From Discrete to Continuous Mixed Populations of Conformists, Nonconformists, and Imitators

TL;DR

This paper investigates how the proportions of conformists, nonconformists, and imitators fluctuate in finite populations and demonstrates that these fluctuations diminish as population size increases, leading to stable equilibria in large populations.

Contribution

It establishes a connection between discrete population dynamics and continuous differential inclusions, proving that fluctuations vanish in large populations.

Findings

Fluctuations decrease as population size grows.

Continuous dynamics always reach equilibrium.

Large populations tend to stabilize strategy proportions.

Abstract

In two-strategy decision-making problems, individuals often imitate the highest earners or choose either the common or rare strategy. Individuals who benefit from the common strategy are conformists, whereas those who profit by choosing the less common one are called nonconformists. The population proportions of the two strategies may undergo perpetual fluctuations in finite, discrete, heterogeneous populations of imitators, conformists, and nonconformists. How these fluctuations evolve as population size increases was left as an open question and is addressed in this paper. We show that the family of Markov chains describing the discrete population dynamics forms a generalized stochastic approximation process for a differential inclusion--the continuous-time dynamics. Furthermore, we prove that the continuous-time dynamics always equilibrate. Then, by leveraging results from the stochastic approximation theory, we show that the amplitudes of fluctuations in the proportions of the two strategies in the population approach zero with probability one when the population size grows to infinity. Our results suggest that large-scale perpetual fluctuations are unlikely in large, well-mixed populations consisting of these three types, particularly when imitators follow the highest earners.