A statistical mechanics model for the emergence of consensus

TL;DR

This paper models how consensus emerges in large populations through a statistical mechanics approach, revealing that interactions can increase the likelihood of transitive majority decisions as the number of alternatives grows.

Contribution

It introduces a multi-component random field Ising model to analyze the emergence of transitive majorities and derives a phase diagram with a tri-critical point.

Findings

Interacting populations can more likely reach consensus as the number of alternatives increases.

Transitivity constraints influence the probability of a transitive majority.

Non transitive voting behavior reduces the likelihood of a transitive majority.

Abstract

The statistical properties of pairwise majority voting over S alternatives is analyzed in an infinite random population. We first compute the probability that the majority is transitive (i.e. that if it prefers A to B to C, then it prefers A to C) and then study the case of an interacting population. This is described by a constrained multi-component random field Ising model whose ferromagnetic phase describes the emergence of a strong transitive majority. We derive the phase diagram, which is characterized by a tri-critical point and show that, contrary to intuition, it may be more likely for an interacting population to reach consensus on a number S of alternatives when S increases. This effect is due to the constraint imposed by transitivity on voting behavior. Indeed if agents are allowed to express non transitive votes, the agents' interaction may decrease considerably the probability of a transitive majority.