On Discrete-Time Replicator Equations with Nonlinear Payoff Functions

An evolutionary game is usually identified by a smooth (possibly nonlinear) payoff function. We propose a model of evolutionary game in which for any two different pure strategies, a nonlinear payoff function of a pure strategy which is frequent in number is better/worse than other one. In order to observe some evolutionary bifurcation diagram, we also control the nonlinear payoff function in two different regimes: positive and negative. One of the interesting feature of the model is that if we switch a controlling parameter from positive to negative regime then a set of local evolutionarily stable strategies (ESSs) changes from one set to another one. We show that the Folk Theorem of Evolutionary Game Theory is true for a discrete-time replicator equation governed by the proposed nonlinear payoff function. In the long-run time, the following scenario can be observed: (i) in the positive regime, the active dominating pure strategies will outcompete other strategies and only they will survive forever; (ii) in the negative regime, all active pure strategies will coexist together and they will survive forever. As an application, we also show that the nonlinear payoff functions defined by discrete population models for a single species such as Beverton–Holt’s model, Hassell’s model, Maynard Smith–Slatkin’s model, Ricker’s model, Skellam’s model satisfy the hypothesis of the proposed model.