Your problem is basically that you’re mixing up the idealised problem with the realistic problem. By “idealised” I mean a general approach to reading this sort of problem where you forget all the other factors that clearly aren’t intended to be considered. For instance, thoughts like “Does he know something about the first number on that sheet being atypical, and is trying to make me pay money to make a wrong guess?”—which could be part of his clever scam. In the idealised problem you assume he’s genuine and honest and so on. The idealised problem is generally the one you’re supposed to think about in these cases, but there’s never a shortage of wiseacres who’ll try and circumvent the whole issue with some “realistic” consideration.
In the idealised problem you’re not told anything about a second opportunity to get more information, therefore it doesn’t exist. Adding a possibility of more information simply creates a new, different idealised problem. In this modified problem the value of the information may be non-zero.
In the “realistic” problem you can consider the possibility of him offering you another number without it being explicitly mentioned. But in that case there’s a wealth of other things to worry about making it all too complicated.
I think you’re also barking up the wrong tree in the first place trying to create some sort of well defined “value” of information that’s independent of your prior (i.e. of other available information). I don’t imagine such a thing exists.
I think you’re also barking up the wrong tree in the first place trying to create some sort of well defined “value” of information that’s independent of your prior (i.e. of other available information). I don’t imagine such a thing exists.
This is (now) my intuition too.
But my old intuition was:
If I think there’s a 1⁄2 chance then I’m in possession of an option worth £500, if I think there’s a 3⁄4 chance then I’m in possession of an option worth £750
so if I think there’s a 1⁄2 chance I should work out all the expected consequences and average over their new value to work out new value after getting the information, and the difference is the price I think that information’s worth.
And I still can’t see what’s wrong with that. Can you?
(What originally prompted me to think of the question was the worry that receiving certain sorts of information would make the expected value of my option go down, and I wanted to play with that. I was completely freaked out when I realized that there were lots of prior beliefs where that method gave £0 as the answer.)
I don’t see why you’d think anything was wrong with that. I even did the math now and agree with your specific value of £125. Your value of 0 is correct in the other case too. About the only thing I don’t agree with is your sense of surprise. There’s plenty of information that’s worth nothing, and no reason it couldn’t later be worth something in combination with other information.
For example, if he told you the d12 numbers were written in red pen (and the others in blue), that’s worth nothing on its own. But suddenly looking at one of the numbers is worth quite a lot more than it was...
Your problem is basically that you’re mixing up the idealised problem with the realistic problem. By “idealised” I mean a general approach to reading this sort of problem where you forget all the other factors that clearly aren’t intended to be considered. For instance, thoughts like “Does he know something about the first number on that sheet being atypical, and is trying to make me pay money to make a wrong guess?”—which could be part of his clever scam. In the idealised problem you assume he’s genuine and honest and so on. The idealised problem is generally the one you’re supposed to think about in these cases, but there’s never a shortage of wiseacres who’ll try and circumvent the whole issue with some “realistic” consideration.
In the idealised problem you’re not told anything about a second opportunity to get more information, therefore it doesn’t exist. Adding a possibility of more information simply creates a new, different idealised problem. In this modified problem the value of the information may be non-zero.
In the “realistic” problem you can consider the possibility of him offering you another number without it being explicitly mentioned. But in that case there’s a wealth of other things to worry about making it all too complicated.
I think you’re also barking up the wrong tree in the first place trying to create some sort of well defined “value” of information that’s independent of your prior (i.e. of other available information). I don’t imagine such a thing exists.
This is (now) my intuition too.
But my old intuition was:
If I think there’s a 1⁄2 chance then I’m in possession of an option worth £500, if I think there’s a 3⁄4 chance then I’m in possession of an option worth £750
so if I think there’s a 1⁄2 chance I should work out all the expected consequences and average over their new value to work out new value after getting the information, and the difference is the price I think that information’s worth.
And I still can’t see what’s wrong with that. Can you?
(What originally prompted me to think of the question was the worry that receiving certain sorts of information would make the expected value of my option go down, and I wanted to play with that. I was completely freaked out when I realized that there were lots of prior beliefs where that method gave £0 as the answer.)
I don’t see why you’d think anything was wrong with that. I even did the math now and agree with your specific value of £125. Your value of 0 is correct in the other case too. About the only thing I don’t agree with is your sense of surprise. There’s plenty of information that’s worth nothing, and no reason it couldn’t later be worth something in combination with other information.
For example, if he told you the d12 numbers were written in red pen (and the others in blue), that’s worth nothing on its own. But suddenly looking at one of the numbers is worth quite a lot more than it was...