False Consensus, Information Theory, and Prediction Markets

TL;DR

This paper uses information theory to analyze how Bayesian agents with private information aggregate their knowledge in prediction markets, showing conditions under which false consensus does not occur and illustrating the framework's power through multiple results.

Contribution

It introduces an information theoretic framework based on interaction information to analyze information aggregation and truthfulness in prediction markets and related settings.

Findings

Agents' private information is aggregated when conditionally independent given the event.

The sign of interaction information relates to super/sub-additivity of shared information.

Reproves known results on agent agreement and information revelation using the framework.

Abstract

We study a setting where Bayesian agents with a common prior have private information related to an event's outcome and sequentially make public announcements relating to their information. Our main result shows that when agents' private information is independent conditioning on the event's outcome whenever agents have similar beliefs about the outcome, their information is aggregated. That is, there is no false consensus. Our main result has a short proof based on a natural information theoretic framework. A key ingredient of the framework is the equivalence between the sign of the ``interaction information'' and a super/sub-additive property of the value of people's information. This provides an intuitive interpretation and an interesting application of the interaction information, which measures the amount of information shared by three random variables. We illustrate the power of this information theoretic framework by reproving two additional results within it: 1) that agents quickly agree when announcing (summaries of) beliefs in round robin fashion [Aaronson 2005]; and 2) results from [Chen et al 2010] on when prediction market agents should release information to maximize their payment. We also interpret the information theoretic framework and the above results in prediction markets by proving that the expected reward of revealing information is the conditional mutual information of the information revealed.