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Large Deviations and Stochastic Stability in the Small Noise Double Limit, II: The Logit Model

Author

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  • Sandholm, William H.

    (Center for Mathematical Economics, Bielefeld University)

  • Staudigl, Mathias

    (Center for Mathematical Economics, Bielefeld University)

Abstract

We describe the large deviations properties, stationary distribution asymptotics, and stochastically stable states of stochastic evolutionary processes based on the logit choice rule, focusing on behavior in the small noise double limit. These aspects of the stochastic evolutionary process can be characterized in terms of solutions to certain minimum cost path problems. We solve these problems explicitly using tools from optimal control theory. The analysis focuses on three-strategy coordination games that satisfy the marginal bandwagon property and that have an interior equilibrium, but our approach can be applied to other classes of games and other choice rules.

Suggested Citation

  • 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.
    • Handle: RePEc:bie:wpaper:506
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    File URL: https://pub.uni-bielefeld.de/download/2901490/2902027
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    References listed on IDEAS

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    11. 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.
    12. Myatt, David P. & Wallace, Chris, 2003. "A multinomial probit model of stochastic evolution," Journal of Economic Theory, Elsevier, vol. 113(2), pages 286-301, December.
    13. Staudigl, Mathias, 2012. "Stochastic stability in asymmetric binary choice coordination games," Games and Economic Behavior, Elsevier, vol. 75(1), pages 372-401.
    14. Alós-Ferrer, Carlos & Netzer, Nick, 2010. "The logit-response dynamics," Games and Economic Behavior, Elsevier, vol. 68(2), pages 413-427, March.
    15. Sandholm, William H., 2001. "Potential Games with Continuous Player Sets," Journal of Economic Theory, Elsevier, vol. 97(1), pages 81-108, March.
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    Citations

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    Cited by:

    1. Arigapudi, Srinivas, 2020. "Transitions between equilibria in bilingual games under logit choice," Journal of Mathematical Economics, Elsevier, vol. 86(C), pages 24-34.
    2. Sung-Ha Hwang & Jonathan Newton, 2017. "Payoff-dependent dynamics and coordination games," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 64(3), pages 589-604, October.
    3. 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.
    4. Arigapudi, Srinivas, 2020. "Exit from equilibrium in coordination games under probit choice," Games and Economic Behavior, Elsevier, vol. 122(C), pages 168-202.
    5. 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.
    6. Sawa, Ryoji & Wu, Jiabin, 2023. "Statistical inference in evolutionary dynamics," Games and Economic Behavior, Elsevier, vol. 137(C), pages 294-316.
    7. 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.
    8. Konstantin Avrachenkov & Vivek S. Borkar, 2019. "Metastability in Stochastic Replicator Dynamics," Dynamic Games and Applications, Springer, vol. 9(2), pages 366-390, June.
    9. 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.
    10. Jonathan Newton, 2018. "Evolutionary Game Theory: A Renaissance," Games, MDPI, vol. 9(2), pages 1-67, May.
    11. Sawa, Ryoji, 2021. "A prospect theory Nash bargaining solution and its stochastic stability," Journal of Economic Behavior & Organization, Elsevier, vol. 184(C), pages 692-711.
    12. Sawa, Ryoji & Wu, Jiabin, 2018. "Reference-dependent preferences, super-dominance and stochastic stability," Journal of Mathematical Economics, Elsevier, vol. 78(C), pages 96-104.
    13. Sawa, Ryoji & Wu, Jiabin, 2018. "Prospect dynamics and loss dominance," Games and Economic Behavior, Elsevier, vol. 112(C), pages 98-124.
    14. 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.
    15. 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.
    16. Williams, Noah, 2022. "Learning and equilibrium transitions: Stochastic stability in discounted stochastic fictitious play," Journal of Economic Dynamics and Control, Elsevier, vol. 145(C).

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