Abstract
In a many-to-one matching model with responsive preferences in which indifferences are allowed, we study three notions of core, three notions of stability, and their relationships. We show that (i) the core contains the stable set, (ii) the strong core coincides with the strongly stable set, and (iii) the super core coincides with the super stable set. We also show how the core and the strong core in markets with indifferences relate to the stable matchings of their associated tie-breaking strict markets.
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Notes
Irving (1994) refers to stable matchings as weakly stable matchings.
“\(R_{f_1}:: w_1, w_4, [w_2, w_3]\)” indicates that \(w_1 P_{f_1} w_4 P_{f_1} w_2 I_{f_1} w_3.\)
Otherwise, if \(w'\in \mu '(f)\) is such that \(\emptyset P_f w'\) we have, by responsiveness, \(\mu '(f){\setminus } \{w'\}P_f \mu '(f)\). Let \(\mu ''(f)=\mu '(f){\setminus } \{w'\}\) and \(\mu ''(a)=\mu '(a)\) for each \(a\in C {\setminus } \{f,w'\}.\) Then, \(\mu ''\) dominates \(\mu\) via \(C{\setminus }\{w'\}.\)
This result also holds for strict preferences (See p. 167 of Roth and Sotomayor 1990, for more details).
By \({\mathcal {X}}^c\) we denote the complement of \({\mathcal {X}}.\)
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We acknowledge financial support from UNSL through Grants 032016 and 030320, from Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET) through Grant PIP 112-200801-00655, and from Agencia Nacional de Promoción Científica y Tecnológica through Grant PICT 2017-2355.
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Bonifacio, A.G., Juarez, N., Neme, P. et al. Core and stability notions in many-to-one matching markets with indifferences. Int J Game Theory 53, 143–157 (2024). https://doi.org/10.1007/s00182-023-00865-0
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DOI: https://doi.org/10.1007/s00182-023-00865-0