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}.
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}.