Emergence of cooperation in nonlinear higher-order public goods games

2026-04-08

AI Summary

This study investigates how cooperative behavior emerges under adverse conditions in nonlinear public goods games involving higher-order group interactions. By constructing higher-order network models based on hypergraphs and integrating evolutionary game theory, nonlinear dynamical analysis, and complex network simulations, the work systematically examines the impact of synergistic enhancement or diminishing returns on cooperation phase transitions in both single-order and mixed-order games. The findings reveal that mixed-order interactions can induce rich dynamical phenomena such as bistability and coexistence of cooperation, while scale-free hypergraph structures significantly promote cooperation, with the correlation between initial cooperator placement and node hyperdegree playing a pivotal role. These results uncover an intrinsic link between higher-order network structure and the continuity of cooperation phase transitions, offering a novel paradigm for understanding the evolution of cooperation in higher-order interaction settings.

Abstract

Evolutionary game theory has provided substantial contributions to explain the emergence of cooperation under unfavourable conditions in ecology, economics, and the social sciences. Recently, inspired by newly available empirical evidence on group interactions, higher-order networks have emerged as a natural framework to properly encode multiplayer games in structured populations. Here, we study the emergence of cooperation in a nonlinear public goods game (PGG) on hypergraphs, where collective reinforcement captures the synergistic or discounting effect associated with each additional cooperator. In well-mixed populations, single-order PGGs, where all games have the same number of players, display a change in the nature of transition from continuous to discontinuous depending on the exact form of nonlinearity. By contrast, mixed-order PGGs, where games with different number of players coexist, exhibit a richer dynamical regime wherein a state of active coexistence of bistability and cooperation can arise. We further find that scale-free hypergraphs promote cooperation, highlighting the crucial role played by both the initial placement of cooperators and the presence of hyperdegree correlations. Overall, our results provide a comprehensive characterization of nonlinear PGGs on hypergraphs and open up new avenues for richer models of evolutionary dynamics of multiplayer interactions on structured populations.

Problem

• cooperation
• nonlinear public goods game
• higher-order interactions
• hypergraphs
• evolutionary dynamics

Innovation

• higher-order interactions
• nonlinear public goods game
• hypergraphs
• evolutionary game theory
• scale-free structure