Variant of Glove Game Suppose there are players, where ` players own a left-hand glove and r players own a right-hand glove. Let N be the set of all players, let L be the subset of N consisting of...

Variant of Glove Game

Suppose there are

players, where ` players own a left-hand glove and r players own a right-hand glove. Let N be the set of all players, let L be the subset of N consisting of the players who own a left-hand glove, and let R be the subset of N consisting of the players who own a right-hand glove. Let
denote an arbitrary coalition and let

denote the number of players in S. For instance,

is the number of players in S who own a right-hand glove. As before, each pair of gloves has worth 1. Find an expression for v.S/, i.e., the maximal profit that S can generate by cooperation of its members.