338.A Hundred-Sided Die With Costly Rerolls
★★★optimal stoppingJane Street
A fair 100-sided die shows a number from 1 to 100. After each roll you may either take that many dollars and stop, or pay to roll again, as often as you like, keeping only your final roll. 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 you should stop exactly when the number in front of you beats that. Write V as an equation in itself.
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