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Social Choice and Welfare

Erschienen in:

01.08.2022 | Original Paper

On the unique core partition of coalition formation games: correction to İnal (2015)

verfasst von: Satoshi Nakada, Ryo Shirakawa

Erschienen in: Social Choice and Welfare | Ausgabe 3/2023

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Abstract

In this note, we consider sufficient conditions for the uniqueness of the core partitions of coalition formation games. İnal (Soc Choice Welf 45:745–763, 2015) introduces a sufficient condition called k-acyclicity and claims that this condition is independent of another sufficient condition called top-coalition property. We show that this claim is incorrect and, in particular, k-acyclicity is equivalent to the common ranking property introduced by Banerjee et al. (Soc Choice Welf 18:135–153, 2001), which is a stronger condition than the top-coalition property.

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Iehlé (2007) provides a necessary and sufficient condition for the existence of a core partition.

The uniqueness of the core allocations is important not only because the multiplicity of the allocations weakens the predictive power of the model but also because it is a necessary condition to construct a suitable allocation rule in a wide range of applications (Sönmez 1999, 1996; Takamiya 2003).

The common ranking property is firstly introduced by Farrell and Scotchmer (1988) in a more specific model and the top-coalition property is firstly introduced by Alcalde (1994) in the roommate problems, in which the condition is called \(\alpha \)-reducibility. Banerjee et al. (2001) generalize these concepts in the context of hedonic games. Casajus (2008) and Abe (2021) relate hedonic games and cooperative games with coalition structures and provide sufficient conditions for allocation rules of the cooperative games which induce a hedonic game that satisfies the common ranking property and the top-coalition property. For other sufficient conditions to guarantee the uniqueness, see also Pápai (2004) and Pycia (2012).

k-acyclicity is initiated to Echenique and Yenmez (2007) in the context of college admission problems with students’ preferences over both colleges and colleagues.

Other results in İnal (2015) are true.

For other \(V \subseteq N\), \(\{2,3\}\) itself is a top-coalition of \(V=\{2,3\}\). Singleton sets are obvious.

Note that this is not a unique order that is consistent with the common ranking property. The proof of Proposition 1 presents an algorithm that produces this linear order from the given preference profile. This is an application of the algorithms in the proof of Theorems 1 and 2 in Banerjee et al. (2001), each of which finds a core partition assuming the (weak) top-coalition property.

Abe T (2021) Stability and values for games with coalition structures. Econ Lett 200:109750CrossRef

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Banerjee S, Konishi H, Sönmez T (2001) Core in a simple coalition formation game. Soc Choice Welf 18:135–153CrossRef

Bogomolnaia A, Jackson MO (2002) The stability of hedonic coalition structures. Games Econ Behav 38:201–230CrossRef

Casajus A (2008) On the stability of coalition structures. Econ Lett 100:271–274CrossRef

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Gale D, Shapley LS (1962) College admissions and the stability of marriage. Am Math Mon 69:9–15CrossRef

Iehlé V (2007) The core-partition of a hedonic game. Math Soc Sci 54:176–185CrossRef

İnal H (2015) Core of coalition formation games and fixed-point methods. Soc Choice Welf 45:745–763CrossRef

Pápai S (2004) Unique stability in simple coalition formation games. Games Econ Behav 48:337–354CrossRef

Pycia M (2012) Stability and preference alignment in matching and coalition formation. Econometrica 80:323–362CrossRef

Shapley L, Scarf H (1974) On cores and indivisibility. J Math Econ 1:23–37CrossRef

Sönmez T (1996) Implementation in generalized matching problems. J Math Econ 26:429–439CrossRef

Sönmez T (1999) Strategy-proofness and essentially single-valued cores. Econometrica 67:677–689CrossRef

Takamiya K (2003) On strategy-proofness and essentially single-valued cores: a converse result. Soc Choice Welf 20:77–83CrossRef

Titel: On the unique core partition of coalition formation games: correction to İnal (2015)
verfasst von: Satoshi Nakada
Ryo Shirakawa
Publikationsdatum: 01.08.2022
Verlag: Springer Berlin Heidelberg
Erschienen in: Social Choice and Welfare / Ausgabe 3/2023
Print ISSN: 0176-1714
Elektronische ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-022-01423-5

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