A Generalized Adaptive Dynamics Framework can Describe the Evolutionary Ultimatum Game

Abstract

Adaptive dynamics describes the evolution of games where the strategies are continuous functions of some parameters. The standard adaptive dynamics framework assumes that the population is homogeneous at any one time. Differential equations point to the direction of the mutant that has maximum payoff against the resident population. The population then moves towards this mutant. The standard adaptive dynamics formulation cannot deal with games in which the payoff is not differentiable. Here we present a generalized framework which can. We assume that the population is not homogeneous but distributed around an average strategy. This approach can describe the long-term dynamics of the Ultimatum Game and also explain the evolution of fairness in a one-parameter Ultimatum Game.