2012 Volume 33 Pages 85-96
It is said that although a sample distribution of the St. Petersburg gambling game was not considered, D. Bernoulli might have solved a paradox upon the gambling game in two ways: One is a solution through a diminishing marginal utility function of the natural logarithm; the other is one through the geometric mean.
However, we would like to show here that both of them may not work at all, by computing not only a few of the Taylor’s formulas including a Maclaurin series for the former but also the harmonic mean for the latter. In addition to it, we can see that the gambling game has a geometric distribution, which has infinite possible outcomes and whose moments depend on a player’s subjective probability θ of obtaining tails per one coin flip.
To do so, we assume in this paper that every player is supposed to be a Bayesian, who employs natural conjugate prior information of a beta distribution for a Bayes’ solution δ* stemmed from a minimization problem of a posterior loss function. Then, we would like to derive the Kelly’s formula and study how such a Bayesian player revises his or her speculative or gambling spirit stirred up in order to flip a coin at the risk of some entrance fee ϕ.
