On the Replication of the Pre-kernel and Related Solutions

Based on the results discussed by Meinhardt (The Pre-Kernel as a Tractable Solution for Cooperative Games: An Exercise in Algorithmic Game Theory, volume 45 of Theory and Decision Library: Series C, Springer, Heidelberg, 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 single-valuedness 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. Hence, a bargaining outcome related to this pre-kernel element is stable. 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 consequently continuous. In addition, we provide sufficient conditions that preserve the pre-nucleolus property for related games even when the default game possesses not a single pre-kernel point. Finally, we apply the same techniques to related solutions of the pre-kernel, namely the modiclus, and anti-pre-kernel, to work out replication results for them.