A Population's Feasible Posterior Beliefs

TL;DR

This paper characterizes which distributions of beliefs are feasible in a population of Bayesian agents sharing a prior, providing conditions and applications in polarization, symmetric distributions, and Bayesian persuasion.

Contribution

It offers a necessary and sufficient condition for the feasibility of belief distributions in a population, advancing understanding in Bayesian belief modeling.

Findings

Derived a feasibility condition for belief distributions

Characterized symmetric product distributions of posteriors

Provided a formula for optimal sender value in Bayesian persuasion

Abstract

We consider a population of Bayesian agents who share a common prior over some finite state space and each agent is exposed to some information about the state. We ask which distributions over empirical distributions of posteriors beliefs in the population are feasible. We provide a necessary and sufficient condition for feasibility. We apply this result in several domains. First, we study the problem of maximizing the polarization of beliefs in a population. Second, we provide a characterization of the feasible agent-symmetric product distributions of posteriors. Finally, we study an instance of a private Bayesian persuasion problem and provide a clean formula for the sender's optimal value.