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On Equilibrium Properties of the Replicator–Mutator Equation in Deterministic and Random Games

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  • Manh Hong Duong

    (University of Birmingham)

  • The Anh Han

    (Teesside University)

Abstract

In this paper, we study the number of equilibria of the replicator–mutator dynamics for both deterministic and random multi-player two-strategy evolutionary games. For deterministic games, using Descartes’ rule of signs, we provide a formula to compute the number of equilibria in multi-player games via the number of change of signs in the coefficients of a polynomial. For two-player social dilemmas (namely the Prisoner’s Dilemma, Snow Drift, Stag Hunt and Harmony), we characterize (stable) equilibrium points and analytically calculate the probability of having a certain number of equilibria when the payoff entries are uniformly distributed. For multi-player random games whose pay-offs are independently distributed according to a normal distribution, by employing techniques from random polynomial theory, we compute the expected or average number of internal equilibria. In addition, we perform extensive simulations by sampling and averaging over a large number of possible payoff matrices to compare with and illustrate analytical results. Numerical simulations also suggest several interesting behaviours of the average number of equilibria when the number of players is sufficiently large or when the mutation is sufficiently small. In general, we observe that introducing mutation results in a larger average number of internal equilibria than when mutation is absent, implying that mutation leads to larger behavioural diversity in dynamical systems. Interestingly, this number is largest when mutation is rare rather than when it is frequent.

Suggested Citation

  • Manh Hong Duong & The Anh Han, 2020. "On Equilibrium Properties of the Replicator–Mutator Equation in Deterministic and Random Games," Dynamic Games and Applications, Springer, vol. 10(3), pages 641-663, September.
    • Handle: RePEc:spr:dyngam:v:10:y:2020:i:3:d:10.1007_s13235-019-00338-8
    DOI: 10.1007/s13235-019-00338-8
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    References listed on IDEAS

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    1. Chaitanya Gokhale & Arne Traulsen, 2014. "Evolutionary Multiplayer Games," Dynamic Games and Applications, Springer, vol. 4(4), pages 468-488, December.
    2. Fudenberg, D. & Harris, C., 1992. "Evolutionary dynamics with aggregate shocks," Journal of Economic Theory, Elsevier, vol. 57(2), pages 420-441, August.
    3. Han, The Anh & Traulsen, Arne & Gokhale, Chaitanya S., 2012. "On equilibrium properties of evolutionary multi-player games with random payoff matrices," Theoretical Population Biology, Elsevier, vol. 81(4), pages 264-272.
    4. Manh Hong Duong & Hoang Minh Tran & The Anh Han, 2019. "On the Expected Number of Internal Equilibria in Random Evolutionary Games with Correlated Payoff Matrix," Dynamic Games and Applications, Springer, vol. 9(2), pages 458-485, June.
    5. Manh Hong Duong & The Anh Han, 2016. "On the Expected Number of Equilibria in a Multi-player Multi-strategy Evolutionary Game," Dynamic Games and Applications, Springer, vol. 6(3), pages 324-346, September.
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    Cited by:

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    2. Manh Hong Duong & The Anh Han, 2021. "Statistics of the number of equilibria in random social dilemma evolutionary games with mutation," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 94(8), pages 1-13, August.
    3. Wang, Chaoqian & Szolnoki, Attila, 2022. "Involution game with spatio-temporal heterogeneity of social resources," Applied Mathematics and Computation, Elsevier, vol. 430(C).
    4. Mansoor Saburov, 2022. "On Discrete-Time Replicator Equations with Nonlinear Payoff Functions," Dynamic Games and Applications, Springer, vol. 12(2), pages 643-661, June.
    5. Chen, Luoer & Deng, Churou & Duong, Manh Hong & Han, The Anh, 2024. "On the number of equilibria of the replicator-mutator dynamics for noisy social dilemmas," Chaos, Solitons & Fractals, Elsevier, vol. 180(C).
    6. Huang, Chaochao & Wang, Chaoqian, 2024. "Memory-based involution dilemma on square lattices," Chaos, Solitons & Fractals, Elsevier, vol. 178(C).

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