What's the Value of Information?

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.

Sure, I suppose. But usually there's only one piece of evidence being discussed explicitly, and I think it makes little sense to use the word "prior" to refer to the probability that results from updating on it.

Vaniver, looks like you were thinking about the problem in the same way that I was, getting repeated chances to buy new numbers.

Sort of. The calculations that I ran are all one-step-ahead calculations, starting with different priors. Consider three different cases:

• You pay X now, and he reads you the first number, and then you guess.
• You pay Y now, then he reads you the first number, then you have the option to buy a second number, and then you guess.
• You pay Z now, then he reads you the first two numbers, and then you guess.

Pricing X is easy; it's £125. Pricing Z is a bit tougher, but still okay. Pricing Y involves coming up with 13 different prices- the twelve possibilities after the first roll, and then Y (which depends on each of those possibilities!). Doing that with arbitrary n is doable but tough! (It's somewhat easier if you have a set price for each successive number, so you can swiftly terminate trees once you've hit the point that it's no longer worth the price.)

And so, even at 9:1 odds, there is some number of numbers he can read off that will have positive VoI. It will be very low- because it's very unlikely you will get that many informative numbers- but it is true that if you aren't perfectly certain, a test that gives you perfect certainty will have positive VoI.

What do you do then? The conclusion 'I literally won't lift a finger to know more numbers' doesn't seem right unless you're certain of the answer already.

The thing to focus on here is both the amount of additional certainty and the effect of additional certainty. The number you get when you're at 9:1 tells you a lot less than the number you get when you're at 1:1. Imagine the next number being a 1- in the first case, it feels like you just got £100, but in the second case it feels like you just got £500. Similarly, when I'm at 1:1, telling me one additional number is expected to change my guess in some cases. When I'm at 9:1, regardless of what he tells me, I still make the same call.

There is such a thing as certain enough when there are tests that aren't informative enough.

(Interestingly, note that you can never reach perfect certainty that it's 2d6, and there will always be a positive VoI for another number because there will always be a positive chance that it's a 1.)

p=rand(); c=0;

[c+=1, rand()<p]

betainc(x,a,b)

betainc(x+.05,a,b)-betainc(x-.05,a,b)

betainc(.65,3,2)-betainc(.55,3,2)=.17200

betainc(.66,3,2)-betainc(.56,3,2)=.17331

betainc(x+.05,a,b)-betainc(x-.05,a,b);

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.

orthonormal's making the more subtle point that decisions are binary, and so certainty is crudely partitioned into two regions. With hedging and other financial instruments, then relative degrees of certainty matter- if I'm 90% sure that it's 1d12 and you're 80% sure that it's 1d12, then we can bet against each other, each thinking that we're picking up free money. (Suppose you pay me $3 if it's 1d12, and I pay you $17 if it's 2d6. Both of us have an expected value of $1 from this bet.) The more accurate my estimate is, the better odds I can make.

With the decision problem, we both decide the same way, and will both win or lose together.

What Vaniver said. I'm claiming that it's artificial as a decision theory problem, not in the sense of being unrealistic, but in the sense of having constrained options that don't allow you to make full use of information.

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...

It could be indeed, but Oswald is known for his kindly and straightforward nature and wouldn't pull that sort of fast one.

It's not a fast one so much as careful attention to the setup. The first number on the sheet would be the first number in the title- which unambiguously specifies which sheet it is. If you change the setup to "the value of the first result," the point will be defused.

Not quite never, and the predictions of your various theories are also priors. So suppose I'm a physicist in the 19th century. And I've got two theories 'Classical Physics' and 'We're wrong about everything'.

This bears no resemblance to the actual history. How much resemblance was it intended to have? You say in another comment:

And I don't, by the way, put this forward as an account of 'how classical physics fell'.

But your reason for that is only:

Those guys were using classical logic.

There were several known problems with classical physics in the late 19th century, and "classical logic" vs. "new improved Bayesian logic" has nothing to do with how they were resolved.

1. The black body spectrum could not be explained.
2. The photoelectric effect (going a few years into the 20th century). It took a certain amount of energy to knock an electron off an atom, but light of arbitrarily low intensity could still do it. Only the wavelength mattered: there was a wavelength threshold but no intensity threshold.
3. EM theory predicted an absolute velocity of light, but Newtonian mechanics defines no preferred frame of reference, and the Michelson-Morley experiment failed to find one.

If you've got an immense prior belief in a theory that can explain anything at all, then yes, that's hard to shift.

Having a prior so immense that it's hard to shift is a problem anyway. But what is "immense", and what is "hard"? I pointed out here that ordinary people are quite capable of updating against 80dB of prior improbability (and if their posterior certainty is of the same order of magnitude then they've updated by around 160dB).

And I don't, by the way, put this forward as an account of 'how classical physics fell'. Those guys were using classical logic.

Probability theory is the generalization of logic to uncertain propositions, which is why it can deal with 'I only ever see white swans' being evidence for 'All swans are white'.

Suppose you're certain 1000:1 that the coin is fair. The only coin flip outcomes worth considering are runs of heads (obviously once you see a single outcome of tails, you're done).

If you ask for k=5 or more coin flips, then you can only be wrong if the coin is fair but ended up doing HHH...HH anyway, which is long enough to convince you. This has probability less than 1/2^k (since Pr[fair coin] < 1). At that point, every additional coin flip is worth a ridiculously tiny amount you can solve for.

If you ask for fewer than 5 coin flips, then no sequence of coin flips you see will convince you that the coin isn't fair with probability over 50%, and you'll just end up betting on that no matter what you see. So these coin flips are worthless unless you will have the chance to buy more.

The "either 2/3 or 1/3" coin is actually worse in the scary-zeroes department. The nice feature of both the 2d6 and the double-heads coins is that they completely lack an outcome, so no matter what your prior is, if you are currently considering betting on these, there is a small chance you'll be convinced not to.

On the other hand, if both coins can come up both heads and tails, then a single extra flip is worthless unless your prior odds are between 2:1 and 1:2 -- no matter what outcome you see, your belief will shift in one direction or the other by exactly one bit. It's like a biased random walk on the number line, and you don't know what the bias is. But no matter where you are, there's always some number of coin flips that will be worth buying (all at once), although potentially the price you'd pay for them would be really tiny.