Riemannian game dynamics

We study a class of evolutionary game dynamics defined by balancing a gain determined by the game's payoffs against a cost of motion that captures the difficulty with which the population moves between states. Costs of motion are represented by a Riemannian metric, i.e., a state-dependent inner product on the set of population states. The replicator dynamics and the (Euclidean) projection dynamics are the archetypal examples of the class we study. Like these representative dynamics, all Riemannian game dynamics satisfy certain basic desiderata, including positive correlation and global convergence in potential games. Moreover, when the underlying Riemannian metric satisfies a Hessian integrability condition, the resulting dynamics preserve many further properties of the replicator and projection dynamics. We examine the close connections between Hessian game dynamics and reinforcement learning in normal form games, extending and elucidating a well-known link between the replicator dynamics and exponential reinforcement learning.Comment: 47 pages, 12 figures; added figures and further simplified the derivation of the dynamic

Stochastic evolutionary game dynamics

In this review, we summarize recent developments in stochastic evolutionary game dynamics of finite populations.Comment: To appear in "Reviews of Nonlinear Dynamics and Complexity" Vol. II, Wiley-VCH, 2009, edited by H.-G. Schuste

Evolutionary game dynamics in phenotype space

Evolutionary dynamics can be studied in well-mixed or structured populations. Population structure typically arises from the heterogeneous distribution of individuals in physical space or on social networks. Here we introduce a new type of space to evolutionary game dynamics: phenotype space. The population is well-mixed in the sense that everyone is equally likely to interact with everyone else, but the behavioral strategies depend on distance in phenotype space. Individuals might behave differently towards those who look similar or dissimilar. Individuals mutate to nearby phenotypes. We study the `phenotypic space walk' of populations. We present analytic calculations that bring together ideas from coalescence theory and evolutionary game dynamics. As a particular example, we investigate the evolution of cooperation in phenotype space. We obtain a precise condition for natural selection to favor cooperators over defectors: for a one-dimensional phenotype space and large population size the critical benefit-to-cost ratio is given by b/c=1+2/sqrt{3}. We derive the fundamental condition for any evolutionary game and explore higher dimensional phenotype spaces.Comment: version 2: minor changes; equivalent to final published versio