International Journal of Game Theory

, Volume 1, Issue 1, pp 111–130 | Cite as

The assignment game I: The core

  • L. S. Shapley
  • M. Shubik
Papers

Abstract

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 thecore 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.

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References

  1. Böhm, V.: The Continuity of the Core. University of Bonn, February 1972.Google Scholar
  2. Böhm-Bawerk, E. von: Positive Theory of Capital (translated by William Smart). G. E. Steckert, New York, 1923 (original publication 1891).Google Scholar
  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).Google Scholar
  4. Dantzig, G. B.: Linear Programming and Extensions. Princeton University Press, Princeton, 1963.Google Scholar
  5. Debreu, G. andH. Scarf: A limit theorem on the core of an economy. Int. Econ. Rev.4, 235–246, 1963.Google Scholar
  6. Edgeworth, F. Y.: Mathematical Psychics. Kegan Paul, London, 1881.Google Scholar
  7. Gale, D.: The Theory of Linear Economic Models. McGraw Hill, New York, 1960.Google Scholar
  8. Gale, D. andL. S. Shapley: College admission and the stability of marriage. Amer. Math. Monthly69, 9–15, 1962.Google Scholar
  9. Henry, C.: Indivisibilités dans une économie d'echanges. Econometrica38, 542–558, 1970.Google Scholar
  10. Shapley, L. S.: Markets as Cooperative Games. The Rand Corporation, P-629, March 1955.Google Scholar
  11. —: The solutions of a symmetric market game. Annals of Mathematics Study40, 145–162, 1959.Google Scholar
  12. -: Values of Large Games V: An 18-Person Market Game, The Rand Corporation, RM-2860, November 1961.Google Scholar
  13. —: Complements and substitutes in the optimal assignment problem. Nav. Res. Log. Q.9, 45–48, 1962.Google Scholar
  14. Shapley, L. S. andM. Shubik: Quasi-cores in a monetary economy with nonconvex preferences. Econometrica34, 805–827, 1966.Google Scholar
  15. —: Pure competition, coalitional power, and fair division. Int. Econ. Rev.10, 337–362, 1969.Google Scholar
  16. -: The kernels and bargaining sets of market games. Without year (forthcoming).Google Scholar
  17. Shitovitz, B.: Oligopoly in Markets with a Continuum of Traders. The Hebrew University, Depart- ment of Mathematics, RM-63, August 1970.Google Scholar
  18. Shubik, M.: Edgeworth market games. Annals of Mathematics Study40, 267–278, 1959.Google Scholar
  19. von Neumann, J. andO. Morgenstern: Theory of Games and Economic Behavior. Princeton University Press, Princeton, 1944.Google Scholar

Copyright information

© Physica-Verlag 1971

Authors and Affiliations

  • L. S. Shapley
    • 1
  • M. Shubik
    • 2
  1. 1.The Rand CorporationSanta MonicaUSA
  2. 2.Yale UniversityNew Haven

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