St Petersburg paradox

Paradox in the theory of probability published by Daniel Bernoulli in 1730 in the Commentarii of the St Petersburg academy. Someone offers you the following opportunity: he will toss a fair coin. If it comes up heads on the first toss he will pay you one dollar; if heads does not appear until the second throw, two dollars, and so on, doubling your winnings each time heads fails to appear on another toss. The game continues until heads is first thrown, when it stops. What is a fair amount for you to pay for this opportunity? The incredible reply is: an infinite amount. For your expectation of gain is given by the series ½ + (2 × 1/4) + (4 × 1/8) +…, which has an infinite sum. This little offer is apparently worth more to you than all the wealth in the world. Yet nobody in their right mind would pay much at all for it. The paradox has been taken to show the incoherence of allowing infinite utilities into decision theory. Once they are allowed, then it is worth staking any finite sum on any indefinitely small chance of an infinite payoff. If it is specified that the St Petersburg game has to stop when, for example, the payoff reaches the size of the National Debt, then the calculation of what you should pay to enter the game remains quite reasonable: if the National debt is, say 2ⁿ dollars, you should pay ½n dollars and no more to enter the game.