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Paid re-rolls

Expected valueMedium

A fair $6$ -sided die is rolled. You may keep the result and be paid that many dollars, or pay $1.5$ dollars to roll again; after each new roll you may again keep it or pay $1.5$ to re-roll, with no limit. The first roll is free. With optimal play, what is the expected net payout?

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Hint

Let the value of facing a fresh roll be unknown and make it satisfy a self-consistency equation.

Solution

Let $W$ be the expected net payout before a roll; after seeing $x$ you get the better of $x$ and $W - \frac{3}{2}$ .
Guess that you keep $3$ , $4$ , $5$ and $6$ , so $W = \frac{18}{6} + \frac{2}{6} \times (W - \frac{3}{2})$ , which gives $\frac{2 W}{3} = \frac{5}{2}$ .
Then $W = \frac{15}{4}$ , so $W - \frac{3}{2} = \frac{9}{4}$ lies between $2$ and $3$ , which confirms the guess.
The optimal expected net payout is $\frac{15}{4}$ .

Answer: $\frac{15}{4}$

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