Exact time-dependent dynamics of discrete binary choice models

TL;DR

This paper introduces a comprehensive method to derive exact time-dependent solutions for binary choice models with interactions, including classical models like voter and ant recruitment, for any number of agents and parameters.

Contribution

The authors develop a generic approach to solve the full dynamics of binary decision models, extending solutions to finite populations and asymmetric cases, and providing analytical results for several key models.

Findings

Exact solutions for the voter model dynamics.

Analytical solutions for ant recruitment models.

Extensions to asymmetric and vacillating voter models.

Abstract

We provide a generic method to find full dynamical solutions to binary decision models with interactions. In these models, agents follow a stochastic evolution where they must choose between two possible choices by taking into account the choices of their peers. We illustrate our method by solving Kirman and F\"ollmer's ant recruitment model for any number $N$ of agents and for any choice of parameters, recovering past results found in the limit $N\to \infty$. We then solve extensions of the ant recruitment model for increasing asymmetry between the two choices. Finally, we provide an analytical time-dependent solution to the standard voter model and a semi-analytical solution to the vacillating voter model.