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.
Highlights
In the eighteenth century, when the paradox was first studied, the main goal of probability theory was to answer questions that involved gambling, especially the problem of defining fair amounts that should be paid to play a certain game or if a game was interrupted before its end
It is intuitive to take the expected gain at a trial of the game or the expected gain of a player if the game were not to end respectively, as moral values, and that is what the first probabilists established as those fair amounts
Applying the same method used by [9], we have, for the game played with a coin with probability θ (0 < θ < 12 ) of heads, the entrance fee presented in the theorem below
Summary
In the eighteenth century, when the paradox was first studied, the main goal of probability theory was to answer questions that involved gambling, especially the problem of defining fair amounts that should be paid to play a certain game or if a game was interrupted before its end. Those amounts were named moral value or moral price and are what we call expected utility nowadays. We generalize his method for the game in which the coin used is not fair, i.e., its probability of heads is θ 6= 1/2, and for the case in which the coin is a random one, i.e., its probability θ of heads is a random variable defined in (0, 1)
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