2019/04/22 by Manh Hong Duong, The Anh Han, Duong, Manh Hong +1
Biochemistry, Genetics and Molecular Biology · Decision Sciences · Social Sciences · #Dynamical Systems (math.DS) #Evolution and Genetic Dynamics #Evolutionary Game Theory and Cooperation #FOS: Biological sciences #FOS: Mathematics #Game Theory and Applications #Populations and Evolution (q-bio.PE)
paper · pdf · doi:10.48550/arxiv.1904.09805
openalex publication_date 2019/04/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we study the number of equilibria of the replicator-mutator\ndynamics for both deterministic and random multi-player two-strategy\nevolutionary games. For deterministic games, using Decartes' rule of signs, we\nprovide a formula to compute the number of equilibria in multi-player games via\nthe number of change of signs in the coefficients of a polynomial. For\ntwo-player social dilemmas (namely, the Prisoner's Dilemma, Snowdrift, Stag\nHunt, and Harmony), we characterize (stable) equilibrium points and\nanalytically calculate the probability of having a certain number of equilibria\nwhen the payoff entries are uniformly distributed. For multi-player random\ngames whose payoffs are independently distributed according to a normal\ndistribution, by employing techniques from random polynomial theory, we compute\nthe expected or average number of internal equilibria. In addition, we perform\nextensive simulations by sampling and averaging over a large number of possible\npayoff matrices to compare with and illustrate analytical results. Numerical\nsimulations also suggest several interesting behaviour of the average number of\nequilibria when the number of players is sufficiently large or when the\nmutation is sufficiently small. In general, we observe that introducing\nmutation results in a larger average number of internal equilibria than when\nmutation is absent, implying that mutation leads to larger behavioural\ndiversity in dynamical systems. Interestingly, this number is largest when\nmutation is rare rather than when it is frequent.\n