Petersburg Game Research Articles

ON SUMS OF INDEPENDENT RANDOM VARIABLES WITH INFINITE MOMENTS AND „FAIR” GAMES

At the origin of this investigation is the well-known result of Feller1 concerning the Petersburg game, in which the player receives $2i if heads first appears at the ith toss of an unbiased coin (i = 1, 2, …). Since the expectation of the player’s gain x is infinite, a problem arises in deciding on the proper fee for the privilege of playing the game; Feller shows that if the player pays variable entrance fees with cumulative fee b n = n log2 n for the first n games, and if s n = x 1 + … + x n denotes his total gain, then the games becomes “fair” in the sense that $$\mathop {\lim }\limits_{n \to \infty } \frac{{{S_n}}}{{{b_n}}} = 1{\text{ in probability}}{\text{.}}$$ (1) .