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

We study cooperative and competitive solutions for a manyto-many generalization of Shapley and Shubik [9]’s assignment game. We consider the Core, three other notions of group stability and two alternative de…nitions of competitive equilibrium. We show that (i) each group stable set is closely related with the Core of certain games de…ned using a proper notion of blocking and (ii) each group stable set contains the set of payo¤vectors associated to the two de…nitions of competitive equilibrium. We also show that all six solutions maintain a strictly nested structure. Moreover, each solution can be identi…ed 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. Keywords: Assignment Game; Competitive Equilibrium; Core; Group Stability. Journal of Economic Literature Classi…cation Numbers: C78; D78. The work of Arribillaga and Neme is partially supported by the Universidad Nacional de San Luis, through grant 319502, and by the Consejo Nacional de Investigaciones Cientí…cas y Técnicas (CONICET), through grant PIP 112-200801-00655. Massó acknowledges …nancial support from the Spanish Ministry of Economy and Competitiveness, through the Severo Ochoa Programme for Centers of Excellence in R&D (SEV-2011-0075) and through grant ECO2008-0475-FEDER (Grupo Consolidado-C), and from the Generalitat de Catalunya, through the prize “ICREA Academia”for excellence in research and grant SGR2009-419. yInstituto de Matemática Aplicada San Luis (UNSL-CONICET). Ejército de los Andes 950. 5700 San Luis, Argentina. E-mails: [email protected], [email protected] zUniversitat Autònoma de Barcelona and Barcelona GSE. Departament d’Economia i d’Història Econòmica. Edi…ci B, UAB. 08193, Bellaterra (Barcelona), Spain. E-mail: [email protected]