Given an n-person game (N, v), a reduced game (T, vT) is the game obtained if some subset T of the players assumes reasonable behavior on the part of the remaining players and uses that as a given so as to bargain within T. This "reasonable" behavior on the part of N-T must be defined in terms of some solution concept, ϕ, and so the reduced game depends on ϕ. Then, the solution concept ϕ is said to be consistent if it gives the same result to the reduced games as it does to the original game. It turns out that, given a symmetry condition on two-person games, the Shapley value is the only consistent solution on the space of TU games. Modification of some definitions will instead give the prekernel, the prenucleolus, or the weighted Shapley values. A generalization to NTU games is given. This works well for the class of hyperplane games, but not quite so well for general games.
Consistency and Potentials in Cooperative TU-Games: Sobolev’s Reduced Game Revived
