Based on simple coin flip; other games:
I am sure there are more small games that have a similar "known" problem space.
What change would you make that results in multiple rounds being required?
For example, if each player flips multiple coins, and then we share probability estimates for "all coins heads" or "majority of coins heads" or expectations for number of heads, in each case the first time I share my summary, I am sharing info that exactly tells the other player what information I have (and vice versa). So we will agree exactly from the second round onwards.
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.
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