On the St. Petersburg paradox

We 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 μ_r = (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.