Not sure… what happens when the ranges are different sizes, or otherwise the type of information learnable by each player is different in non symmetric ways?
Anyways, thanks, upon another reading of your comment, I think I’m starting to get it a bit.
Different size ranges in Hal’s example? Nothing in particular happens. It’s ok for different random variables to have different ranges.
Otoh, if the players get different ranges about a single random variable, then they could have problems.
Suppose there is one d6. Player A learns whether it is in 1-2, 3-4, or 5-6. Player B learns whether it is in 1-3 or 4-6. And suppose the actual value is 1. Then A knows it’s 1-2. So A knows B knows it’s 1-3. But A reasons that B reasons that if it were 3 then A would know it’s 3-4, so A knows B knows A knows it’s 1-4. But A reasons that B reasons that A reasons that if it were 4 then B would know it’s 4-6, so A knows B knows A knows B knows it’s 1-6. So there is no common knowledge, i.e. I∧J=Ω. (Omitting the argument w, since if this is true then it’s true for all w.)
And if it were a d12, with ranges still size 2 and 3, then the partitions line up at one point, so the meet stops at {1-6, 7-12}.
Not sure… what happens when the ranges are different sizes, or otherwise the type of information learnable by each player is different in non symmetric ways?
Anyways, thanks, upon another reading of your comment, I think I’m starting to get it a bit.
Different size ranges in Hal’s example? Nothing in particular happens. It’s ok for different random variables to have different ranges.
Otoh, if the players get different ranges about a single random variable, then they could have problems. Suppose there is one d6. Player A learns whether it is in 1-2, 3-4, or 5-6. Player B learns whether it is in 1-3 or 4-6.
And suppose the actual value is 1.
Then A knows it’s 1-2. So A knows B knows it’s 1-3. But A reasons that B reasons that if it were 3 then A would know it’s 3-4, so A knows B knows A knows it’s 1-4. But A reasons that B reasons that A reasons that if it were 4 then B would know it’s 4-6, so A knows B knows A knows B knows it’s 1-6. So there is no common knowledge, i.e. I∧J=Ω. (Omitting the argument w, since if this is true then it’s true for all w.)
And if it were a d12, with ranges still size 2 and 3, then the partitions line up at one point, so the meet stops at {1-6, 7-12}.