Evolutionary games on minimally structured populations

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.

💡 Research Summary

The paper introduces a parsimonious framework for studying evolutionary games on structured populations, stripping away the details of specific spatial embeddings or complex interaction networks. The authors argue that many previously reported effects—such as the promotion of cooperation, maintenance of biodiversity, or emergence of global oscillations—are highly sensitive to the exact topology of the interaction graph. To isolate the truly structural influences, they propose a “minimally structured population” consisting of two hierarchical levels of mixing.

At the lower level, individuals are grouped into local sub‑populations of effective size (N_{\ell}). Within each sub‑population interactions are assumed to be well‑mixed, i.e., any two members can meet with equal probability. At the higher level, sub‑populations exchange members with a global pool at a rate controlled by a single parameter (\phi) (the relative strength of local dynamics versus global mixing). When (\phi=0) the system is a collection of isolated islands; when (\phi=1) the whole population behaves as a single well‑mixed group. Thus the entire topology is captured by just two numbers, (N_{\ell}) and (\phi).

The authors start from a stochastic individual‑based description of strategy updating (either replicator dynamics or imitation) and perform two successive mean‑field approximations. The first approximation homogenises each local island, yielding a deterministic equation for the local frequency (x_{\ell}(t)) that still retains a dependence on the finite size (N_{\ell}) (through demographic noise terms). The second approximation treats the exchange between islands as a coupling to the global average frequency (X(t)). The resulting macroscopic dynamics can be written as

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