On Extensions to Aumann’s Theory of Common Knowledge

In 1976, Robert Aumann published his widely influential mathematical formulation of the idea of “common knowledge” in [2]. This paper had a great impact in game theory, control theory, economics, and related fields because of the conclusions that it drew. Aumann’s main theorem stated that if two agents began with identical prior beliefs (i.e. unbiased relative to each other) and their posterior beliefs following some event were common knowledge, then these posteriors would necessarily be identical. In Aumann’s terms, two such agents could not “agree to disagree”. Later, Geanakouplos and Polemarchakis [4] established a reasonable method by which such posteriors (or generally any kind of information) could “become” common knowledge. They showed, using Aumann’s framework, that if two such agents communicated some piece of information back and forth repeatedly, then eventually this information would become common knowledge. One reason Aumann’s result is so intriguing is that the kind of behaviour he predicted is simply not consistent with real-world observations. Indeed, it is very often the case that two or more rationally-minded people who appear to both understand a common set of facts do agree to disagree. This discrepancy led to much analysis of Aumann’s assumptions and to refinements of his results. For instance, one might suppose that the convergence of beliefs predicted in [4] might simply require too much computational power or too much information transmission. Indeed, this question was brought up by Aaronson [1]; however, he concluded that this was not the case. Perhaps one of the more obvious differences between Aumann’s model and the real world is the fact that the model was developed over countable probability spaces. In the present paper, we shall discuss an extension of this model to general state spaces put forward by Nielsen [7]. Nielsen gives a similar conclusion as Aaronson by showing that Aumann’s theorem is not truly restricted by the assumption of countability. Nielsen also generalizes Geanakouplos and Polemarchakis’s results. In further support of the Aumann-Nielsen model, we will discuss a paper of Brandenburger and Dekel [3], in which they give a rather different—but perhaps intuitively more approachable—