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The assignment game. I: The core. (English) Zbl 0236.90078

Summary: The assignment game is a model for a two-sided market in which a product that comes in large, indivisible units (e.g., houses, cars, etc.) 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 here that the outcomes in the core of such a game – i.e., those that cannot be improved upon by any subset of players – are the solutions of a certain linear programming problem dual to the optimal assignment problem, and that these outcomes correspond exactly to the price-lists that competitively balance supply and demand. The geometric structure of the core is then described and interpreted in economic terms, with explicit attention given to the special case (familiar in the classic literature) in which there is no product differentiation – i.e., in which the units are interchangeable. Finally, a critique of the core solution reveals an insensitivity to some of the bargaining possibilities inherent in the situation, and indicates that further analysis would be desirable using other game-theoretic solution concepts.

MSC:

91A40 Other game-theoretic models
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References:

[1] Böhm, V.: The Continuity of the Core. University of Bonn, February 1972.
[2] Böhm-Bawerk, E. von: Positive Theory of Capital (translated by William Smart). G. E. Steckert, New York, 1923 (original publication 1891). · JFM 22.0241.01
[3] Cournot, A. A.: Researches into the Mathematical Principles of the Theory of Wealth (translated by N. T. Bacon). Macmillan and Co., New York, 1897 (original publication 1838). · JFM 28.0211.07
[4] Dantzig, G. B.: Linear Programming and Extensions. Princeton University Press, Princeton, 1963. · Zbl 0108.33103
[5] Debreu, G. andH. Scarf: A limit theorem on the core of an economy. Int. Econ. Rev.4, 235–246, 1963. · Zbl 0122.37702 · doi:10.2307/2525306
[6] Edgeworth, F. Y.: Mathematical Psychics. Kegan Paul, London, 1881. · Zbl 0005.17402
[7] Gale, D.: The Theory of Linear Economic Models. McGraw Hill, New York, 1960. · Zbl 0114.12203
[8] Gale, D. andL. S. Shapley: College admission and the stability of marriage. Amer. Math. Monthly69, 9–15, 1962. · Zbl 0109.24403 · doi:10.2307/2312726
[9] Henry, C.: Indivisibilités dans une économie d’echanges. Econometrica38, 542–558, 1970. · Zbl 0211.23002 · doi:10.2307/1909559
[10] Shapley, L. S.: Markets as Cooperative Games. The Rand Corporation, P-629, March 1955.
[11] –: The solutions of a symmetric market game. Annals of Mathematics Study40, 145–162, 1959. · Zbl 0085.13808
[12] -: Values of Large Games V: An 18-Person Market Game, The Rand Corporation, RM-2860, November 1961.
[13] –: Complements and substitutes in the optimal assignment problem. Nav. Res. Log. Q.9, 45–48, 1962. · doi:10.1002/nav.3800090106
[14] Shapley, L. S. andM. Shubik: Quasi-cores in a monetary economy with nonconvex preferences. Econometrica34, 805–827, 1966. · Zbl 0154.45303 · doi:10.2307/1910101
[15] –: Pure competition, coalitional power, and fair division. Int. Econ. Rev.10, 337–362, 1969. · doi:10.2307/2525647
[16] -: The kernels and bargaining sets of market games. Without year (forthcoming).
[17] Shitovitz, B.: Oligopoly in Markets with a Continuum of Traders. The Hebrew University, Depart- ment of Mathematics, RM-63, August 1970.
[18] Shubik, M.: Edgeworth market games. Annals of Mathematics Study40, 267–278, 1959. · Zbl 0084.36403
[19] von Neumann, J. andO. Morgenstern: Theory of Games and Economic Behavior. Princeton University Press, Princeton, 1944. · Zbl 0063.05930
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