On the structure of cooperative and competitive solutions for a generalized assignment game

The Spanish Ministry of Economy and Competitiveness, through the Severo Ochoa Programme for Centres of Excellence in R&D (SEV-2011-0075) and FEDER grant ECO2008-04756 (Grupo Consolidado-C), and from the Generalitat de Catalunya, through the prize "ICREA Academia" for excellence in research and grants SGR2009-419 and SGR2014-515We 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