The St. Petersburg lottery or St. Petersburg paradox[1] is a paradox related to probability and decision theory in economics. It is based on a particular (theoretical) lottery game that leads to a random variable with infinite expected value (i.e., infinite expected payoff) but nevertheless seems to be worth only a very small amount to the participants. The St. Petersburg paradox is a situation where a naive decision criterion which takes only the expected value into account predicts a course of action that presumably no actual person would be willing to take. Several resolutions are possible.

The paradox is named from Daniel Bernoulli's presentation of the problem and his solution, published in 1738 in the Commentaries of the Imperial Academy of Science of Saint Petersburg (Bernoulli 1738). However, the problem was invented by Daniel's cousin Nicolas Bernoulli who first stated it in a letter to Pierre Raymond de Montmort on September 9, 1713 (de Montmort 1713).

A casino offers a game of chance for a single player in which a fair coin is tossed at each stage. The pot starts at 2 dollars and is doubled every time a head appears. The first time a tail appears, the game ends and the player wins whatever is in the pot. Thus the player wins 2 dollars if a tail appears on the first toss, 4 dollars if a head appears on the first toss and a tail on the second, 8 dollars if a head appears on the first two tosses and a tail on the third, 16 dollars if a head appears on the first three tosses and a tail on the fourth, and so on. In short, the player wins 2k dollars, where k equals number of tosses. What would be a fair price to pay the casino for entering the game?

To answer this, we need to consider what would be the average payout: with probability 1/2, the player wins 2 dollars; with probability 1/4 the player wins 4 dollars; with probability 1/8 the player wins 8 dollars, and so on. The expected value is thus

$E= \frac{1}{2}\cdot 2+\frac{1}{4}\cdot 4 + \frac{1}{8}\cdot 8 + \frac{1}{16}\cdot 16 + \cdots$
$= 1 + 1 + 1 + 1 + \cdots$
$=\infty \,.$

Works cited

### Bibliography

• Haigh, John (1999). Taking Chances. Oxford,UK: Oxford University Press. p. 330. ISBN 0198526636.(Chapter 4)