Evolutionary games on higher-order networks

Traditional evolutionary game theory often represents population structure with dyadic networks, although many applications require payoffs defined over groups. Hypergraphs and simplicial complexes retain group membership explicitly, but their effects must be distinguished from nonlinearities already present in multiplayer payoff functions. This structured scoping review separates these two sources of variation. A multiplayer nonlinear-payoff effect arises from thresholds, synergy, saturation, or heterogeneous group benefits on a fixed interaction structure. Such effects can occur in well-mixed or spatial groups without higher-order topology. A higher-order topological effect arises from explicit group membership, hyperedge overlap, inter-order organization, or simplicial closure. Its assessment requires the payoff, update rule, group-size distribution, and selection regime to be fixed or closely matched. A documented search through July 30, 2026 identified 85 core works: 80 primary studies and 5 conceptual or review articles. Across this evidence, nonlinear payoffs generate frequency dependence, internal equilibria, invasion thresholds, and, in some models, multistability. Higher-order topology instead changes group sharing, spillover between groups, and the routes to invasion or fixation. Discontinuous transitions and hysteresis are model outcomes whose origins depend on the complete payoff, topology, update, and feedback specification. Related graph-based research on fairness, trust, AI safety races, open-data stewardship, and cheap talk defines a broader agenda. Social welfare must also be evaluated separately from cooperation frequency. Direct empirical tests and factorial comparisons with matched baselines remain limited.