Average lengths for the two-player Name Game

Extract

In the Name Game, letters of the alphabet are drawn at random, and a player marks off all occurrences of the letter in his name as it is called. The winner is the player whose name is deleted first; but a tie can occur when players' names have letters in common. For the two-player game, the probability of a player winning depends not only on the length of his own name but on how many letters occur only in the other player's name. (See [1] for probabilities involving more players.) For example, if Stephanie plays against Georges, then there are 11 letters in the union, 2 in the intersection, 3 that are in Georges but not Stephanie, and 6 that are in Stephanie but not Georges. In this case, the probability of a tie is 2/11, the probability of Stephanie winning is 3/11, and the probability of Georges winning is 6/11. These probabilities are easily derived by considering the 11! permutations of the letters in the union.