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Fixation Probabilities of Strategies for Bimatrix Games in Finite Populations

Author

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  • Takuya Sekiguchi

    (Japan Society for the Promotion of Science
    SOKENDAI (The Graduate University for Advanced Studies))

  • Hisashi Ohtsuki

    (SOKENDAI (The Graduate University for Advanced Studies))

Abstract

Recent developments in stochastic evolutionary game theory in finite populations yield insights that complement the conventional deterministic evolutionary game theory in infinite populations. However, most studies of stochastic evolutionary game theory have investigated dynamics of symmetric games, although not all social and biological phenomena are described by symmetric games, e.g., social interactions between individuals having conflicting preferences or different roles. In this paper, we describe the stochastic evolutionary dynamics of two-player $$2 \times 2$$ 2 × 2 bimatrix games in finite populations. The stochastic process is modeled by a frequency-dependent Moran process without mutation. We obtained the fixation probability that the evolutionary dynamics starting from a given initial state converges to a specific absorbing state. Applying the formula to the ultimatum game, we show that evolutionary dynamics favors fairness. Furthermore, we present two novel concepts of stability for bimatrix games, based on our formula for the fixation probability, and demonstrate that one of the two serves as a criterion for equilibrium selection.

Suggested Citation

  • Takuya Sekiguchi & Hisashi Ohtsuki, 2017. "Fixation Probabilities of Strategies for Bimatrix Games in Finite Populations," Dynamic Games and Applications, Springer, vol. 7(1), pages 93-111, March.
    • Handle: RePEc:spr:dyngam:v:7:y:2017:i:1:d:10.1007_s13235-015-0170-2
    DOI: 10.1007/s13235-015-0170-2
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    Cited by:

    1. Takuya Sekiguchi, 2023. "Fixation Probabilities of Strategies for Trimatrix Games and Their Applications to Triadic Conflict," Dynamic Games and Applications, Springer, vol. 13(3), pages 1005-1033, September.
    2. Fukutomi, Masao & Kurokawa, Shun, 2018. "How much cost should reciprocators pay in order to distinguish the opponent's cooperation from the opponent's defection?," Applied Mathematics and Computation, Elsevier, vol. 336(C), pages 301-314.
    3. Sekiguchi, Takuya, 2023. "Abundance of strategies for trimatrix games in finite populations," Applied Mathematics and Computation, Elsevier, vol. 448(C).
    4. Marta C. Couto & Saptarshi Pal, 2023. "Introspection Dynamics in Asymmetric Multiplayer Games," Dynamic Games and Applications, Springer, vol. 13(4), pages 1256-1285, December.
    5. Lv, Shaojie & Zhao, Changheng & Li, Jiaying, 2022. "Generosity in public goods game with the aspiration-driven rule," Chaos, Solitons & Fractals, Elsevier, vol. 165(P2).

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