The Pre-Kernel as a Tractable Solution for Cooperative Games

Meinhardt, Holger Ingmar

doi:10.1007/978-3-642-39549-9

Cited by 3 publications

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“…Proposition 3.1 (Meseguer-Artola (1997); Meinhardt (2013)). For a TU game with indirect function π, a pre-imputation x ∈ I 0 (v) is in the pre-kernel of N, v for the coalition structure B = {B 1 , .…”

Section: A Dual Pre-kernel Representationmentioning

confidence: 99%

“…In order to advance our understanding about TU games and game classes which possess an unique pre-kernel element, we propose an alternative approach to investigate this issue while applying results and techniques recently provided in the book by Meinhardt (2013). There, it was shown that the pre-kernel of the grand coalition can be characterized by a finite union of solution sets of a family of quadratic and convex functions (Theorem 7.3.1).…”

Section: Introductionmentioning

confidence: 99%

“…In the next step, the notion of lexicographically smallest most effective coalitions is introduced in order to identify payoff equivalence classes on the domain of the objective function from which a pre-kernel element can be determined. Moreover, relevant concepts from Meinhardt (2013) are reconsidered. Section 4 studies the uniqueness of the pre-kernel for related games.…”

Section: Introductionmentioning

confidence: 99%

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On the Single-Valuedness of the Pre-Kernel

Meinhardt¹

2015

Preprint

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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.

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Section: A Dual Pre-kernel Representationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…

…”

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confidence: 99%

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On the Single-Valuedness of the Pre-Kernel

Meinhardt¹

2015

Preprint

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“…We remark that a (5 + 5)-player assignment game with a non-convex kernel was given by Granot and Granot (1992). The challenging task of computing the (pre)kernel in a general TU game is the topic of the recent book by Meinhardt (2014).…”

Section: Introductionmentioning

confidence: 99%

The kernel is in the least core for permutation games

Solymosi

2014

Cent Eur J Oper Res

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Permutation games are totally balanced transferable utility cooperative games arising from certain sequencing and re-assignment optimization problems. It is known that for permutation games the bargaining set and the core coincide, consequently, the kernel is a subset of the core. We prove that for permutation games the kernel is contained in the least core, even if the latter is a lower dimensional subset of the core. By means of a 5-player permutation game we demonstrate that, in sense of the lexicographic center procedure leading to the nucleolus, this inclusion result can not be strengthened. Our 5-player permutation game is also an example (of minimum size) for a game with a non-convex kernel.JEL classification code: C71. AMS 2010 classification. Primary: 91A12; Secondary: 91A40.

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On the Replication of the Pre-kernel and Related Solutions

Meinhardt

2023

Comput Econ

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Based on results discussed by Meinhardt (2013), which presents a dual characterization of the prekernel 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 therefore continuous. In addition, 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. 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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The Pre-Kernel as a Tractable Solution for Cooperative Games

Cited by 3 publications

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On the Single-Valuedness of the Pre-Kernel

On the Single-Valuedness of the Pre-Kernel

The kernel is in the least core for permutation games

On the Replication of the Pre-kernel and Related Solutions

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