On the Single-Valuedness of the Pre-Kernel

TL;DR

This paper investigates conditions under which the pre-kernel of a cooperative game is unique, using a novel approach that involves linear mappings and game class relations, extending understanding of pre-kernel single-valuedness.

Contribution

It introduces a new method to establish the uniqueness of the pre-kernel by analyzing related game classes and linear transformations, without relying solely on existing game class extensions.

Findings

Pre-kernel singleton property is preserved under related game transformations.

The pre-kernel correspondence is shown to be single-valued and continuous on a convex hull subset.

Sufficient conditions are provided for the pre-nucleolus property to hold in related games.

Abstract

Based on results given in the recent book by Meinhardt (2013), which presents a dual characterization of the pre-kernel by a finite union of solution sets of a family of quadratic and convex objective functions, we could derive some results related to the uniqueness of the pre-kernel. Rather than extending the knowledge of game classes for which the pre-kernel consists of a single point, we apply a different approach. We select a game from an arbitrary game class with a single pre-kernel element satisfying the non-empty interior condition of a payoff equivalence class, and then establish that the set of related and linear independent games which are derived from this pre-kernel point of the default game replicates this point also as its sole pre-kernel element. In the proof we apply results and techniques employed in the above work. Namely, we prove in a first step that the linear mapping of a pre-kernel element into a specific vector subspace of balanced excesses is a singleton. Secondly, that there cannot exist a different and non-transversal vector subspace of balanced excesses in which a linear transformation of a pre-kernel element can be mapped. Furthermore, we establish that on the restricted subset on the game space that is constituted by the convex hull of the default and the set of related games, the pre-kernel correspondence is single-valued, and therefore continuous. Finally, we provide sufficient conditions that preserve the pre-nucleolus property for related games even when the default game has not a single pre-kernel point.