The Evil Twin Strategy for a Football Pool

Abstract

At Emmy's we have a weekly football pool in which each person puts $1 in the pot and picks the winning team in about ten football games. The one who picks the most winners gets the entire pot. A point spread is provided, and the favored team must win by the number of points specified. The effect of the point spread is to make the probability of picking the winner of a particular game approximately 1/2. Joe observed that most of us were just guessing and that some were even tossing a coin to decide which team to pick. He asked if he could contribute $2 to the pot and submit two entries. Joe proposed to submit his choices and those of his evil twin. For each game, his evil twin picks the winner to be the team that Joe picks to lose. This request led to interesting moral, legal, and mathematical discussions and to the following problem. Assume that all the participants in the football pool except Joe submit one entry and toss a fair coin to make their choices. Joe submits two entries: one for himself and one for his evil twin. For his choices, Joe also tosses a coin. For each game, Joe's evil twin picks as winner the team that Joe picks to lose. The player with the maximum score wins the entire pot. If there is more than one player with the maximum score, the winning players split the pot evenly. Is this a favorable game to Joe? That is, is his expected winning greater than $2? If Joe and his evil twin are competing only with Dana and there is only one game, the computation is easy. Either Joe or his evil twin will be correct. If Dana is wrong he wins nothing, and if he is correct he shares the $3 pot with either Joe or his evil twin. Thus Dana's expected winning is