  For a game with positive expectation and some negative profit, a unique price
exists, at which the optimal proportion of investment reaches its maximum. For
a game with parallel translated profit, the ratio of this price to its
expectation tends to converge toward less than or equal to 1/2 if its
expectation converges to 0. In this paper, we will investigate such properties
by using the integral representations of a complete Bernstein function and
establish several Abelian and Tauberian theorems.
