.elsevier.com/locate/inca/622836

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  1. Ritzberger, Klaus & Weibull, Jorgen W, 1995. "Evolutionary Selection in Normal-Form Games," Econometrica, Econometric Society, vol. 63(6), pages 1371-99, November.
  2. Glen Ellison, 2010. "Learning, Local Interaction, and Coordination," Levine's Working Paper Archive 391, David K. Levine.
  3. Arthur J Robson & Fernando Vega-Redondo, 1999. "Efficient Equilibrium Selection in Evolutionary Games with Random Matching," Levine's Working Paper Archive 2112, David K. Levine.
  4. Kandori, M. & Mailath, G.J., 1991. "Learning, Mutation, And Long Run Equilibria In Games," Papers 71, Princeton, Woodrow Wilson School - John M. Olin Program.
  5. Martin J. Osborne & Ariel Rubinstein, 1994. "A Course in Game Theory," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262650401, January.
  6. Bergin, James & Lipman, Barton L, 1996. "Evolution with State-Dependent Mutations," Econometrica, Econometric Society, vol. 64(4), pages 943-56, July.
  7. Fudenberg, D. & Kreps, D.M., 1992. "Learning Mixed Equilibria," Working papers 92-13, Massachusetts Institute of Technology (MIT), Department of Economics.
  8. Noeldecke,Georg & Samuelson,Larry, . "An evolutionary analysis of backward and forward induction," Discussion Paper Serie B 228, University of Bonn, Germany.
  9. 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.
  10. Young, H Peyton, 1993. "The Evolution of Conventions," Econometrica, Econometric Society, vol. 61(1), pages 57-84, January.
  11. repec:fth:coluec:607 is not listed on IDEAS
  12. Jordan J. S., 1993. "Three Problems in Learning Mixed-Strategy Nash Equilibria," Games and Economic Behavior, Elsevier, vol. 5(3), pages 368-386, July.
  13. Joerg Oechssler, 1994. "Decentralization and the Coordination Problem," Game Theory and Information 9403004, EconWPA.
  14. Crawford, Vincent P., 1985. "Learning behavior and mixed-strategy Nash equilibria," Journal of Economic Behavior & Organization, Elsevier, vol. 6(1), pages 69-78, March.
  15. Joerg Oechssler, 1993. "Competition among Conventions," Game Theory and Information 9312001, EconWPA, revised 04 Dec 1993.
  16. repec:att:wimass:9122 is not listed on IDEAS
  17. Rosenthal, R W, 1979. "Sequences of Games with Varying Opponents," Econometrica, Econometric Society, vol. 47(6), pages 1353-66, November.
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Citations

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Cited by:
  1. Demichelis, Stefano & Germano, Fabrizio, 2002. "On (un)knots and dynamics in games," Games and Economic Behavior, Elsevier, vol. 41(1), pages 46-60, October.
  2. Jorg Oechssler & Karl Schlag, 1997. "An Evolutionary Analysis of Bagwell's Example," Game Theory and Information 9704001, EconWPA, revised 11 Apr 1997.
  3. Thomas Norman, 2010. "Cycles versus equilibrium in evolutionary games," Theory and Decision, Springer, vol. 69(2), pages 167-182, August.
  4. 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.
  5. Oechssler, Jörg & Schlag, Karl H., 1997. "Loss of commitment? An evolutionary analysis of Bagwell's example," SFB 373 Discussion Papers 1997,39, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
  6. 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.
  7. 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.
  8. DeMichelis, S. & Germano, F., 2000. "On Knots and Dynamics in Games," Papers 2-2000, Tel Aviv.
  9. Carlos Alós-Ferrer, 2000. "Finite Population Dynamics and Mixed Equilibria," Vienna Economics Papers 0008, University of Vienna, Department of Economics.
  10. Marie-Laure Cabon-Dhersin & Nathalie Etchart-Vincent, 2013. "Wording and gender effects in a Game of Chicken. An explorative experimental study," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-00796708, HAL.
  11. Joerg Oechssler, 1994. "Decentralization and the Coordination Problem," Game Theory and Information 9403004, EconWPA.

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