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

Manh Hong Duong, The Anh Han

Research output: Contribution to journalArticle

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.
Original languageEnglish
Pages (from-to)1-23
JournalDynamic Games and Applications
Early online date27 Nov 2019
DOIs
Publication statusE-pub ahead of print - 27 Nov 2019

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Polynomials
Game
Normal distribution
Snow
Dynamical systems
Mutation
Sampling
Computer simulation
Social Dilemma
René Descartes
Internal
Random Polynomials
Prisoners' Dilemma
Evolutionary Game
Equilibrium Point
Averaging
Gaussian distribution
Dynamical system
Calculate
Numerical Simulation

Cite this

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title = "On Equilibrium Properties of the Replicator–Mutator Equation in Deterministic and Random Games",
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.",
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On Equilibrium Properties of the Replicator–Mutator Equation in Deterministic and Random Games. / Duong, Manh Hong; Han, The Anh.

In: Dynamic Games and Applications, 27.11.2019, p. 1-23.

Research output: Contribution to journalArticle

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