Geometric Properties of the Kernel, Nucleolus, and Related Solution Concepts

TLDR

The kernel and nucleolus are studied in relation to other concepts, especially the core, and the prekernel—closely related to the kernel—has a simpler definition and analytic properties. The unifying technical idea is to analyze the behavior of the strong ε‑core as ε varies. A notion of quasi‑zero‑monotonicity is developed to enlarge the class of games where kernel considerations can be replaced by prekernel considerations, the nucleolus is approached through a new geometrical definition equivalent to Schmeidler's but yielding elementary proofs of existence, unicity, and other properties, and the intuitive interpretations of the two solution concepts are clarified: the kernel as a multi‑bilateral bargaining equilibrium without interpersonal utility comparisons, and the nucleolus as the result of an arbitrator's desire to minimize the dissatisfaction of the most dissatisfied coalition. One of the central results is that the portion of the prekernel that falls within the core, or any other strong ε‑core, depends only on the latter's geometrical shape.

Abstract

Two solution concepts for cooperative games in characteristic-function form, the kernel and the nucleolus, are studied in their relationship to a number of other concepts, most notably the core. The unifying technical idea is to analyze the behavior of the strong ϵ-core as ϵ varies. One of the central results is that the portion of the prekernel that falls within the core, or any other strong ϵ-core, depends only on the latter's geometrical shape. The prekernel is closely related to the kernel and often coincides with it, but has a simpler definition and simpler analytic properties. A notion of “quasi-zero-monotonicity” is developed to aid in enlarging the class of games in which kernel considerations can be replaced by prekernel considerations. The nucleolus is approached through a new, geometrical definition, equivalent to Schmeidler's original definition but providing very elementary proofs of existence, unicity, and other properties. Finally, the intuitive interpretations of the two solution concepts are clarified: the kernel as a kind of multi-bilateral bargaining equilibrium without interpersonal utility comparisons, in which each pair of players bisects an interval which is either the battleground over which they can push each other aided by their best allies (if they are strong) or the no-man's-land that lies between them (if they are weak); the nucleolus as the result of an arbitrator's desire to minimize the dissatisfaction of the most dissatisfied coalition.