Wednesday, May 02, 2007

St. Petersburg Paradox

This paradox (for further details, see here and here) is nearly 300 years old now, but it can still challenge a naive idea of what it is to be rational, despite its simplicity. For a simple version of the paradox, imagine God offering you the following deal, on His fair tossing of a fair coin. You can tell somehow that it is God talking to you, and so you rightly believe all that He tells you.

Firstly God will pay you a certain amount, as below, and then God will repeatedly toss the coin, until it lands heads up. Each time it lands tails up (if it does) you will owe Him a certain amount of money, as follows. For the first tail 2 cents, for the second another 2 cents, but for the third another 4 cents, and so forth, so that for a total of N tails before the first head a total of 2-to-the-power-of-N cents. God helpfully explains what that formula means mathematically. The chance of Him getting a tail on the first toss is 50%, and the chance of Him also getting a tail on the second toss is 25% (since there are four equally likely possibilities, HH, HT, TH and this one, TT) and so forth. So the mathematical expectation is (2/2 + 4/4 + 8/8 + …) cents = (1 + 1 + 1 + …) cents = an infinite number of cents, or dollars (or trillions of dollars).

In view of that fact, God feels that it would not be irrational to offer you the entire wealth of the Universe. In effect, you would become God’s appointed Queen (or King) of the Universe in exchange for you owing Him a number of cents thus determined, by the fair toss of a fair coin. To simplify matters, assume that God has shown you that, whether or not you take him up on this deal, you will live forever in some form or another (e.g. as an immortal soul), and that the wealth of the Universe includes alien medical technology that can prolong your natural life indefinitely, and also teleportation devices (so that you could actually spend all that wealth). Of course, conversely God could (if necessary) make you pay Him arbitrary amounts over and above your new wealth, were you to end up owing Him money (were you that unlucky), by getting you to work for Him at a very reasonable rate of pay, in some relatively pleasant part of Purgatory. But still, there is no trickery involved here (it is God talking, not the Devil). God will not be offended if you do not take the deal, it is only being offered because He suspects that you might wish to take it. Would you?

Well, what is your chance of losing much? It is extremely small, because for you to return as much as 10 dollars, from the vast wealth of the Universe, God would have to throw at least 10 tails in a row. And if God threw less than 47 tails before throwing his first head, which seems almost certain to occur, you would not even have to return a paltry trillion dollars. Your chance of having to work for trillions of years in the afterlife is much, much smaller. Would it be rational to reject such an offer just because of the remote possibility of something that almost certainly won’t happen? I mean, smell the coffee! What would actually happen, if the above scenario were offered, and you took God up on His deal? You would own most of the Universe.

Still, your mathematical expectation (the mean, not the mode) is of an infinite loss, and perhaps rationality should have such a mathematical precision (e.g. via coherent betting quotients). According to such a view, your intuitions about what is rational would, if you took God up on this deal, be letting you down. Nonetheless it does also (as above) seem rational to accept this deal. So, could both options be rational? Incidentally, I don't think that the infinitude of this scenario is the underlying source of the paradox, even though our intuitions are often confounded by infinite scenarios. That is because God's mathematical expectation did not need to be of an infinite gain, only of a gain greater than what He offered you. A paradox would therefore remain even if, after a number of tails equal to, say, twice the wealth of the Universe in cents, further tails would entail no further losses.

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