Abstract
This paper considers von Neumann-Morgenstern (vNM) stable sets in marriage games. Ehlers (Journal of Economic Theory 134: 537–547, 2007) showed that if a vNM stable set exists in a marriage game, the set is a maximal distributive lattice of matchings that includes all core matchings. To determine what matchings form a vNM stable set, we propose a polynomial-time algorithm that finds a man-optimal matching and a woman-optimal matching in a vNM stable set of a given marriage game. This algorithm also generates a modified preference profile such that a vNM stable set is obtained as the core of a marriage game with the modified preference profile. It is well known that cores of marriage games are nonempty. However, the nonemptiness of cores does not imply the existence of a vNM stable set. It is proved using our algorithm that there exists a unique vNM stable set for any marriage game.
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The author thanks for very helpful comments from Lars Ehlers, Flip Klijn, Shigeo Muto, Alvin Roth, Marilda Sotomayor, Akihisa Tamura, and an anonymous reviewer and David Manlove of ICALP 2008 workshop ‘Matching Under Preferences’. He also thanks Hisao Endo for a discussion on the medical matching in Japan, and Onur Kesten for permitting the author to quote his example.
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Wako, J. A Polynomial-Time Algorithm to Find von Neumann-Morgenstern Stable Matchings in Marriage Games. Algorithmica 58, 188–220 (2010). https://doi.org/10.1007/s00453-010-9388-y
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DOI: https://doi.org/10.1007/s00453-010-9388-y