20.Fixing Your Number in the Pairs Game
★★★expected valueJane Street
Four players each hold a different whole number from 1 to 50. The player holding the highest number and the player holding the lowest number form one team; the two players holding the middle numbers form the other. The team with the larger total pays the team with the smaller total the difference between the two totals. You may fix your own number to any value from 1 to 50 before the other three are drawn, and those three are then a uniformly random set of three distinct values from the remaining 49. Which value should you fix in order to maximise your expected winnings?
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Split by which team you land on. One of the two cases turns out to be worth exactly nothing on average, no matter what you hold — find it using the expected order statistics, and the problem collapses to the other case.
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