On cooperative solutions of a generalized assignment game: Limit theorems to the set of competitive equilibria

We study two cooperative solutions of a market with indivisible goods modeled as a generalized assignment game: Set-wise stability and Core. We …rst establish that the Set-wise stable set is contained in the Core and it contains the non-empty set of competitive equilibrium payo¤s. We then state and prove three limit results for replicated markets. First, the sequence of Cores of replicated markets converges to the set of competitive equilibrium payo¤s when the number of replicas tends to in…nity. Second, the Set-wise stable set of a two-fold replicated market already coincides with the set of competitive equilibrium payo¤s. Third, for any number of replicas there is a market with a Core payo¤ that is not a competitive equilibrium payo¤. Support for the research of J. Massó was received through the prize “ICREA Acadèmia” for excellence in research, funded by the Generalitat de Catalunya. He also acknowledges the support of MOVE, where he is an a¢ liated researcher, and of the Barcelona Graduate School of Economics (through its Research Recognition Programme), where he is an a¢ liated professor. His work is also supported by the Spanish Ministry of Science and Innovation, through grants ECO2008-04756 (Grupo Consolidado-C) and CONSOLIDER-INGENIO 2010 (CDS2006-00016), and by the Generalitat de Catalunya, through grant SGR2009-419 and a BE-2008 fellowship to visit the Instituto de Matemática Aplicada de San Luis (Argentina). The work of A. 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. yDepartament d’Economia i d’Història Econòmica and CODE. Universitat Autònoma de Barcelona. 08193, Bellaterra (Barcelona), Spain. E-mail: [email protected] zInstituto de Matemática Aplicada de San Luis. Universidad Nacional de San Luis and CONICET. Ejército de los Andes 950. 5700, San Luis, Argentina. E-mail: [email protected] Journal of Economic Literature Classi…cation Numbers: C78; D78.