Abstract
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.
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Notes
For an overview of the most recent developments in this highly dynamic research field we refer the reader to Algaba et al. (2020, Chap. 6. and 7.). Even there, the application of the theory of linear algebraic groups reveals to us that the Borel-groups (minimal parabolic groups) are acting on the bases of TU games, and the Shapley value remains stable whenever the change of basis is located in the same orbit. In the same vein Hernández-Lamoneda et al. (2007) were able to compute a decomposition for the space of cooperative games under the action of the symmetric group \(\text {Sym}(N)\) to identify all irreducible subspaces that are relevant to study symmetric linear solutions, this result was extended by Hernández-Lamoneda et al. (2009) for games in partition function form.
In fact, they computed with their method the kernel of all weighted majority games with 5 and fewer players as well as all extreme zero-sum games with 5 players.
In the sense of Aumann (1961).
In the sense of Hart and Kurz (1983).
Again, the figure has been generated with out Mathematica Package TuGames implemented within Meinhardt (2023a).
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Meinhardt, H.I. On the Replication of the Pre-kernel and Related Solutions. Comput Econ (2023). https://doi.org/10.1007/s10614-023-10428-w
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DOI: https://doi.org/10.1007/s10614-023-10428-w
Keywords
- Transferable utility game
- Pre-kernel
- Pre-nucleolus
- Anti-pre-nucleolus
- Modiclus
- Computational methods
- Uniqueness of the pre-kernel
- Convex analysis
- Fenchel-moreau conjugation
- Indirect function
- Stability analysis