Anders Martin-Löf: A survey of the theory of the Petersburg game

This year we can celebrate the 300th anniversary of the invention of the Petersburg game by Daniel Bernoulli, where the gain is 2,4,8... with probability 1/2, 1/4, 1/8,.... The fact that the expected gain is infinite has given rise to numerous discussions of the relevance of this quantity compared to some more reasonable utility. The recent studies of the game take the attitude that the gain is relevant and studies the total gain in a large number of games and tries to finde interesting limit theorems for the distribution of this quantity. It turns out that an interesting class of limit distributions is obtained, named semistable by Levy. Luckily quite a simple asymptotic expression for the probability of a large gain can be found. Also variations of the game where interest on the capital is taken into account can be studied in a similar way.