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Journal of Applied Mathematics
Volume 2014 (2014), Article ID 190614, 20 pages
http://dx.doi.org/10.1155/2014/190614
Research Article

On the Structure of Cooperative and Competitive Solutions for a Generalized Assignment Game

1Instituto de Matemática Aplicada San Luis (UNSL-CONICET), Ejército de los Andes 950, 5700 San Luis, Argentina
2Departament d'Economia i d'Història Econòmica, Universitat Autònoma de Barcelona and Barcelona GSE, Edifici B, Bellaterra, 08193 Barcelona, Spain

Received 5 December 2013; Revised 14 February 2014; Accepted 15 February 2014; Published 30 April 2014

Academic Editor: Pu-yan Nie

Copyright © 2014 R. Pablo Arribillaga et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study cooperative and competitive solutions for a many-to-many generalization of Shapley and Shubik’s (1971) assignment game. We consider the Core, three other notions of group stability, and two alternative definitions of competitive equilibrium. We show that (i) each group stable set is closely related to the Core of certain games defined using a proper notion of blocking and (ii) each group stable set contains the set of payoff vectors associated with the two definitions of competitive equilibrium. We also show that all six solutions maintain a strictly nested structure. Moreover, each solution can be identified with a set of matrices of (discriminated) prices which indicate how gains from trade are distributed among buyers and sellers. In all cases such matrices arise as solutions of a system of linear inequalities. Hence, all six solutions have the same properties from a structural and computational point of view.