Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Azadeh Aghaeeyan, Pouria Ramazi

TL;DR

This work analyzes a well-mixed population where imitators, conformists, and nonconformists participate in a two-strategy game. By casting the finite, stochastic update rules as a generalized stochastic approximation process for a differential inclusion, the authors connect discrete Markov dynamics to a deterministic continuous-time flow that always reaches equilibria or continua of equilibria. They prove that, as population size $N$ grows, the amplitude of endogenous fluctuations in strategy adoption vanishes almost surely, establishing a rigorous link between finite dynamics and the large-population limit. These results provide a principled framework for predicting adoption dynamics in large social-ecological systems and highlight how imitation of the highest earners interacts with threshold-based conformity and anticonformity to shape long-run outcomes.

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.

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

TL;DR

This work analyzes a well-mixed population where imitators, conformists, and nonconformists participate in a two-strategy game. By casting the finite, stochastic update rules as a generalized stochastic approximation process for a differential inclusion, the authors connect discrete Markov dynamics to a deterministic continuous-time flow that always reaches equilibria or continua of equilibria. They prove that, as population size grows, the amplitude of endogenous fluctuations in strategy adoption vanishes almost surely, establishing a rigorous link between finite dynamics and the large-population limit. These results provide a principled framework for predicting adoption dynamics in large social-ecological systems and highlight how imitation of the highest earners interacts with threshold-based conformity and anticonformity to shape long-run outcomes.

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.
Paper Structure (7 sections, 17 theorems, 41 equations, 2 figures, 1 table)

This paper contains 7 sections, 17 theorems, 41 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Background
  4. The link to the continuous-time dynamics
  5. Main results
  6. Supporting Results
  7. Concluding Remarks

Key Result

Theorem 1

roth2013stochastic Let $\mathbf{X}^{\epsilon}$ be a family of GSAPs for a differential inclusion $\dot{\bm{x}} \in \bm{\mathcal{V}}(\bm{x})$. Then for any $T>0$ and any $\alpha >0$, we have uniformly in $\bm{x}_0 \in {\bm{\mathcal{X}}}$.

Figures (2)

  • Figure 1: Evolution of discrete mixed population dynamics for three population sizes in Example \ref{['example']} and that of the continuous-time. Each panel shows the evolution of the population proportions over time, where $x_{IM}$ is the proportion of imitators playing strategy $\mathtt{A}$; $x_i$ for $i = 1,2,3$ (resp. $x'_1$) denotes the proportion of $\mathtt{A}$-playing nonconformists (resp. conformists) of type $i$, and $x$ is the proportion of $\mathtt{A}$-players. Panel (a) corresponds to $\mathsf{N} = 100$ units over $600$ time steps; panels (b) and (c) are associated with $\mathsf{N}$ equal to $200$ and $400$, respectively. As the $\mathsf{N}$ increases, the amplitude of fluctuations in $x$ (black curve) decreases. Panel (d) depicts the evolution of the corresponding continuous-time population dynamics.
  • Figure 2: Stochastic approximation theory provides a link between the behavior of finite populations governed by the discrete-time dynamics and the corresponding continuous-time system. Analysis of the one-dimensional abstract dynamics facilitates that of the mean dynamics. Convergence of the abstract dynamics implies convergence of the mean dynamics, which, in turn, implies that the discrete population dynamics converge in probability.

Theorems & Definitions (29)

  • Definition 1
  • Example 1
  • Definition 2: roth2013stochastic
  • Theorem 1
  • Theorem 2: aghaeeyan2023discrete
  • Proposition 1
  • Definition 3
  • Definition 4
  • Lemma 1
  • Proposition 2
  • ...and 19 more