Common Knowledge of an Aggregate of Expectations

By L. T. Nielsen, A. Brandenburger, J. Geanakoplos, R. McKelvey, and T. Page

Econometrica 58, 5 (1990), 1235--1239

Introduction

When individuals make economic decisions based on diverse sources of information, market aggregates – such as prices – will to a greater or lesser extent reflect the pieces of information received. By observing a market aggregate, each person may learn something about the information possessed by others. In some cases, enough information is reflected to induce everyone to agree on a consensus estimate of the relevant uncertainty. This theme has been extensively explored in the literature on rational expectations.

In this note we provide a theorem giving a set of conditions under which public observation of an aggregate of individual expectations leads to consensus. The key concept underlying the result is that of common knowledge, as formalized by Aumann (1976. (An event is said to be common knowledge if everyone knows it, everyone knows that everyone knows it, and so on ad infinitum.) We suppose that n individuals start with a common prior and then form expectations, conditioned on their diverse information sources, of some random variable of interest. The theorem states that if a stochastically monotone aggregate of the n conditional expectations is common knowledge, then the expectations must be identical.

The work presented here builds on a theorem due to McKelvey and Page (1986). First, we extend their result from conditional probabilities of an event to conditional expectations of a random variable. Second, we provide a simple and brief proof of the theorem on conditional expectations. Third, we cover the general case in which information is represented by sigma-fields rather than by partitions. The original statement of the theorem on conditional expectations is due to Nielsen (1987a); versions of the present proof were derived independently in three working papers, namely Brandenburger and Geanakoplos (1987), McKelvey and Page (1987), and Nielsen (1987b).

Section 2 reviews the formalization of common knowledge developed by Aumann (1976) and restates Aumann's fundamental theorem on the impossibility of "agreeing ot disagree." Section 3 develops the main result of this note.