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Redrawing a uniform payout

CalculusHard

You are paid $U$ dollars, where $U$ is uniform on $[0, 1]$ . After seeing a draw you may reject it and take a fresh independent uniform draw instead; you may reject at most $2$ times, so you must accept the third draw if you reach it. With optimal play, what is your expected payment, in dollars? Give the answer as a fraction in lowest terms.

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Hint

Work backwards: with $n$ draws left, accept a draw exactly when it beats the expected value of continuing.

Solution

With only the last draw left, the expected payment is $\frac{1}{2}$ .
With $2$ draws left, accept when $U > \frac{1}{2}$ ; the value is $\frac{1}{2} \times \frac{3}{4} + \frac{1}{2} \times \frac{1}{2} = \frac{5}{8}$ .
With $3$ draws left, accept when $U > \frac{5}{8}$ ; the value is the integral of $u$ from $\frac{5}{8}$ to $1$ plus $\frac{5}{8} \times \frac{5}{8}$ .
That integral is $\frac{1 - \frac{25}{64}}{2} = \frac{39}{128}$ , and $\frac{5}{8} \times \frac{5}{8} = \frac{50}{128}$ .
So the expected payment is $\frac{39}{128} + \frac{50}{128} = \frac{89}{128}$ .

Answer: $\frac{89}{128}$

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