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Unlimited paid rerolls

Expected valueHard

You roll a fair $6$ -sided die. After each roll you may stop and take the face as payoff, or pay a fee of $2$ and roll again. There is no limit on the number of rerolls. The net payoff is the last face minus the total fees paid. Playing optimally, what is the expected net payoff before the first roll?

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Hint

Let $V$ be the value before a roll; rerolling restarts the game at a cost, giving an equation for $V$ itself.

Solution

Let $V$ be the optimal expected value before a roll. After seeing face $f$ you either keep $f$ or pay $2$ and restart, worth $V - 2$ .
So you keep $f$ exactly when $f \geq V - 2$ , and $V$ satisfies $V = E [max (f, V - 2)]$ .
Guess that faces $2$ to $6$ are kept: $V = \frac{20 + ( V - 2 )}{6}$ , so $5 V = 18$ .
Then $V - 2 = 1.6$ , so face $1$ is rerolled and face $2$ is kept, which confirms the guess.
The expected net payoff is $\frac{18}{5}$ .

Answer: $\frac{18}{5}$

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