In probability theory, the St. Petersburg paradox exhibits a random variable whose value is very probably very small, and yet has an infinite expected value. This poses a situation where decision theory may superficially appear to recommend a course of action that no rational person would be willing to take. That appearance evaporates when utilities are taken into account. It was first ennunciated by Daniel Bernoulli in 1738.
In a game of chance, you pay a fixed fee to enter, and then a coin will be tossed repeatedly until a "head" first appears. You win 1 cent if a head appears on the first toss, 2 cents if on the second, 4 cents if on the third, 8 cents if on the fourth, etc. It doubles with every toss. In short, you win
2k−1 cents if the coin must be tossed k times.
How much would you be willing to pay to enter the game?
The probability that the first head occurs on the kth toss is:
The probability that you win more than $10.24 (i.e., 210 cents) is less than one in a thousand. The probability that you win more than $1 is less than one in a hundred. Nonetheless, the expected amount that you win is infinite! Here is how it is calculated:
This sum diverges to infinity. Thus, now matter how much you pay to enter (imagine paying $1 billion each time, and winning only a few cents on nearly all occasions when you have paid that fee for the privilege) you will come out ahead in the long run. That is because on the very rare occasions when a large payoff comes along, it will far more than repay all the hundreds of trillions of dollars you have paid to play.
Decision theory applied naively without taking utility into account would suggest that any fee, no matter how high, would be worth paying for this opportunity. In practice, no reasonable person would pay more than a few cents to enter.
Encounter with the paradox leads to a deeper understanding of a variety of issues in economics and decision theory, in particular:
- Utility;
- Diminishing marginal utility of money;
- Risk aversion; and
- The gestalt of factors that are not simply represented in mathematical models but which provide human decision-making with its context.
For a fuller treatment see:
A translation of Daniel Bernoulli's original presentation is found in:
- Bernoulli, Daniel: 1738, "Exposition of a New Theory on the Measurement of Risk", Econometrica vol 22 (1954), pp23-36.
