Your conclusion that if you have a prior p(1d12) of 0.9, you should not spend even £1 to see the piece of paper is entirely correct, if counterintuitive. The reason is as follows (n is the number, 2d6 is the chance of 2d6 landing on that number, 1d12 is the chance of 1d12 landing on that number, p(6|n)/p(12|n) is the odds ratio, and p(6|n) is the probability that you would give that the 2d6 was correct given a prior of 0.1 for the 2d6 (0.9 for the 1d12).:
As you can see, no matter what information you get, you will never get any piece of information that will convince you to pick differently, and so the value of information is 0. If, however, you had information that could bring p( 2d6 | n ) above 0.5, the value of information would be nonzero. However, for that you would either need a lower prior (in this case, p( 1d12 ) ≤ 2⁄3) or stronger evidence (such as 4 slips of paper: nothing less can possibly change your mind in this setup, even if all the slips were 7s).
You’re getting confused on the word “same” I think. Omega is only offering you the same deal if your prior is 0.9 and you get only one shot. If you get multiple chances to update, that changes the nature of the game and you need to know how many chances you will have (or what the probability of getting another chance is). As is, you’re missing necessary information though.
Information that can’t be any use to me is worth nothing.
But that information can become worthwhile if combined with future information which may or may not become available.
So the price should be non-zero if Omega says ‘What will you pay me for the first number on the sheet, I may also sell you further numbers later’.
But if that’s true, then shouldn’t the £125 value for a 50:50 prior also be affected by what happens afterwards?
In the original question, that’s exactly what I was imagining happening. You’d buy the first number for anything less than £125, and maybe it’s a 7, so you’re now back in the same situation with a new prior of 25:75, and so what will you pay for the second number, and so on.… And I was hoping that it would all converge nicely.
I think the reason I’m finding it paradoxical is that we’ve all jumped straight to the conclusion that the fair price is £125 without feeling that we needed to ask ‘And what happens next?’, and found that unproblematic.
But then when we look at the 9:1 case, where the value calculated this way is 0 and looks a bit suspicious, and we all start thinking ‘Ahh, but don’t we need to know more about what happens next in order to price this’.
But if that reasoning affects the £0 price, why wouldn’t it also affect the £125 price?
Which is why I’m asking ‘Is my question ill-posed, and if not, what is the answer’.
The reasoning does affect the £125 price. In the case where you get an arbitrarily large number of pieces of information, the value converges on £1000 - (current EV). This makes sense, as an arbitrarily large number of papers gives you an arbitrarily high level of confidence that you will get the £1000. So with no information, the current EV is £500, so the possible value of information is £1000 - £500 = £500. In the case where you’ve got a prior of 0.9 on the 1d12, your EV is already £900 (90% chance of winning £1000) so the EV of infinite information is still only £100 (£1000 - £900).
In reference to your original question, you should be willing to pay somewhat more than £125, and less than £500 for that first piece of information (I would have to calculate the exact amount). The amount would vary based on how many more opportunities to buy information you would have.
Your conclusion that if you have a prior p(1d12) of 0.9, you should not spend even £1 to see the piece of paper is entirely correct, if counterintuitive. The reason is as follows (n is the number, 2d6 is the chance of 2d6 landing on that number, 1d12 is the chance of 1d12 landing on that number, p(6|n)/p(12|n) is the odds ratio, and p(6|n) is the probability that you would give that the 2d6 was correct given a prior of 0.1 for the 2d6 (0.9 for the 1d12).:
As you can see, no matter what information you get, you will never get any piece of information that will convince you to pick differently, and so the value of information is 0. If, however, you had information that could bring p( 2d6 | n ) above 0.5, the value of information would be nonzero. However, for that you would either need a lower prior (in this case, p( 1d12 ) ≤ 2⁄3) or stronger evidence (such as 4 slips of paper: nothing less can possibly change your mind in this setup, even if all the slips were 7s).
You’re getting confused on the word “same” I think. Omega is only offering you the same deal if your prior is 0.9 and you get only one shot. If you get multiple chances to update, that changes the nature of the game and you need to know how many chances you will have (or what the probability of getting another chance is). As is, you’re missing necessary information though.
So, I think we’re both thinking, in the 0.9 case:
Information that can’t be any use to me is worth nothing.
But that information can become worthwhile if combined with future information which may or may not become available.
So the price should be non-zero if Omega says ‘What will you pay me for the first number on the sheet, I may also sell you further numbers later’.
But if that’s true, then shouldn’t the £125 value for a 50:50 prior also be affected by what happens afterwards?
In the original question, that’s exactly what I was imagining happening. You’d buy the first number for anything less than £125, and maybe it’s a 7, so you’re now back in the same situation with a new prior of 25:75, and so what will you pay for the second number, and so on.… And I was hoping that it would all converge nicely.
I think the reason I’m finding it paradoxical is that we’ve all jumped straight to the conclusion that the fair price is £125 without feeling that we needed to ask ‘And what happens next?’, and found that unproblematic.
But then when we look at the 9:1 case, where the value calculated this way is 0 and looks a bit suspicious, and we all start thinking ‘Ahh, but don’t we need to know more about what happens next in order to price this’.
But if that reasoning affects the £0 price, why wouldn’t it also affect the £125 price?
Which is why I’m asking ‘Is my question ill-posed, and if not, what is the answer’.
The reasoning does affect the £125 price. In the case where you get an arbitrarily large number of pieces of information, the value converges on £1000 - (current EV). This makes sense, as an arbitrarily large number of papers gives you an arbitrarily high level of confidence that you will get the £1000. So with no information, the current EV is £500, so the possible value of information is £1000 - £500 = £500. In the case where you’ve got a prior of 0.9 on the 1d12, your EV is already £900 (90% chance of winning £1000) so the EV of infinite information is still only £100 (£1000 - £900).
In reference to your original question, you should be willing to pay somewhat more than £125, and less than £500 for that first piece of information (I would have to calculate the exact amount). The amount would vary based on how many more opportunities to buy information you would have.