Natural Gradient Ascent in Evolutionary Games

We study evolutionary games with a continuous trait space in which replicator dynamics are restricted to the manifold of multidimensional Gaussian distributions. We demonstrate that the replicator equations are natural gradient flow for maximization of the mean fitness. Our findings extend previous results on information-geometric aspects of evolutionary games with a finite strategy set. Throughout the paper we exploit the information-geometric approach and the relation between evolutionary dynamics and Natural Evolution Strategies, the concept that has been developed within the framework of black-box optimization. This relation sheds a new light on the replicator dynamics as a compromise between maximization of the mean fitness and preservation of diversity in the population.