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.
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.