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.
Highlights
The replicator–mutator dynamics has become a powerful mathematical framework for the modelling and analysis of complex biological, economical and social systems
We explore further connections between classical/random polynomial theory and evolutionary game theory developed in [4,5,6,7] to study equilibrium properties of the replicator–mutator dynamics
We recall that finding an equilibrium point of the replicator–mutator dynamics for d-player two-strategy games is equivalent to finding a positive root of the polynomial (17) with coefficients given in (18)
Summary
The replicator–mutator dynamics has become a powerful mathematical framework for the modelling and analysis of complex biological, economical and social systems. We have provided explicit formulas for the computation of the expected number and the distribution of internal equilibria for the replicator dynamics with multi-player games by employing techniques from both classical and random polynomial theory [4,5,6,7]. That is when there is no mutation, the first condition in (2) means that all the strategies have the same fitness which is the average fitness of the whole population This benign property is no longer valid in the presence of mutation making the mathematical analysis harder. For multi-player two-strategy random games whose pay-offs are independently distributed according to a normal distribution, we obtain explicit formulas to compute the expected number of equilibria by relating it to the expected number of positive roots of a random polynomial.
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