Evolutionary game theory on measure spaces

An attempt is made to find a comprehensive mathematical framework in which to investigate the problems of wellposedness, asymptotic analysis and numerical analysis for fully nonlinear evolutionary game theoretic models. 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 as bases, a general model is formulated as a dynamical system on the state space of finite signed measures. Wellposedness is established, and then it is shown that by choosing appropriate payoff kernels this model includes classical density models, both selection and mutation and discrete models all in a continuous manner. It is then demonstrated how to perform completions of this measure space and how that these completions allow one to form weak (generalized) asymptotic limits and finite dimensional approximations and hence to perform numerical analysis and asymptotic analysis. In particular it is shown that the model has a compact attractor and that for pure replicator dynamics a certain weak solution converges to a Dirac measure centered at the fittest strategy; thus this Dirac measure is a globally attractive equilibrium point which is termed a continuously stable strategy (CSS). It is also shown that in the discrete case for pure replicator dynamics and even for small perturbation of pure replicator dynamics there exists a globally asymptotically stable equilibrium.