Solution
Left Gloves and Right Gloves
Show the problem again
Five players each own one glove: three of them own a left glove and two own a right glove. A coalition's worth is the number of matched (left, right) pairs it can form, so v(S) = min(number of left gloves in S, number of right gloves in S), and each pair sells for 1 unit. Find the Shapley value of one of the right-glove owners, as a reduced fraction.
Worked solution
Work with one right-glove owner R. Order the five players uniformly at random; R's marginal contribution is 1 exactly when the players before R contain more lefts than rights. The other four players are 3 lefts and 1 right. If k players precede R (k = 0,...,4, each with probability 1/5), they form a uniformly random k-subset of those four. k = 0: contribution 0. k = 1: it must be a left, probability 3/4. k = 2: both lefts (probability 1/2) works, left plus right ties and fails: 1/2. k = 3: three lefts (1/4) or two lefts and the right (3/4) both give more lefts than rights: 1. k = 4: 3 lefts against 1 right: 1. Averaging: (0 + 3/4 + 1/2 + 1 + 1)/5 = 13/20. As a check, efficiency gives 3x + 2y = 2 with y = 13/20, so each left owner gets x = 7/30. The scarce side earns more: each right-glove owner has Shapley value **13/20**.
Source: The glove game is a standard textbook example of a cooperative game (see Shapley and Shubik, 1969, on market games). Numbers chosen for AxiomIQ. Statement written for AxiomIQ.