The Nucleolus, the Kernel, and the Bargaining Set: An Update

Iñarra, Elena; Serrano, Roberto; Shimomura, Ken-Ichi

doi:10.3917/reco.712.0225

Cited by 8 publications

(5 citation statements)

References 113 publications

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“…In cooperative game theory, some solutions describe stable compromises in noncooperative situations among coalitions (for surveys, see Mas‐Colell, 1987; Vohra, 1997). Although there have been several studies of the bargaining set of NTU games (see Iñarra et al, 2020, p. 260), little research has been done on the bargaining set of NTU games with coalition structures. Our model, as well as Zhou's (1994), describes coalition formation as an outcome of bargaining.…”

Section: Discussionmentioning

confidence: 99%

“…Although there have been several studies of the bargaining set of NTU games (see Iñarra et al . , 2020, p. 260), little research has been done on the bargaining set of NTU games with coalition structures. Our model, as well as Zhou's (1994), describes coalition formation as an outcome of bargaining.…”

Section: Discussionmentioning

confidence: 99%

“…Since the introduction of the bargaining set by Aumann and Maschler (1964), no intersection requirement has appeared in the definitions of alternative versions of the bargaining set, including

italicMB

(for details, see Zhou, 1994, sections 2 and 3; Iñarra et al . , 2020, pp. 254–258).…”

Section: The Zhou Bargaining Set and The Steady Bargaining Setmentioning

confidence: 99%

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The bargaining set and coalition formation

Shimomura

2021

Int J of Economic Theory

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We address the problem of predicting how rational agents will form coalitions in a nontransferable utility game, and within each coalition how they will allocate the gains obtained through cooperation. To answer these questions, we propose solution concepts according to which the coalition structure and the payoff allocations are simultaneously determined. We prove the nonemptiness and partial efficiency of the steady bargaining set, a refinement of the Zhou bargaining set, for at least one coalition structure under the restrictive non-crossing condition. In addition, we show the nonemptiness and possible inefficiency of the Mas-Colell bargaining set if this condition is not assumed.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

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The bargaining set and coalition formation

Shimomura

2021

Int J of Economic Theory

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“…On the other hand, Iñarra et al [15] provided an update on the bargaining set, the kernel and the nucleolus, respectively, by investigating consistency, non-emptiness, and uniqueness of the nucleolus. Inspired by Iñarra et al [15], it is reasonable that some more different solutions and related axiomatic results could be extended to multi-choice NTU games, such as the bargaining set, the kernel, the nucleolus, and so on. This is left to the readers.…”

Section: Discussionmentioning

confidence: 99%

A Solution Concept and Its Axiomatic Results under Non-Transferable-Utility and Multi-Choice Situations

Hwang

Liao

2020

Mathematics

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In real situations, agents might take different energy levels to participate. On the other hand, agents always face an increasing need to focus on non-transferable-utility situations efficiently in their operational processes. Thus, we introduce the replicated core under non-transferable-utility situations, and analyze non-emptiness of the replicated core by means of a balanced result. In order to express the rationality of the replicated core, we also define different reduced games to axiomatize the replicated core.

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“…However, for the class of average-convex games, there is only some evidence that both solution concepts coalesce. For getting an overview of the recent developments in this field, we refer the inclined reader to Iñarra et al (2020).…”

Section: Introductionmentioning

confidence: 99%

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 Nucleolus, the Kernel, and the Bargaining Set: An Update

Abstract: One of the many important contributions in David Schmeidler's distinguished career was the introduction of the nucleolus. This paper is an update on the nucleolus and its two related supersolutions, i.e., the kernel and the bargaining set. JEL classification numbers: C71, C72.

Cited by 8 publications

References 113 publications

The bargaining set and coalition formation

The bargaining set and coalition formation

A Solution Concept and Its Axiomatic Results under Non-Transferable-Utility and Multi-Choice Situations

On the Replication of the Pre-kernel and Related Solutions

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