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.
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.