Polarization and coherence in mean field games driven by private and social utility

TL;DR

This paper analyzes a mean field game modeling opinion dynamics with social conformity and stubbornness, revealing complex phase diagrams with multiple equilibria and polarization phenomena, and compares mean field results with finite-player simulations.

Contribution

It provides a complete analytical description of the phase diagram and equilibrium behavior in a mean field game with binary states, incorporating social and private utilities.

Findings

Multiple Nash equilibria can exist depending on parameters.

The phase diagram includes polarized and unpolarized, coherent and incoherent equilibria.

Finite-player simulations show the selected equilibrium often benefits the underdog subpopulation.

Abstract

We study a mean field game in continuous time over a finite horizon, T, where the state of each agent is binary and where players base their strategic decisions on two, possibly competing, factors: the willingness to align with the majority (conformism) and the aspiration of sticking with the own type (stubbornness). We also consider a quadratic cost related to the rate at which a change in the state happens: changing opinion may be a costly operation. Depending on the parameters of the model, the game may have more than one Nash equilibrium, even though the corresponding N-player game does not. Moreover, it exhibits a very rich phase diagram, where polarized/unpolarized, coherent/incoherent equilibria may coexist, except for T small, where the equilibrium is always unique. We fully describe such phase diagram in closed form and provide a detailed numerical analysis of the N-player counterpart of the mean field game. In this finite dimensional setting, the equilibrium selected by the population of players is always coherent (favoring the subpopulation whose type is aligned with the initial condition), but it does not necessarily minimize the cost functional. Rather, it seems that, among the coherent ones, the equilibrium prevailing is the one that most benefits the underdog subpopulation forced to change opinion.