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On the Expected Number of Internal Equilibria in Random Evolutionary Games with Correlated Payoff Matrix

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  • Manh Hong Duong

    (Imperial College London)

  • Hoang Minh Tran

    (Data Analytics Department, Esmart Systems)

  • The Anh Han

    (Teesside University)

Abstract

The analysis of equilibrium points in random games has been of great interest in evolutionary game theory, with important implications for understanding of complexity in a dynamical system, such as its behavioural, cultural or biological diversity. The analysis so far has focused on random games of independent payoff entries. In this paper, we overcome this restrictive assumption by considering multiplayer two-strategy evolutionary games where the payoff matrix entries are correlated random variables. Using techniques from the random polynomial theory, we establish a closed formula for the mean numbers of internal (stable) equilibria. We then characterise the asymptotic behaviour of this important quantity for large group sizes and study the effect of the correlation. Our results show that decreasing the correlation among payoffs (namely, of a strategist for different group compositions) leads to larger mean numbers of (stable) equilibrium points, suggesting that the system or population behavioural diversity can be promoted by increasing independence of the payoff entries. Numerical results are provided to support the obtained analytical results.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:dyngam:v:9:y:2019:i:2:d:10.1007_s13235-018-0276-4
    DOI: 10.1007/s13235-018-0276-4
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    References listed on IDEAS

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    1. Daniel Friedman, 1998. "On economic applications of evolutionary game theory," Journal of Evolutionary Economics, Springer, vol. 8(1), pages 15-43.
    2. Chaitanya Gokhale & Arne Traulsen, 2014. "Evolutionary Multiplayer Games," Dynamic Games and Applications, Springer, vol. 4(4), pages 468-488, December.
    3. Francisco C. Santos & Marta D. Santos & Jorge M. Pacheco, 2008. "Social diversity promotes the emergence of cooperation in public goods games," Nature, Nature, vol. 454(7201), pages 213-216, July.
    4. Fudenberg, D. & Harris, C., 1992. "Evolutionary dynamics with aggregate shocks," Journal of Economic Theory, Elsevier, vol. 57(2), pages 420-441, August.
    5. 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.
    6. Altenberg, Lee, 2010. "Proof of the Feldman–Karlin conjecture on the maximum number of equilibria in an evolutionary system," Theoretical Population Biology, Elsevier, vol. 77(4), pages 263-269.
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    Cited by:

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    2. D. Timothy Bishop & Mark Broom & Richard Southwell, 2020. "Chris Cannings: A Life in Games," Dynamic Games and Applications, Springer, vol. 10(3), pages 591-617, September.
    3. Pi, Jinxiu & Yang, Guanghui & Tang, Wei & Yang, Hui, 2022. "Stochastically stable equilibria for evolutionary snowdrift games with time costs," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 604(C).

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