124.Four Coins With a Costly Re-Toss
★★★optimal stoppingJane Street
Four fair coins are tossed and you are paid for every tail showing. Instead of taking that payout you may pay to toss all four coins again, and you may keep doing this as often as you like, paying each time. Playing optimally, what is the value of the game?
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Let be the value of the game just before a toss. A re-toss returns you to exactly the same position one dollar poorer, so after seeing the result you are choosing between two known quantities — write as the average of that comparison and solve for it.
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