Two rerolls, but each reroll costs a dollar

Question

You roll a fair six-sided die and are paid its face value. After the first or second roll you may reroll, but **each reroll costs $\$ 1 $**; you must accept the third roll if you take it (up to$ 3$ rolls total).

What is the fair value of the game under optimal play?

Accepted Answer

Backward induction again, but now the option to reroll comes with a \1 charge, which lowers each keep-threshold. Last roll (no options left): E0 = 3.5. One reroll left: rerolling is worth E0 - 1 = 2.5 (the final roll's value minus the fee). Keep the current roll iff it beats 2.5, i.e. keep \{3,4,5,6\}, reroll \{1,2\}: E1 = \frac{2}{6}(2.5) + \frac{1}{6}(3+4+5+6) = \frac{5}{6} + \frac{18}{6} = \frac{23}{6} \approx 3.833. Two rerolls left (first roll): rerolling is now worth E1 - 1 = \frac{23}{6} - 1 = \frac{17}{6} \approx 2.833. Keep iff the roll beats \frac{17}{6}, i.e. keep \{3,4,5,6\}, rero…

Two rerolls, but each reroll costs a dollar

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