Suppose that a person has a given fortune A > 0 and can
Question and Solution
Suppose that a person has a given fortune A > 0 and can bet any amount b of this fortune in a certain game (0 ? b ? A).
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Suppose that a person has a given fortune A > 0 and can bet any amount b of this fortune in a certain game (0 ? b ? A). If he wins the bet, then his fortune becomes A + b; if he loses the bet, then his fortune becomes A ? b. In general, let X denote his fortune after he has won or lost. Assume that the probability of his winning is p (0 < p < 1) and the probability of his losing is 1 ? p. Assume also that his utility function, as a function of his final fortune x, is U (x) = log x for x > 0. If the person wishes to bet an amount b for which the expected utility of his fortune E[U (X)] will be a maximum, what amount b should he bet?
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