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.
Highlights
Gale and Shapley [1] introduce ordinal two-sided matching models to study assignment problems between two disjoint sets of agents
We study cooperative and competitive solutions for a many-to-many generalization of Shapley and Shubik’s (1971) assignment game
It is assumed that each agent has strict ordinal preferences over the set of agents that he does not belong to plus the prospect of remaining unmatched
Summary
Gale and Shapley [1] introduce ordinal two-sided matching models to study assignment problems between two disjoint sets of agents. We show that each one of the sets of payoffs corresponding to the three group stability notions can be directly identified with the union of Cores of particular cooperative games with transferable utility, where the blocking power of coalitions is inherited from the corresponding nature of the sale contracts between buyers and sellers (unit-by-unit, goodby-good, or global). Second and using this identification, we show that the three notions of group stability are supported by a Cartesian product structure between a given set of matrices of prices and the set of optimal assignments; all payoff vectors in any of the sets corresponding to the three group stability notions are fully identified by a set of matrices of prices; all payoff vectors in any of the sets corresponding to the three group stability notions are completely identified with the solutions of a system of bounded linear inequalities. The Appendices include the proofs of three results omitted in the main text
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