When Does Population Diversity Matter? A Unified Framework for Binary-Choice Dynamics

TL;DR

This paper introduces a unified framework for binary-choice dynamics that accounts for individual preference diversity, identifying conditions under which diversity influences collective behavior or can be ignored.

Contribution

It provides a systematic approach to compare annealed and quenched dynamics and identifies a key constraint making them equivalent, simplifying analysis.

Findings

Identifies a constraint making annealed and quenched dynamics equivalent.

Shows that under this constraint, the system reduces to one dimension.

Ruling out oscillatory behaviors when the constraint is satisfied.

Abstract

We propose a modeling framework for binary-choice dynamics in which agents update their states using two mechanisms selected based on individual preference drawn from an arbitrary distribution. We compare annealed dynamics, where preferences change over time, and quenched dynamics, where they remain fixed. Our framework bridges gaps between existing models and provides a systematic approach to assess when individual-level diversity affects collective dynamics and when it can be effectively ignored. We identify a constraint on transition probabilities that makes annealed and quenched dynamics equivalent. We show that when this condition is satisfied, the quenched dynamics reduces to a one-dimensional system, ruling out oscillatory behavior that may otherwise emerge.