In this note, we extend Aumann’s agreement theorem to a framework where beliefs are modelled by conditional probability systems à la Battigalli, P., and M. Siniscalchi. 1999. “Hierarchies of Conditional Beliefs and Interactive Epistemology in Dynamic Games.” Journal of Economic Theory 88: 188–230. We prove two independent generalizations of the agreement theorem, one where the agents share some common conditioning event, and one where they may not.
Proof of Proposition 1.
First of all observe that covers , since is a partition of . Then, consider an arbitrary collection of positive reals such that . For each define
Verifying that is a probability measure in is trivial. Moreover, notice that for every and every with , it is the case that
with eq. (10) following from since either or .
Proof of Theorem 1.
Define , which is by hypothesis nonempty. Since the event is measurable with respect to the partition , either or . Now, for an arbitrary , it is necessarily the case that . Hence, for each ,
The second equality follows from , which is by definition true for all . Now, define the set . Hence,
Finally, since does not depend on , it is the case that .
Proof of Theorem 2.
Step 1. Define , which is by hypothesis nonempty. Then, define the set . Then, following Tsakas and Voorneveld (2011), Proof of Lemma, we define as the coarsest partition of that generates , i.e., formally for every , the element of that contains is defined by . Thus, we obtain
with eq. (11) following from the fact that is balanced. Before moving forward, let us point out that or might in principle be equal to 0 for some or respectively. In such cases, the conditional probabilities given these null events take arbitrary values.
Step 2. Now, similarly to the proof of Theorem 1, it is the case that for all , thus implying for all and all .
Step 3. Now take an arbitrary such that . Then, there exists some such that , and therefore since is a common prior, we obtain for all with . Hence, by eq. (12) it follows that
with eq. (13) following again from the fact that is balanced. Indeed, the conditional probability can be seen as an indicator function, with if and otherwise. Finally, notice that does not depend on , thus completing the proof.
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I am indebted to three anonymous referees and the associate editor for their valuable comments. This note supersedes two previous papers by the same author, titled “Strong belief and agreeing to disagree” and “Hierarchies of conditional beliefs derived from commonly known priors” respectively.
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