We introduce a mathematical model that combines the concepts of complex contagion with payoff-biased imitation, to describe how social behaviors spread through a population. Traditional models of social learning by imitation are based on simple contagion -- where an individual may imitate a more successful neighbor following a single interaction. Our framework generalizes this process to incorporate complex contagion, which requires multiple exposures before an individual considers adopting a different behavior. We formulate this as a discrete time and state stochastic process in a finite population, and we derive its continuum limit as an ordinary differential equation that generalizes the replicator equation, the most widely used dynamical model in evolutionary game theory. When applied to linear frequency-dependent games, our social learning with complex contagion produces qualitatively different outcomes than traditional imitation dynamics: it can shift the Prisoner's Dilemma from a unique all-defector equilibrium to either a stable mixture of cooperators and defectors in the population, or a bistable system; it changes the Snowdrift game from a single to a bistable equilibrium; and it can alter the Coordination game from bistability at the boundaries to two internal equilibria. The long-term outcome depends on the balance between the complexity of the contagion process and the strength of selection that biases imitation towards more successful types. Our analysis intercalates the fields of evolutionary game theory with complex contagions, and it provides a synthetic framework that describes more realistic forms of behavioral change in social systems.

翻译：本文提出一个数学模型，将复杂传染概念与收益偏向模仿相结合，用以描述社会行为在群体中的传播机制。传统的模仿型社会学习模型基于简单传染——个体可能在单次交互后模仿更成功的邻居。我们的框架将这一过程推广至复杂传染情形，即个体需要多次接触后才考虑采纳不同行为。我们将该过程表述为有限群体中的离散时间离散状态随机过程，并推导出其连续极限——一个推广了演化博弈论中最广泛使用的动力学模型（复制者方程）的常微分方程。当应用于线性频率依赖博弈时，复杂传染下的社会学习会产生与传统模仿动力学截然不同的结果：它能使囚徒困境从唯一的全背叛者均衡转变为群体中合作者与背叛者的稳定混合态或双稳态系统；将雪堆博弈从单稳态转变为双稳态；并能将协调博弈从边界双稳态调整为两个内部均衡点。长期结果取决于传染过程的复杂程度与偏向高收益类型的自然选择强度之间的平衡。我们的分析将演化博弈论与复杂传染理论相融合，为社会系统中更现实的行为变化形式提供了综合性的理论框架。