Social learning with complex contagion

TL;DR

This paper introduces a unified model that merges complex contagion with payoff-biased imitation to describe social learning. By deriving a continuum-limit ODE that generalizes the replicator equation, it shows how contagion complexity (via a threshold distribution with parameter $\gamma$) qualitatively reshapes outcomes in classic games: Prisoner’s Dilemma can yield stable mixtures or bistability, Snowdrift can become bistable, and Coordination can host internal equilibria. Analytically, the authors provide explicit interior equilibria for the Donation, Snowdrift, and Coordination games at specific $\gamma$ values and validate the continuum model with Monte Carlo simulations. The framework bridges complex contagion and evolutionary game theory, revealing how conformity-like thresholds interact with payoff biases to drive more nuanced, realistic behavioral evolution in social systems.

Abstract

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.

Figures (5)

• Figure 1: Complex contagion can qualitatively change outcomes of behavioral evolution. We plot the scaled payoff difference $\bar{D}(x)$ as a function of the frequency $x$ of strategy 1, for three different games. A stable internal equilibrium, where both strategies co-exist, occurs when $\bar{D}$ intersects the $x$-axis with negative slope. The equilibrium at $x=0$ ($x=1$) is stable if $\bar{D}$ is negative (positive) at that boundary. Under simple contagion ($\gamma=1$), the only stable outcomes are pure defection in the Prisoner's Dilemma, a mixture of both strategies in the Snowdrift game, and pure coordination in the Coordination game. But complex contagion ($\gamma>1$ or $\gamma<1$) leads to qualitatively different outcomes, including, e.g., a stable mixture of cooperators and defectors in a Prisoner's dilemma. Parameters: $\alpha=0.15$, $b=5$, $c=2$, $p=4.5$, $q=4$.
• Figure 2: Strategy evolution in the Rock-Paper-Scissor game with simple and complex contagion. Each vertex represents the monomorphic population of each strategy. When $\gamma=1$ (simple contagion and classic replicator equation), the dynamics exhibit a cyclic nature. Whereas the population is attracted to each monomorphic state when $\gamma>1$, and it attracted to a stable mixture of strategies when $\gamma<1$. Parameters: $s=0.3$.
• Figure S1: Three-strategy Prisoner's Dilemma Each vertex represents the monomorphic population of each strategy. The right and left bottom verteces represent full and moderate cooperators, and the top vertex represents defectors. When $\gamma=1$ (classic replicator equation), the dynamics exhibits a cyclic nature. It is attracted to each monomorphic population when $\gamma>1$ and attracted to the mixed strategy population when $\gamma<1$. Parameters: $s=3,b_{1}=3,b_{2}=4,c_{1}=1,c_{2}=3$.
• Figure S2: Rock-Paper-Scissor game under the alternative dynamics Each vertex represents the monomorphic population of each strategy. When $\kappa=1$ (classic replicator equation), the dynamics exhibits a cyclic nature. It is attracted to each monomorphic population when $\kappa>1$. Parameter: $s=0.3$.
• Figure S3: Three-strategy Prisoner's Dilemma under the alternative dynamics Each vertex represents the monomorphic population of each strategy. The right and left bottom verteces represent full and moderate cooperators, and the top vertex represents defectors. When $\kappa=1$ (replicator equation), the dynamics is attracted toward the all-defector population. It is attracted to each monomorphic population when $\kappa>1$ with different sizes of attraction, largest for the all-defector population and smallest for the all-full-cooporator population. Parameters are the same as in Figure \ref{['fig:3-strategy PD']}.

Theorems & Definitions (2)

• Theorem
• Theorem