St. Petersburg paradox

The St. Petersburg paradox or St. Petersburg lottery is a paradox involving the game of flipping a coin where the expected payoff of the theoretical lottery game approaches infinity but nevertheless seems to be worth only a very small amount to the participants. The St. Petersburg paradox is a situation where a naïve decision criterion that 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 problem was invented by Nicolas Bernoulli, who stated it in a letter to Pierre Raymond de Montmort on September 9, 1713. However, the paradox takes its name from its analysis by Nicolas' cousin Daniel Bernoulli, one-time resident of Saint Petersburg, who in 1738 published his thoughts about the problem in the Commentaries of the Imperial Academy of Science of Saint Petersburg. 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 the current stake. 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 dollars, where is the number of consecutive head tosses. What would be a fair price to pay the casino for entering the game? To answer this, one needs to consider what would be the expected payout at each stage: 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. Assuming the game can continue as long as the coin toss results in heads and, in particular, that the casino has unlimited resources, the expected value is thus Assuming the game can continue indefinitely, this sum grows without bound, and so the expected win is an infinite amount of money.