The swap duel from one to twenty

Question

You and an opponent each secretly receive a number drawn uniformly at random from $1$ to $20$ (independent draws). You see yours. You may keep it or swap it for a fresh uniform draw from $1$ to $20$ , which you must then keep. The opponent never swaps. Higher number wins; a tie counts as a loss for you.

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

Accepted Answer

Holding a number x, you win only when the opponent is strictly below you, and their number is uniform on 1 to 20: P(	ext{win} \mid 	ext{keep } x) = \frac{x - 1}{20}. Swapping gives a fresh uniform Y, so your win chance averages P(	ext{win} \mid 	ext{swap}) = E\!\left[\frac{Y-1}{20}ight] = \frac{10.5 - 1}{20} = \frac{9.5}{20} = 0.475. Swap whenever 	frac{x-1}{20} < 0.475, that is when x < 10.5. Keep 11 through 20, swap 1 through 10. Combining: half the time you swap and win with chance 0.475; half the time you keep an 11 to 20, whose win chances 	frac{10}{20}, 	frac{11}{20}, \ldots, \…

The swap duel from one to twenty

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