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Barcelona GSE Working Paper No. 740

Title: On the Structure of Cooperative and Competitive Solutions for a Generalized Assignment Game
Authors: Pablo Arribillaga, Jordi Massó, Alejandro Neme
Date: 07-10-2013
Keywords: assignment game, competitive equilibrium, core, group stability
JEL Codes: C78, D78
We study cooperative and competitive solutions for a many- to-many generalization of Shapley and Shubik (1972)’s 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 with 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 to 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.
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