On the St. Petersburg paradox

We¹ ask what is the appropriate price c to pay to receive an amount X ≥ 0; X ∼ F(x). This is known as the St. Petersburg paradox if Pr{X = 2^k} = 2^−k, k = 1, 2, … Here EX = ∞. Is any price c aceptable? We consider this distribution as well as the distribution Pr{X = 2^2k} = 2^−k, k = 1, 2, …, which might be called the super St. Petersburg paradox in which not only is EX equal to infinity, but E log X is infinity as well. Let μr = (EX^r)^1/r denote the r^th mean of X. We identify three critical costs, μ−1 = 1/E(1/X), μ0 = e^{E lnX}, and μ1 = EX, and conclude that we want some of X if c ≤ μ1, and that taking all of X is growth optimal if c ≤ μ−1. Thus all prices c are attractive in the St. Petersburg game.