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Paid reroll decision

Expected valueMedium

You roll a fair $6$ -sided die and see the face. You may keep it, or pay a fee of $1$ to reroll once, after which you must keep the new face. Your net payoff is the face kept minus any fee paid. Playing optimally, what is the expected net payoff?

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Hint

A reroll is worth the mean of a fresh roll minus the fee; keep the face when it beats that.

Solution

A reroll has net value $3.5 - 1 = 2.5$ .
So reroll on faces below $2.5$ , that is $1$ and $2$ , and keep $3$ , $4$ , $5$ and $6$ .
Keeping contributes $\frac{3 + 4 + 5 + 6}{6} = 3$ .
Rerolling happens with probability $\frac{2}{6}$ and contributes $\frac{2}{6} \times 2.5 = \frac{5}{6}$ .
The expected net payoff is $3 + \frac{5}{6} = \frac{23}{6}$ .

Answer: $\frac{23}{6}$

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