The Consistent Shapley Value for Games without Side Payments

Abstract

In [Maschler and Owen, 1989], a new generalization of the Shapley value for a class of NTU games was introduced. The motivation was a desire to preserve as much as possible the consistency property of the Shapley value for TU games, in the sense of [Hart and Mas-Colell, 1989]. It turned out that the new value resulted from an intuitive dynamic process which was interesting also for the class of TU games. Unfortunately, the class of NTU games was quite narrow; namely, the class of hyperplane games. The purpose of this paper is to extend the definition to the general class of NTU games, whose coalition functions satisfy (essentially) the usual requirements. 1. Introduction . In [Maschler and Owen, 1989] we introduced an extension of the Shapley value, which we called "the cnnsistent Shapley value", to the class of hyperplane games. This class consists of non-transferable-utility (NTU) games whose coalitional [characteristic] function is a half space of)Rs for each coalition S. The consistent value was justified by several nice properties: (1) It generalized in some sense the [Hart and Mas-Colell] reduced game property of the Shapley value for transferable utility (TU) games (see Hart and Mas-Colell [1989]). (2) It was an obvious generalization of the random order procedure to get the Shapley value of TU games. (We could therefore call this solution concept also "a random order Shapley value" .) (3) An intuitively justifiable dynamic procedure-interesting also in the TU case-leads the players from any arbitrary imputation to the value. Unfortunately, the random order procedure of our [1989] paper did not produce satisfactory results for other NTU games. Usually it yielded outcomes which were not even Pareto-optimal. The reason is that, unlike the case of a hyperplane game, a convex combination of Pareto-optimal points need not be Pareto-optimal. It became necessary, therefore, to get a more satisfactory solution for NTU games--one that will coincide with the consistent value when the games happen to be hyperplane games. That is the purpose of this chapter.