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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Notes
Iehlé (2007) provides a necessary and sufficient condition for the existence of a core partition.
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.
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Acknowledgements
We thank an associate editor and anonymous referees for providing us with valuable comments. We also thank the participants of SING 15 and Summer Workshop on Game Theory 2019 at Sendai for helpful comments. Nakada and Shirakawa acknowledge the financial support from Japan Society for the Promotion of Science KAKENHI: Grant Number 19K13651 (Nakada) and 22J20710 (Sirakawa). All remaining errors are our own.
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Nakada, S., Shirakawa, R. On the unique core partition of coalition formation games: correction to İnal (2015). Soc Choice Welf 60, 517–521 (2023). https://doi.org/10.1007/s00355-022-01423-5
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DOI: https://doi.org/10.1007/s00355-022-01423-5