Bridge, the card game. Bidding is the process of two players exchanging information about the cards they hold via the very limited communications channel (bids). The play itself is also used to transfer more information about which cards remain in the hand.
I don't know if that will work as a demonstration of the Aumann's Theorem, though, bridge gets very complicated very fast :-/
That's an excellent practical example, though it doesn't really have the explicit probability math I was hoping for.
In particular, I like that you'll see stuff like which player thinks the partnership has the better contract flips back and forth, especially around auctions involving controls, stops, or other specific invitational questions. The concept of evaluating your hand within a window ("My hand is now very weak, given that I opened") is also explicitly reasoning about what your partner infers based on what you told them.
I think the most important thing here might be that bridge requires multiple rounds because bidding is limited bandwidth, whereas giving a full-precision probability estimate is not.
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.
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