The Assignment Game I

The Core

by Lloyd S. Shapley, Martin Shubik

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The assignment game is a model for a two-sided market in which a product that comes in large, indivisible units is exchanged for money, and in which each participant either supplies or demands exactly one unit. The units need not be alike, and the same unit may have different values to different participants. It is shown that the outcomes in the core of such a game are the solutions of a certain linear programming problem dual to the optimal assignment problem; these outcomes correspond exactly to the price lists that competitively balance supply and demand. The geometric structure of the core is described and interpreted in economic terms, with explicit attention given to the special case in which there is no product differentiation. A critique of the core solution reveals an insensitivity to some of the bargaining possibilities inherent in the situation, and indicates that other game-theoretic solution concepts should be investigated.

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