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.

翻译：演化博弈论为解释生态学、经济学及社会科学中不利条件下合作行为的涌现做出了重要贡献。近期，受新近可获取的群体互动经验证据启发，高阶网络已成为在结构化群体中恰当编码多人博弈的自然框架。本文研究超图上非线性公共物品博弈（Public Goods Game, PGG）中合作行为的涌现，其中集体强化机制捕捉了每增加一个合作者所带来的协同或折扣效应。在均匀混合群体中，单阶PGG（所有博弈参与人数相同）展现了相变特性从连续型到间断型的转变，这一转变取决于非线性的具体形式。相比之下，混合阶PGG（不同参与人数的博弈共存）呈现出更丰富的动力学行为，其中双稳态与合作可主动共存。我们进一步发现无标度超图能够促进合作，凸显了合作者初始分布与超度相关性所发挥的关键作用。总体而言，我们的结果为超图上非线性PGG提供了全面刻画，并为结构化群体中多人交互演化动力学更丰富模型的建立开辟了新途径。