Social learning with complex contagion.

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. But real-world contagion processes are often complex, meaning that multiple exposures may be required before an individual considers changing their type. We introduce a framework that combines the concepts of simple payoff-biased imitation with complex contagion, to describe how social behaviors spread through a population. We formulate this model 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, a widely used dynamical model in evolutionary game theory. When applied to linear frequency-dependent games, 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 toward more successful types. Our analysis intercalates the fields of evolutionary game theory with complex contagions, and it provides a synthetic framework to describe more realistic forms of behavioral change in social systems.