I’m not sure what you mean. The algorithm I described gives the same formula f=(p+q)/2 as Marcello gave, although I arrived at it with a different justification. According to that formula, if EY and NB are both 100% sure that the bill was EY’s, then EY gets the entire $20.
If each of them is 100% sure that the bill is his own, then they split the bill 50-50.
If EY is 100% sure that the bill is his, while NB thinks that there is a significant chance that the bill is his (e.g., 50%), then EY doesn’t get the entire $20. Do you see a compelling reason why he should?
I meant that if EY is 100% sure, while NB is 50% that EY should get the entire 20.
I don’t think that I see anything you don’t on this. To me, it seems that if EY knows something, and NB trusts him, that NB should update to know it too, but it looks like you disagree.
Perhaps I am working by analogy. I think that the equivalent property for the g function is more clearly true. If A is a theorem, and B is a conjecture, and we prove they are equivalent, then B as a theorem as well.
However, it at least seems that EY should get close to all of it. If there is 60 in the pot If NB just threw money in and has no idea whether he put 20 or 40 in, while EY knows (99.99%) that he put exactly 40 into the pot, EY should get a lot more than 15$
It’s very hard to say what will happen if they are each going to update based on each other’s probabilities.
Aumann’s theorem doesn’t apply directly, because they do not have common knowledge of their posteriors, even after they exchange probabilities. For, each will know that the other will have updated, but he won’t know what the other’s new posterior is. It’s not clear to me that their probabilities will begin to converge even after they go through many iterations of exchanging posteriors. If their probabilities converge, then that convergence value will depend subtlely on what each knows about the other, what each knows that the other knows about him, and so on ad infinitum.
Nonetheless, you’re right about what will happen if EY starts out 100% confident (which he never would). In that case, no matter what, if their posteriors converge, then they would have to converge on 100% certainty that the money belongs to Eliezer. If EY starts out 100% confident, no amount of confidence on NB’s part couldn’t ever make him budge from that absolute certainty. I’m not sure what conditions would guarantee that their probabilities would converge. (They certainly won’t if NB starts out 100% certain that the money is his.) But, if they could somehow establish that their probabilities would converge, then, yes, they may as well give all the money to EY.
But, in general, I don’t know how to analyze the problem if you allow them to update based on each other’s posteriors. I don’t know how they could determine whether their posteriors will converge, nor what that value of convergence might be.
If they aren’t allowed to update, if the 20$ must be apportioned based on their initial probabilities, then Marcello’s f=(p+q)/2 formula seems to me to be the best way to go.
Reposting:
I got the same answer as Marcello by assuming that each of them should get the same expected utility out of the split.
Say that Nick keeps x and Eliezer keep y. Then the expected utility for Nick is
0.85 x − 0.15 ($20 − x),
while the expected utility for Eliezer is
0.8 y − 0.2 ($20 − y).
Setting these equal to each other, and using x + y = $20, yields that Nick should keep x = $9.50, leaving y = $10.50 for Eliezer.
Do you stand behind that decision, even though it does not give the entire 20 to EY when he is 100% sure he deserves it?
I’m not sure what you mean. The algorithm I described gives the same formula f=(p+q)/2 as Marcello gave, although I arrived at it with a different justification. According to that formula, if EY and NB are both 100% sure that the bill was EY’s, then EY gets the entire $20.
If each of them is 100% sure that the bill is his own, then they split the bill 50-50.
If EY is 100% sure that the bill is his, while NB thinks that there is a significant chance that the bill is his (e.g., 50%), then EY doesn’t get the entire $20. Do you see a compelling reason why he should?
I meant that if EY is 100% sure, while NB is 50% that EY should get the entire 20. I don’t think that I see anything you don’t on this. To me, it seems that if EY knows something, and NB trusts him, that NB should update to know it too, but it looks like you disagree. Perhaps I am working by analogy. I think that the equivalent property for the g function is more clearly true. If A is a theorem, and B is a conjecture, and we prove they are equivalent, then B as a theorem as well.
However, it at least seems that EY should get close to all of it. If there is 60 in the pot If NB just threw money in and has no idea whether he put 20 or 40 in, while EY knows (99.99%) that he put exactly 40 into the pot, EY should get a lot more than 15$
It’s very hard to say what will happen if they are each going to update based on each other’s probabilities.
Aumann’s theorem doesn’t apply directly, because they do not have common knowledge of their posteriors, even after they exchange probabilities. For, each will know that the other will have updated, but he won’t know what the other’s new posterior is. It’s not clear to me that their probabilities will begin to converge even after they go through many iterations of exchanging posteriors. If their probabilities converge, then that convergence value will depend subtlely on what each knows about the other, what each knows that the other knows about him, and so on ad infinitum.
Nonetheless, you’re right about what will happen if EY starts out 100% confident (which he never would). In that case, no matter what, if their posteriors converge, then they would have to converge on 100% certainty that the money belongs to Eliezer. If EY starts out 100% confident, no amount of confidence on NB’s part couldn’t ever make him budge from that absolute certainty. I’m not sure what conditions would guarantee that their probabilities would converge. (They certainly won’t if NB starts out 100% certain that the money is his.) But, if they could somehow establish that their probabilities would converge, then, yes, they may as well give all the money to EY.
But, in general, I don’t know how to analyze the problem if you allow them to update based on each other’s posteriors. I don’t know how they could determine whether their posteriors will converge, nor what that value of convergence might be.
If they aren’t allowed to update, if the 20$ must be apportioned based on their initial probabilities, then Marcello’s f=(p+q)/2 formula seems to me to be the best way to go.