Rigorous computer analysis of the Chow-Robbins game

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.

💡 Research Summary

The paper addresses the classic optimal‑stopping problem known as the Chow‑Robbins game, in which a fair coin is tossed repeatedly and the player may stop at any time, receiving a payoff equal to the proportion of heads observed so far. The central object of study is V(a,n), the maximal expected payoff when the player has seen a heads after n tosses. A naïve dynamic‑programming recurrence,
V(a,n)=max{a/n, ½