  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.
