Consider the following four-person game in coalitional form, with transferable utility
v({i}) = 0, ∀i ∈ {1,2,3,4},
v({1,2}) = v({1,3}) = v({2,4}) = v({3,4}) = 1,
v({1,4}) = v({2,3}) = 0,
v({2,3,4}) = v({1,3,4}) = v({1,2,4}) = 1,
v({1,2,3}) = 2 = v({1,2,3,4}).
(Notice that the worth of each coalition except {1,2,3} is equal to the number of disjoint pairs that consist of one player from {1,4} and one player from {2,3} that can be formed among the coalition's members.)
a. Show that the core of this game consists of a single allocation vector.
b. Compute the Shapley value of this game. (Notice the symmetry between players 2 and 3.)
c. Suppose that the worth of {1,2-,3} were changed to v({1,2,3}) = 1. Characterize the core of the new game, and show that all of the new allocations in the core are strictly better for player 1 than the single allocation in the core of the original game.
d. Compute the Shapley value of the new game from (c).