Entrance Fees and a Bayesian Approach to the St. Petersburg Paradox

Abstract: In An Introduction to Probability Theory and its Applications, W. Feller established a way of ending the St. Petersburg paradox by the introduction of an entrance fee, and provided it for the case in which the game is played with a fair coin. A natural generalization of his method is to establish the entrance fee for the case in which the probability of heads is θ (0 < θ < 1/2). The deduction of those fees is the main result of Section 2. We then propose a Bayesian approach to the problem. When the probability of heads is θ (1/2 < θ < 1) the expected gain of the St. Petersburg game is finite, therefore there is no paradox. However, if one takes θ as a random variable assuming values in (1/2, 1) the paradox may hold, which is counter-intuitive. In Section 3 we determine necessary conditions for the absence of paradox in the Bayesian approach and in Section 4 we establish the entrance fee for the case in which θ is uniformly distributed in (1/2, 1), for in this case there is a paradox.