05.A Sixty-Four-Sided Die With a Fee
★★★optimal stoppingJane Street
A fair 64-sided die shows a number from 1 to 64. After each roll you may take that many dollars and stop, or pay to roll again, as often as you like. Playing optimally, what is the expected payout?
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Let V be the value of the game. Rolling again is worth V minus the fee, so stop exactly when the roll beats that. Write V as an equation in itself.
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