Roll a 20-sided die, keep it or re-roll once

Question

You roll a fair twenty-sided die (faces $1$ through $20$ ) and are paid its face value in dollars. You may either keep it, or discard it and roll once more, taking whatever the second roll shows.

What is the optimal strategy, and what is the fair value of the game?

Accepted Answer

The re-roll is worth the expected face of a fair d20: E[	ext{d20}] = \frac{1 + 20}{2} = 10.5. So keep the first face if and only if it beats 10.5: keep 11 through 20; re-roll 1 through 10. Exactly half the faces clear the bar. Value of the game: V = 	frac{10}{20} \cdot 	frac{11 + 20}{2} + 	frac{10}{20} \cdot 10.5 = 	frac12(15.5) + 	frac12(10.5) = 7.75 + 5.25 = 13.0. So the game is worth \13.00, up \2.50 from the plain-roll value of 10.5. The re-roll option is worth more on a d20 than on a d6 (which gains only 0.75), because a wider die has more upside to reach for when your first face i…

Roll a 20-sided die, keep it or re-roll once

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