Exercise 16.4 [St-Petersburg-exercise]
In 1713, Nicolas Bernoulli stated a puzzle, now called the St. Petersburg paradox, which works as follows. You have the opportunity to play a game in which a fair coin is tossed repeatedly until it comes up heads. If the first heads appears on the $n$th toss, you win $2^n$ dollars.
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Show that the expected monetary value of this game is infinite.
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How much would you, personally, pay to play the game?
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Nicolas’s cousin Daniel Bernoulli resolved the apparent paradox in 1738 by suggesting that the utility of money is measured on a logarithmic scale (i.e., $U(S_{n}) = a\log_2 n +b$, where $S_n$ is the state of having $n$). What is the expected utility of the game under this assumption?
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What is the maximum amount that it would be rational to pay to play the game, assuming that one’s initial wealth is $k$?