The swap duel with dice

Question

You and an opponent each secretly roll a fair die (independent rolls). You see yours. You may keep it or reroll once, in which case you must keep the new face. The opponent never rerolls. Higher face wins; a tie counts as a loss for you.

What is your optimal reroll rule, and what is your probability of winning?

Accepted Answer

If you hold a face x, you win only when the opponent's face is strictly below x. Since their face is uniform on 1 to 6, P(	ext{win} \mid 	ext{keep } x) = \frac{x - 1}{6}. If you reroll, your fresh face Y is uniform, so your win chance averages P(	ext{win} \mid 	ext{reroll}) = E\!\left[\frac{Y-1}{6}ight] = \frac{3.5 - 1}{6} = \frac{2.5}{6} \approx 0.417. Reroll whenever 	frac{x-1}{6} < 	frac{2.5}{6}, that is when x < 3.5. Keep 4, 5, 6, reroll 1, 2, 3. Combining: half the time you reroll and win with chance 	frac{2.5}{6}; half the time you keep a 4, 5, 6 with win chances 	frac{3}{6}, …

The swap duel with dice

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