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A Multinomial Probit Model of Stochastic Evolution

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  • David P. Myatt
  • Chris Wallace

Abstract

A strategy revision process in symmetric normal form games is proposed. Following Kandori, Mailath, and Rob (1993), members of a population periodically revise their strategy choice, and choose a myopic best response to currently observed play. Their payoffs are perturbed by normally distributed Harsanyian (1973) trembles, so that strategies are chosen according to multinomial probit probabilities. As the variance of payoffs is allowed to vanish, the graph theoretic methods of the earlier literature continue to apply. The distributional assumption enables a convenient closed form characterisation for the weights of the rooted trees. An illustration of the approach is offered, via a consideration of the role of dominated strategies in equilibrium selection.

Suggested Citation

  • David P. Myatt & Chris Wallace, 2002. "A Multinomial Probit Model of Stochastic Evolution," Economics Series Working Papers 90, University of Oxford, Department of Economics.
    • Handle: RePEc:oxf:wpaper:90
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    References listed on IDEAS

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    1. John C. Harsanyi & Reinhard Selten, 1988. "A General Theory of Equilibrium Selection in Games," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262582384, April.
    2. Glenn Ellison, 2000. "Basins of Attraction, Long-Run Stochastic Stability, and the Speed of Step-by-Step Evolution," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 67(1), pages 17-45.
    3. Morris, Stephen & Rob, Rafael & Shin, Hyun Song, 1995. "Dominance and Belief Potential," Econometrica, Econometric Society, vol. 63(1), pages 145-157, January.
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    5. Maruta, Toshimasa, 1997. "On the Relationship between Risk-Dominance and Stochastic Stability," Games and Economic Behavior, Elsevier, vol. 19(2), pages 221-234, May.
    6. 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.
    7. Fernando Vega-Redondo, 1997. "The Evolution of Walrasian Behavior," Econometrica, Econometric Society, vol. 65(2), pages 375-384, March.
    8. Bergin, James & Lipman, Barton L, 1996. "Evolution with State-Dependent Mutations," Econometrica, Econometric Society, vol. 64(4), pages 943-956, July.
    9. Myatt, David P. & Wallace, Chris C., 2004. "Adaptive play by idiosyncratic agents," Games and Economic Behavior, Elsevier, vol. 48(1), pages 124-138, July.
    Full references (including those not matched with items on IDEAS)

    Citations

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    Cited by:

    1. Alós-Ferrer, Carlos & Netzer, Nick, 2010. "The logit-response dynamics," Games and Economic Behavior, Elsevier, vol. 68(2), pages 413-427, March.
    2. Myatt, David P. & Wallace, Chris, 2008. "An evolutionary analysis of the volunteer's dilemma," Games and Economic Behavior, Elsevier, vol. 62(1), pages 67-76, January.
    3. Dokumacı, Emin & Sandholm, William H., 2011. "Large deviations and multinomial probit choice," Journal of Economic Theory, Elsevier, vol. 146(5), pages 2151-2158.
    4. William H. Sandholm & Mathias Staudigl, 2018. "Sample Path Large Deviations for Stochastic Evolutionary Game Dynamics," Mathematics of Operations Research, INFORMS, vol. 43(4), pages 1348-1377, November.
    5. Newton, Jonathan & Sawa, Ryoji, 2015. "A one-shot deviation principle for stability in matching problems," Journal of Economic Theory, Elsevier, vol. 157(C), pages 1-27.
    6. Klaus, Bettina & Newton, Jonathan, 2016. "Stochastic stability in assignment problems," Journal of Mathematical Economics, Elsevier, vol. 62(C), pages 62-74.
    7. 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.
    8. Hwang, Sung-Ha & Rey-Bellet, Luc, 2021. "Positive feedback in coordination games: Stochastic evolutionary dynamics and the logit choice rule," Games and Economic Behavior, Elsevier, vol. 126(C), pages 355-373.
    9. Staudigl, Mathias, 2011. "Potential games in volatile environments," Games and Economic Behavior, Elsevier, vol. 72(1), pages 271-287, May.
    10. Arigapudi, Srinivas, 2020. "Exit from equilibrium in coordination games under probit choice," Games and Economic Behavior, Elsevier, vol. 122(C), pages 168-202.
    11. Sawa, Ryoji, 2021. "A stochastic stability analysis with observation errors in normal form games," Games and Economic Behavior, Elsevier, vol. 129(C), pages 570-589.
    12. Simon Weidenholzer, 2010. "Coordination Games and Local Interactions: A Survey of the Game Theoretic Literature," Games, MDPI, vol. 1(4), pages 1-35, November.
    13. Bilancini, Ennio & Boncinelli, Leonardo & Nax, Heinrich H., 2021. "What noise matters? Experimental evidence for stochastic deviations in social norms," Journal of Behavioral and Experimental Economics (formerly The Journal of Socio-Economics), Elsevier, vol. 90(C).
    14. Roberto Rozzi, 2021. "Competing Conventions with Costly Information Acquisition," Games, MDPI, vol. 12(3), pages 1-29, June.
    15. Arigapudi, Srinivas, 2024. "Transitions between equilibria in Bilingual Games under Probit Choice," Journal of Mathematical Economics, Elsevier, vol. 111(C).
    16. Staudigl, Mathias, 2012. "Stochastic stability in asymmetric binary choice coordination games," Games and Economic Behavior, Elsevier, vol. 75(1), pages 372-401.
    17. Kim, Chongmin & Wong, Kam-Chau, 2010. "Long-run equilibria with dominated strategies," Games and Economic Behavior, Elsevier, vol. 68(1), pages 242-254, January.
    18. Sawa, Ryoji & Wu, Jiabin, 2018. "Reference-dependent preferences, super-dominance and stochastic stability," Journal of Mathematical Economics, Elsevier, vol. 78(C), pages 96-104.
    19. 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.
    20. Kevin Hasker, 2014. "The Emergent Seed: A Representation Theorem for Models of Stochastic Evolution and two formulas for Waiting Time," Levine's Working Paper Archive 786969000000000954, David K. Levine.
    21. Sandholm, William H. & Staudigl, Mathias, 2016. "Large Deviations and Stochastic Stability in the Small Noise Double Limit, II: The Logit Model," Center for Mathematical Economics Working Papers 506, Center for Mathematical Economics, Bielefeld University.
    22. Daniel Christopher Opolot, 2022. "On the relationship between p-dominance and stochastic stability in network games," International Journal of Game Theory, Springer;Game Theory Society, vol. 51(2), pages 307-351, June.
    23. Akira Okada & Ryoji Sawa, 2016. "An evolutionary approach to social choice problems with q-quota rules," KIER Working Papers 936, Kyoto University, Institute of Economic Research.

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    More about this item

    Keywords

    multinational probit; stochastic evolution; idiosyncrasy; dominated strategies;
    All these keywords.

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • C73 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Stochastic and Dynamic Games; Evolutionary Games

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