Perhaps you should assign some probability to being offered enough information to change your mind? There must be some nonzero chance that after you’ve bought the Nth number, Omega will offer to sell you the (N+1)th number; and if in fact your 9:1 assignment was wrong, a sufficiently long chain of such offers should change your mind. So there is still some chance of buying the first number leading to a change of mind, even if no information about the first number is itself enough to do so.
No, I don’t think it’s ill-posed. You’ve found a specific prior for which the value of one specific piece of information is indeed zero. I don’t see why this should make the more general case, where you have a different prior or are offered more information, ill-posed.
Consider a man who has two different, lethal cancers. Omega comes along and asks what he’ll pay for the cure for one of them. Assume that the cancers are unique in the history of all mankind, so there’s no altruistic benefit; then he will, presumably, pay nothing, since he’ll still die and may as well use the money to amuse himself while waiting. But what will he pay for the cure to both cancers? Ah, a very different question! Likewise, you’ve constructed a situation where one piece of information is without value, but two pieces of information are not. That doesn’t make the question ill-posed; the value of one piece of information is perfectly well-defined, namely zero.
Well, in the particular case he posed with a prior of 0.9 on the 1d12, 2 pieces of information are also useless. In fact, you need 4 pieces of paper to have nonzero value of information (and even then, I think the expected value of the 4 is < £1).
Sure, but I don’t see where that changes the analysis. The probability of you getting 4 pieces of information, contingent on getting the first one, has got to be larger than the probability of getting 4, contingent on not getting the first one. (In fact the latter seems to be a contradiction, which presumably has probability zero.) So the first one still has some value, even if it’s perhaps rather smaller than the value of the time it takes to do the formal calculation of the value.
You’re right. This seems like an interesting exercise in programming, actually: build a tool that tells you the VOI of a certain number of guesses. I know 0-3 have an EV of 0, but when I try to plug in 4, I realize why a recursive function might have been a bad idea.
Perhaps you should assign some probability to being offered enough information to change your mind? There must be some nonzero chance that after you’ve bought the Nth number, Omega will offer to sell you the (N+1)th number; and if in fact your 9:1 assignment was wrong, a sufficiently long chain of such offers should change your mind. So there is still some chance of buying the first number leading to a change of mind, even if no information about the first number is itself enough to do so.
But doesn’t that imply that the original question is ill-posed? And if so, what sort of questions can we calculate the answer to?
No, I don’t think it’s ill-posed. You’ve found a specific prior for which the value of one specific piece of information is indeed zero. I don’t see why this should make the more general case, where you have a different prior or are offered more information, ill-posed.
Consider a man who has two different, lethal cancers. Omega comes along and asks what he’ll pay for the cure for one of them. Assume that the cancers are unique in the history of all mankind, so there’s no altruistic benefit; then he will, presumably, pay nothing, since he’ll still die and may as well use the money to amuse himself while waiting. But what will he pay for the cure to both cancers? Ah, a very different question! Likewise, you’ve constructed a situation where one piece of information is without value, but two pieces of information are not. That doesn’t make the question ill-posed; the value of one piece of information is perfectly well-defined, namely zero.
Well, in the particular case he posed with a prior of 0.9 on the 1d12, 2 pieces of information are also useless. In fact, you need 4 pieces of paper to have nonzero value of information (and even then, I think the expected value of the 4 is < £1).
Sure, but I don’t see where that changes the analysis. The probability of you getting 4 pieces of information, contingent on getting the first one, has got to be larger than the probability of getting 4, contingent on not getting the first one. (In fact the latter seems to be a contradiction, which presumably has probability zero.) So the first one still has some value, even if it’s perhaps rather smaller than the value of the time it takes to do the formal calculation of the value.
You’re right. This seems like an interesting exercise in programming, actually: build a tool that tells you the VOI of a certain number of guesses. I know 0-3 have an EV of 0, but when I try to plug in 4, I realize why a recursive function might have been a bad idea.