Aumann's agreement theorem

Theorem (Aumann, 1976)

Let $(N, Y, (\mathcal{F}_i)_{i \in N}, \mathfrak{s}, \mathbf{P})$ be an Aumann model of incomplete information with beliefs, and suppose $n=2$ (i.e. 2 players). Let $A \subseteq Y$ be an event and let $\omega \in Y$ be a state of the world. If the event "Player I ascribes probability $q_\textrm{I}$ to $A$" is common knowledge in $\omega$, and the event "Player II ascribes probability $q_\textrm{II}$ to $A$" is also common knowledge in $\omega$, then $q_\textrm{I} = q_\textrm{II}$.

References

1. M. Maschler, E. Solan, and Shmuel Zamir, Game Theory, Cambridge University Press, 2013, p. 339.
2. Robert J. Aumann, The Annals of Statistics, Vol. 4, No. 6 (Nov., 1976), pp. 1236-1239. https://www.jstor.org/stable/2958591
3. https://www.princeton.edu/~bayesway/Dick.tex.pdf
4. https://blogs.cornell.edu/info2040/2014/10/30/information-cascades-and-aumanns-agreement-theorem/
5. https://www.lesswrong.com/tag/aumann-s-agreement-theorem
6. https://scottaaronson.blog/?p=2410