“This bill clearly belongs to exactly one of the two us; I think it is 80% likely to be yours and 20% likely to be mine, and you think it is 85% likely to be mine and 15% likely to be yours. In order to fairly divide it, we weigh our beliefs equally (by symmetry), and divide it according to the ratio 80+15:20+85 you:me; that’s 95:105, or 47.5:52.5. That’s a 52.5% chance that I get the bill, and a 47.5% chance that you get it.” (Same expected values as 2, from equivalent math)
I don’t think there’s a general case for the function g. Consider the case where hypotheses B, C, and D are mutually exclusive. Proposition A is equivalent to “not C” but B is only known to be mutually exclusive with C. On discovering that D is false, A becomes equivalent to B, but there is also potentially new information about C.
Variant 1: In a departure from Monty Hall; I put a bean under one of the cups B,C,D after rolling a d10. (Time a) Then I show you what is under cup D. There is not a bean. (Time b)
If the odds ratio at time a is 2:4:4, then the odds at time b are 2:4:0. Proposition A went from 6⁄10 likely to 1⁄3 likely, while B went from 2⁄10 to 1⁄3.
Variant 2: I roll the die; if the result is prime, I put the bean under cup d. Otherwise, if the result is greater than five, I put the bean under cup c. Otherwise I put the bean under cup b. At time a, the odds are 2:4:4. Then I tell you that the number I rolled is a perfect square. You should update to 2:1:0
In both cases, you updated that B and not-C were equivalent, but in the latter case you gained new information about B and C. In the general case, if there is no new information about B and C, then the ratio B:C should remain constant; I intuit that that will probably mean a fairly simple transformation.
“This bill clearly belongs to exactly one of the two us; I think it is 80% likely to be yours and 20% likely to be mine, and you think it is 85% likely to be mine and 15% likely to be yours. In order to fairly divide it, we weigh our beliefs equally (by symmetry), and divide it according to the ratio 80+15:20+85 you:me; that’s 95:105, or 47.5:52.5. That’s a 52.5% chance that I get the bill, and a 47.5% chance that you get it.” (Same expected values as 2, from equivalent math)
I don’t think there’s a general case for the function g. Consider the case where hypotheses B, C, and D are mutually exclusive. Proposition A is equivalent to “not C” but B is only known to be mutually exclusive with C. On discovering that D is false, A becomes equivalent to B, but there is also potentially new information about C.
Variant 1: In a departure from Monty Hall; I put a bean under one of the cups B,C,D after rolling a d10. (Time a) Then I show you what is under cup D. There is not a bean. (Time b)
If the odds ratio at time a is 2:4:4, then the odds at time b are 2:4:0. Proposition A went from 6⁄10 likely to 1⁄3 likely, while B went from 2⁄10 to 1⁄3.
Variant 2: I roll the die; if the result is prime, I put the bean under cup d. Otherwise, if the result is greater than five, I put the bean under cup c. Otherwise I put the bean under cup b. At time a, the odds are 2:4:4. Then I tell you that the number I rolled is a perfect square. You should update to 2:1:0
In both cases, you updated that B and not-C were equivalent, but in the latter case you gained new information about B and C. In the general case, if there is no new information about B and C, then the ratio B:C should remain constant; I intuit that that will probably mean a fairly simple transformation.