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St. Petersburg Paradox Entry Fee

Medium · Betting Games · Quant Trader interview question · probability, expected-value, utility-theory, paradox

You are offered to play the St. Petersburg game. A fair coin is flipped repeatedly until the first time it lands on heads. The payoff is $2^n$, where $n$ is the number of flips it takes to get the first head. What is the maximum amount you would rationally pay to play this game once, assuming your utility function is $U(x) = \sqrt{x}$, where $x$ is the payoff?