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), μ
₀
= e
^{E lnX}
, and μ
₁
= EX, and conclude that we want some of X if c ≤ μ
₁
, and that taking all of X is growth optimal if c ≤ μ
_-1
. Thus all prices c are attractive in the St. Petersburg game.