Stochastic Consensus and the Shadow of Doubt

TL;DR

This paper introduces a stochastic opinion exchange model in networks where agents reinforce beliefs through communication, showing convergence to a consensus that can be significantly influenced by a small number of misinformed agents, with the limit belief distribution possibly following a beta distribution.

Contribution

It presents a novel stochastic model of opinion dynamics with reinforcement, demonstrating convergence properties and the impact of misinformed agents on the consensus.

Findings

Convergence to a stable state occurs almost surely for any network.

The limit belief distribution can be full-support and is influenced by misinformed agents.

The average limit belief appears independent of network structure, possibly following a beta distribution.

Abstract

We propose a stochastic model of opinion exchange in networks. A finite set of agents is organized in a fixed network structure. There is a binary state of the world and each agent receives a private signal on the state. We model beliefs as urns where red balls represent one possible value of the state and blue balls the other value. The model revolves purely around communication and beliefs dynamics. Communication happens in discrete time and, at each period, agents draw and display one ball from their urn with replacement. Then, they reinforce their urns by adding balls of the colors drawn by their neighbors. We show that for any network structure, this process converges almost-surely to a stable state. Futher, we show that if the communication network is connected, this stable state is such that all urns have the same proportion of balls. This result strengthens the main convergence properties of non-Bayesian learning models. Yet, contrary to those models, we show that this limit proportion is a full-support random variable. This implies that an arbitrarily small proportion of misinformed agents can substantially change the value of the limit consensus. We propose a set of conjectures on the distribution of this limit proportion based on simulations. In particular, we show evidence that the limit belief follows a beta distribution and that its average value is independent from the network structure.