The St. Petersburg Paradox

The St. Petersburg paradox is nearly 300 years old, but it can tell us something about rational decision making. Imagine, for example, that God wants to play a game with you. Suppose that you can tell that it is God talking to you. You can believe everything that He tells you (or They tell you) and He will pay you an amount equal to the entire wealth of the universe to play the following game:

He will repeatedly toss a fair coin, in a perfectly fair way, until it lands heads up. If it lands heads up straightaway, then you will just owe Him two cents (which you could return to Him from the vast wealth of the universe). If it lands heads up on the second toss, you will owe Him four cents. If it lands heaps up on the third toss, you will owe Him eight cents. And so on: the amount doubles with each toss.

Before you decide, God shows you that the wealth of the universe includes alien medical technology that can prolong your natural life indefinitely, and also teleportation devices, enabling you to spend whatever wealth you have after the game on trillions of inhabited planets. He also shows you that were you to be extraordinarily unlucky in the game, He could make you pay Him arbitrarily large amounts by having you work for Him at a very reasonable rate of pay in some relatively pleasant part of purgatory for arbitrarily long amounts of time. So, would you want to play this game?

Note that your chance of owing anything to God because of playing this game is extremely small. Even if God threw 47 tails before throwing his first head, you would only have to return a paltry trillion dollars to Him from the enormous wealth that He will have already given you to play this game. Your chance of having to do any work in the afterlife because of this game is negligible. What would actually happen if you played this game? You would become the master of the universe!

Because the chance of God getting a tail on the first toss is 50%, or 1/2, and the chance of Him also getting a tail on the second toss is 25%, or 1/4 (there being 4 equally likely possibilities for two tosses, one of which is two tails), and the chance of Him getting a third tail is 1/8, and so forth, God's mathematical expectation is 2/2 + 4/4 + 8/8 + … cents. That is 1 + 1 + 1 + … cents, an infinite number of cents. But God's realistic expectation was of you doing no work for Him. And there was no chance of you working for Him forever in purgatory because of that game (unless your having the vast wealth of the universe increased your chance of ending up there for other reasons). So, this fictional God created infinitely many universes, and offered one person in each universe the chance to play that game. Each of those people would of course decide to play, and with infinitely many of them playing, God's realistic expectation was the mathematical expectation.

What can that tell us about rational decision making? Well, note that there would still be a paradoxical game even if, after a number of tails equal to, say, ten times the wealth of the universe in cents, there were no further coin tosses. God's mathematical expectation would still be of a massive gain. And for that to be a realistic expectation, there would not need to be infinitely many games. So, it would not even need to be God playing. This is a paradox, not of infinity, but of rational decision making in large and complicated betting environments.