St. Petersburg paradox

The St. Petersburg paradox or St. Petersburg lottery[1] is a paradox related to probability and decision theory in economics. It is based on a 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 to the paradox have been proposed.

The paradox takes its name from its analysis by Daniel Bernoulli, one-time resident of the eponymous Russian city, who published his arguments 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,[2] who first stated it in a letter to Pierre Raymond de Montmort on September 9, 1713 (de Montmort 1713).[3]

A casino offers a game of chance for a single player in which a fair coin is tossed at each stage. The initial stake begins at 2 dollars and is doubled every time heads appears. The first time tails appears, the game ends and the player wins whatever is in the pot. Thus the player wins 2 dollars if tails appears on the first toss, 4 dollars if heads appears on the first toss and tails on the second, 8 dollars if heads appears on the first two tosses and tails on the third, and so on. Mathematically, the player wins 2 k + 1 {\displaystyle 2^{k+1}} dollars, where k {\displaystyle k} is the number of consecutive head tosses. What would be a fair price to pay the casino for entering the game?

Leave a Comment