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Rigorous Computer Analysis of the Chow–Robbins Game

Olle Häggström and Johan Wästlund
The American Mathematical Monthly
Vol. 120, No. 10 (December 2013), pp. 893-900
DOI: 10.4169/amer.math.monthly.120.10.893
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Page Count: 8
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Rigorous Computer Analysis of the Chow–Robbins Game
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Abstract Flip a coin repeatedly, and stop whenever you want. Your payoff is the proportion of heads, and you wish to maximize this payoff in expectation. This so-called Chow–Robbins game is amenable to computer analysis, but while simple-minded number crunching can show that it is best to continue in a given position, establishing rigorously that stopping is optimal seems at first sight to require “backward induction from infinity”. We establish a simple upper bound on the expected payoff in a given position, allowing efficient and rigorous computer analysis of positions early in the game. In particular, we confirm that with 5 heads and 3 tails, stopping is optimal.

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