Chapter 18 The bargaining set, kernel, and nucleolus

Abstract

Publisher Summary This chapter focuses on the bargaining set, kernel, and nucleolus. The theory of the bargaining set answers a more modest question: How would or should the players share the proceeds, given that a certain coalition structure (c.s.) has formed? From a normative point of view, the reason for asking such a question stems from the need to let the players know what to expect from each coalition structure so that they can then make up their mind about the coalitions they want to join, and in what configuration. The kernel was introduced as an auxiliary solution concept, the main task of which was to illuminate properties of the bargaining set and to compute at least part of this set. Kernel had many interesting mathematical properties that reflected in various ways the structure of the game. Kernel [prekernel] is covariant with respect to strategic equivalence. Both are finite unions of polytopes. It is almost as difficult to compute the kernel as to compute the bargaining set. Being a point in the kernel, the nucleolus point has all the nice properties of the kernel points. Becauseit is a solution concept that does not depend on the names of the players, it preserves all the symmetries of the game. The nucleolus has an advantage over the Shapley value. Because the nucleolus is not empty even if the core is empty, it can be stated that the nucleolus is the location of the latent position of the core. The chapter also discusses the axiomatic foundation of the prekernel and the prenucleolus, ideas embodied in the bargaining set, kernel, and nucleolus that spawned many other related solution concepts, and dynamic processes that lead the participants in a cooperative game to reach the bargaining set— or the kernel, or the nucleolus, or many other bargaining sets— via a sequence of steps that make good intuitive sense.