Multiplayer social dilemmas in completely mixed populations and networks of mixing communities

Human and non-human animals often form highly structured social systems in which individuals interact preferentially within specific groups, shaped by habitat structure, movement patterns, or social preferences. We consider a model that captures the interplay between social structure and emerging behaviour. Individuals are probabilistically distributed across the nodes of a spatial network representing interaction sites such as resource patches, territorial areas, or social hubs. Individuals present at the same node interact through a multiplayer game, so group structure is not imposed but instead emerges from movement under spatial constraints. As a benchmark, we analyse the completely mixed population, a null model in which individuals are indistinguishable, analogous to the classical well-mixed assumption, and further have no preference for particular network nodes. We derive exact expressions for fundamental properties of this population under arbitrary payoff functions, including the group size distribution, fitness functions, and fixation probabilities. We consider five multiplayer social dilemmas capturing the tension between individual and collective interests, such as those arising in public goods production, collective defence, and communal resource use. For these dilemmas, we derive analytical conditions for the evolution of cooperation in completely mixed populations. Using these as a reference, we reassess the effects of community structure. We show that asymptotically isolated communities expand the parameter range favouring cooperation, particularly under low population density. Finally, we explore intermediate regimes between completely mixed populations and isolated community structure, demonstrating that this expansion of the cooperative region remains robust even under higher levels of mixing between adjacent communities.