  Population structure induced by both spatial embedding and more general
networks of interaction, such as model social networks, have been shown to have
a fundamental effect on the dynamics and outcome of evolutionary games. These
effects have, however, proved to be sensitive to the details of the underlying
topology and dynamics. Here we introduce a minimal population structure that is
described by two distinct hierarchical levels of interaction. We believe this
model is able to identify effects of spatial structure that do not depend on
the details of the topology. We derive the dynamics governing the evolution of
a system starting from fundamental individual level stochastic processes
through two successive meanfield approximations. In our model of population
structure the topology of interactions is described by only two parameters: the
effective population size at the local scale and the relative strength of local
dynamics to global mixing. We demonstrate, for example, the existence of a
continuous transition leading to the dominance of cooperation in populations
with hierarchical levels of unstructured mixing as the benefit to cost ratio
becomes smaller then the local population size. Applying our model of spatial
structure to the repeated prisoner's dilemma we uncover a novel and
counterintuitive mechanism by which the constant influx of defectors sustains
cooperation. Further exploring the phase space of the repeated prisoner's
dilemma and also of the "rock-paper-scissor" game we find indications of rich
structure and are able to reproduce several effects observed in other models
with explicit spatial embedding, such as the maintenance of biodiversity and
the emergence of global oscillations.
