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On the Expected Number of Equilibria in a Multi-player Multi-strategy Evolutionary Game

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  • Manh Hong Duong

    (University of Warwick)

  • The Anh Han

    (Teesside University)

Abstract

In this paper, we analyze the mean number $$E(n,d)$$ E ( n , d ) of internal equilibria in a general $$d$$ d -player $$n$$ n -strategy evolutionary game where the agents’ payoffs are normally distributed. First, we give a computationally implementable formula for the general case. Next, we characterize the asymptotic behavior of $$E(2,d)$$ E ( 2 , d ) , estimating its lower and upper bounds as $$d$$ d increases. Then we provide a closed formula for $$E(n,2)$$ E ( n , 2 ) . Two important consequences are obtained from this analysis. On the one hand, we show that in both cases, the probability of seeing the maximal possible number of equilibria tends to zero when $$d$$ d or $$n$$ n , respectively, goes to infinity. On the other hand, we demonstrate that the expected number of stable equilibria is bounded within a certain interval. Finally, for larger $$n$$ n and $$d$$ d , numerical results are provided and discussed.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:dyngam:v:6:y:2016:i:3:d:10.1007_s13235-015-0148-0
    DOI: 10.1007/s13235-015-0148-0
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    Cited by:

    1. 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.
    2. 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.
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    4. Samuel C. Wiese & Torsten Heinrich, 2022. "The Frequency of Convergent Games under Best-Response Dynamics," Dynamic Games and Applications, Springer, vol. 12(2), pages 689-700, June.

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