2017/03/31 by Varun Pandit, Archan Mukhopadhyay, Sagar Chakraborty
Biochemistry, Genetics and Molecular Biology · Mathematics · Medicine · Physics and Astronomy · Social Sciences · #Applied mathematics #Chaotic #Computer science #Evolution and Genetic Dynamics #Evolutionary Game Theory and Cooperation #Evolutionary game theory #Game theory #Mathematical and Theoretical Epidemiology and Ecology Models #Mathematical economics #Mathematical optimization #Mathematics #Normal-form game #Population #Repeated game #Replicator equation #Stochastic game #nlin.CD #physics.bio-ph
paper · pdf · doi:10.1063/1.5011955
published as Chaos 28, 033104 (2018) · Published in Chaos. This version has some typos corrected
openalex publication_date 2018/03/01 · arxiv created 2021/02/17 · arxiv updated 2021/02/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Replicator equation-a paradigm equation in evolutionary game dynamics-mathematizes the frequency dependent selection of competing strategies vying to enhance their fitness (quantified by the average payoffs) with respect to the average fitnesses of the evolving population under consideration. In this paper, we deal with two discrete versions of the replicator equation employed to study evolution in a population where any two players' interaction is modelled by a two-strategy symmetric normal-form game. There are twelve distinct classes of such games, each typified by a particular ordinal relationship among the elements of the corresponding payoff matrix. Here, we find the sufficient conditions for the existence of asymptotic solutions of the replicator equations such that the solutions-fixed points, periodic orbits, and chaotic trajectories-are all strictly physical, meaning that the frequency of any strategy lies inside the closed interval zero to one at all times. Thus, we elaborate on which of the twelve types of games are capable of showing meaningful physical solutions and for which of the two types of replicator equation. Subsequently, we introduce the concept of the weight of fitness deviation that is the scaling factor in a positive affine transformation connecting two payoff matrices such that the corresponding one-shot games have exactly same Nash equilibria and evolutionary stable states. The weight also quantifies how much the excess of fitness of a strategy over the average fitness of the population affects the per capita change in the frequency of the strategy. Intriguingly, the weight's variation is capable of making the Nash equilibria and the evolutionary stable states, useless by introducing strict physical chaos in the replicator dynamics based on the normal-form game.