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.