Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

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

These authors contributed equally to this work.
Version 1 : Received: 16 February 2017 / Approved: 17 February 2017 / Online: 17 February 2017 (07:18:58 CET)

A peer-reviewed article of this Preprint also exists.

Marcondes, D.; Peixoto, C.; Souza, K.; Wechsler, S. Entrance Fees and a Bayesian Approach to the St. Petersburg Paradox. Philosophies 2017, 2, 11. Marcondes, D.; Peixoto, C.; Souza, K.; Wechsler, S. Entrance Fees and a Bayesian Approach to the St. Petersburg Paradox. Philosophies 2017, 2, 11.

## Abstract

In his best-selling book 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 head 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 head 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, what is counter-intuitive. On Section 3 we determine a necessary and sufficient condition for the absence of paradox on the Bayesian approach and on 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 paradox.

## Keywords

St. Petersburg Paradox; entrance fees; Bayesian analysis

## Subject

Business, Economics and Management, Econometrics and Statistics