Wednesday, March 03, 2010

Two Envelopes

Consider, to begin with, a simple sort of Two Envelopes scenario: Mr. E. writes out a cheque for £18, another for £12, and puts each inside an envelope. Showing the two envelopes to his friend, Miss Take, he tells her that they contain money, and asks her to take one as a gift. Miss Take does so, and finds that she has £12. Then Mr. E. tells her that one of the cheques was for 50% more than the other, and asks her if she wants to swap her cheque for the other one. Now, Miss Take knows that either the amount in the other envelope is £12 + £6 = £18, or else it is £8 (since £8 + £4 = £12). So she knows that by swapping she would either gain £6 or lose £4. And she also knows that by swapping she is as likely to gain as to lose, since she has no idea which envelope contained the larger amount. So it seems to her that if she swaps she is 50% likely to gain £6 and 50% likely to lose £4. In other words, the mathematical expectation from swapping is 50% of £6 minus 50% of £4, which is £1. Since that is a positive amount, Miss Take decides to swap.
......She feels justified when she ends up with £18 but of course, she had no real reason to swap because her original choice was blind, and by swapping she is in effect just making that choice again. Still, when she swapped she did have more information about what was in the envelopes, and she seems to many to have had some mathematical reason to swap. Such is the Two Envelopes problem, which has been much discussed following M. Kraitchik’s two wallets (in 1953; for a translation of the relevant passage, see previous link). The discussion usually revolves around the problem of randomly choosing the amounts in the envelopes (with one being twice the other, usually), but it clearly does not matter how Mr. E. came by the amounts of £18 and £12. Maybe it was the 18th of December, and Miss Take’s birthday; or maybe Mr. E. was a fan of Tchaikovsky’s 1812. The main thing was that Miss Take did not know both amounts, and that the envelopes did not reveal which contained the larger amount.
......To give us some perspective on Miss Take’s mistake, Mr. E. then offers an identical pair of envelopes to his colleague, Miss Tree, who also picks the one containing £12. But this time, when he offers her the chance to swap envelopes, he tells her that the square of one third of the larger amount is twelve plus twice the smaller amount (in £s). Miss Tree calculates that either the amount in the other envelope is £18 (since 6 is one third of 18, and 6 squared is 12 plus twice 12), or else it is £2 (since 4 is one third of 12, and 4 squared is 12 plus twice 2). So swapping would either gain her £6, or else it would lose her £10. And for all she knows, a gain is as likely as a loss. So now the mathematical expectation appears to be of a loss, of £2. Even so Miss Tree swaps envelopes, precisely because a gain is as likely as a loss. Having read P. N. Johnson-Laird, Miss Tree knows how easily we can be misled by disjunctions; and she suspects that, since only one disjunct of any true disjunction—such as either a gain of £6 or else a loss of £10—needs to be true, so Mr. E. could, by telling her something true about how the two amounts are related, have led her to calculate practically any mathematical expectation.
......The problem of how we should apply mathematical expectations is neither unimportant nor uninteresting, a clearer way to consider randomly choosing a natural number may therefore be via the following variant of a paradox attributed to P. Levy (by F. P. Cantelli, in 1935), which also resembles Kraitchik’s puzzle: Suppose that a god selects two natural numbers at random—i.e. any of them might be chosen, with each being neither more nor less likely to be chosen than any other—and then writes two notes promising the bearer upon demand that many days in paradise, puts them into two identical envelopes, and asks you to pick an envelope. The puzzle is that, whichever you pick, the other is almost certain to contain a far greater gift. That is because, given any natural number, there are only finitely many natural numbers that are smaller than it, and infinitely many that are bigger. But of course, each envelope is as likely as the other to contain the larger amount. Such a random choice therefore seems to be impossible.
......Paradoxically there are, quite plausibly, random real numbers, if the standard set-theoretic axiom of infinity is not too unrealistic; and that is paradoxical because if a god could choose real numbers at random then, since any countable list of real numbers amounts to a correlation between some real numbers and the natural numbers, so a god who knew several such lists might decide that if two randomly chosen real numbers were on the same list then he would present you with the corresponding two envelopes. Regarding the aforementioned plausibility, note that any particle that could decay is, if considered one half-life into the future, mathematically akin to a perfectly fair coin-toss, whence an endless sequence of such particular half-lives could instantiate a random real number in binary notation (with decays corresponding to 1s and non-decays to 0s); and such sequences plausibly exist if space is infinite, or if it lasts forever, or if there are other universes alongside ours (in a multiverse), and so forth...
(PS: This post is linked to in the Philosophers' Carnival 105:)


Jeff said...

Wow. This was interesting to the point of making my brain hurt.

My only thought is regarding the God picking two random numbers... And perhaps you are implying this and I'm just a little slow...
But is it coherent to suggest that it is possible to randomly select a number between 0 (or 1) and infinity?

I can easily imagine what it would be like to randomly pick a number from a list that has an end. (say between 0 and 100) I am not sure that it is possible to randomly pick a number between 0 and infinity. Part of the reason for this is because most numbers are so large that we can't properly concieve of them and therefore aren't a live option of being selected.
It might be objected that the God can randomly choose a number from an infinite list. And while this might be true, I suspect that the paradox lies in the suspicion that we can understand this divine way of thinking, which clearly we can't.

enigMan said...

Many would agree with you; but the standard objection to your suspicion is that we can conceive of a being who is able to do something twice as quickly as he has just done it. He could therefore do something in half a minute, a similar thing in a quarter of a minute, another in an eigth, and so on, and on, endlessly. We can imagine each bit of that, and after a minute he will have done infinitely many things. There was no last thing, but during that minute he was always doing something conceivable. And then we suppose that what he was doing was counting, from 1 (very slowly in the first half minute) to infinity. What I think is that the more we think about that, the more paradoxical it becomes. But such is the standard objection (via supertasks:)

gwern said...

Jeff: your intuition is telling you about the distinction between 'constructive'/'intuitionistic' math and unconstructive/regular math.

Briefly, there is no possible algorithm which can select, with the same probability, any random integer between 0 and infinity. You can do *biased* random selection, like a number _n_ has 1/n chance of selection, but you can't compute a selection where each number has 1/infinity chance of being selected. What is 1/infinity? 0? How does an algorithm which must run in finite (if unbounded) space-time work on infinite data?

It's still well-defined and sensible given non-intuitionistic axioms - like the axiom of choice. But then you give up being able to use the Curry-Howard isomorphism to turn the proof into an algorithm; you'll compute up to the point where the axiom of choice is invoked and then go 'huh? *How* do I pick this arbitrary element?' (It's kind of circular.)

We humans can only do what algorithms do (plausibly, anyway), but since God is defined as magic, no reason why he couldn't do non-algorithmic tasks.

(Any of his standard properties would do the trick: he has infinite power, so he could create a Turing machine which runs infinitely fast; he has infinite knowledge, so he can just ask 'if I could so pick, what would I pick right now?' and use the answer to his counterfactual, etc.)