Evolutionary Game Theory on Measure Spaces: Well-Posedness

TL;DR

This paper develops a comprehensive mathematical framework using measure spaces to analyze the well-posedness and long-term behavior of nonlinear evolutionary game models, unifying classical models and addressing convergence issues.

Contribution

It introduces a measure space-based dynamical system for evolutionary games, establishing well-posedness and encompassing classical models with various nonlinearities and strategy spaces.

Findings

Established well-posedness of the measure-based model

Unified classical density models within the framework

Demonstrated convergence to Dirac measures in certain limits

Abstract

An attempt is made to find a comprehensive mathematical framework in which to investigate the problems of well-posedness and asymptotic analysis for fully nonlinear evolutionary game theoretic models. The model should be rich enough to include all classical nonlinearities, e.g., Beverton-Holt or Ricker type. For several such models formulated on the space of integrable functions, it is known that as the variance of the payoff kernel becomes small the solution converges in the long term to a Dirac measure centered at the fittest strategy; thus the limit of the solution is not in the state space of integrable functions. Starting with the replicator-mutator equation and a generalized logistic equation as bases, a general model is formulated as a dynamical system on the state space of finite signed measures. Well-posedness is established, and then it is shown that by choosing appropriate payoff kernels this model includes all classical density models, both selection and mutation, and discrete and continuous strategy (trait) spaces.