The Nucleolus

The core of a game with transferable utility can be a large set, but it can also be empty. The Shapley value assigns to each game a unique point, which, however, does not have to be in the core.

The nucleolus (Schmeidler [116]) assigns to each game with a nonempty imputation set a unique element of that imputation set; moreover, this element is in the core if the core of the game is nonempty. The pre-nucleolus always exists (and does not have to be an imputation, even if this set is nonempty), but for balanced games it coincides with the nucleolus.

In this chapter, which is partially based on the treatment of the subject in [100] and [98], we consider both the nucleolus and the pre-nucleolus. The reader is advised to read the relevant part of Chap. 9 first.

In Sect. 19.1 we start with an example illustrating the (pre-)nucleolus and Kohlberg’s balancedness criterion (Kohlberg [65]). Section 19.2 introduces the lexicographic order, on which the definition of the (pre-)nucleolus in Sect. 19.3 is based. Section 19.4 presents the Kohlberg criterion, which is a characterization of the (pre-) nucleolus in terms of balanced collections of coalitions. Computational aspects are discussed in Sect. 19.5, while Sect. 19.6 presents Sobolev’s [127] characterization of the pre-nucleolus based on a reduced game property.