Observing Each Other's Observations in the Electronic Mail Game

TL;DR

This paper analyzes a Bayesian coordination game where agents have private information and signals about each other's private info, revealing how information precision affects equilibrium multiplicity and strategic reasoning.

Contribution

It introduces a model where private signals about each other's information influence equilibrium outcomes, highlighting endogenous prediction precision and multiple equilibria.

Findings

Multiple equilibria arise at high private signal precision.

Equilibria differ in how agents weight private information.

Prediction precision becomes endogenous despite exogenous signal accuracy.

Abstract

We study a Bayesian coordination game where agents receive private information on the game's payoff structure. In addition, agents receive private signals on each other's private information. We show that once agents possess these different types of information, there exists a coordination game in the evaluation of this information. And even though the precisions of both signal types is exogenous, the precision with which agents predict each other's actions at equilibrium turns out to be endogenous. As a consequence, we find that there exist multiple equilibria if the private signals' precision is high. These equilibria differ with regard to the way that agents weight their private information to reason about each other's actions.