International Journal of Game Theory

, Volume 1, Issue 1, pp 11–26 | Cite as

Cores of convex games

  • Lloyd S. Shapley
Papers

Abstract

The core of ann-person game is the set of feasible outcomes that cannot be improved upon by any coalition of players. A convex game is defined as one that is based on a convex set function. In this paper it is shown that the core of a convex game is not empty and that it has an especially regular structure. It is further shown that certain other cooperative solution concepts are related in a simple way to the core: The value of a convex game is the center of gravity of the extreme points of the core, and the von Neumann-Morgenstern stable set solution of a convex game is unique and coincides with the core.

Keywords

Economic Theory Game Theory Extreme Point Solution Concept Regular Structure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aumann, R. J., andL. S. Shapley: Values of Non-Atomic Games I, II, III, IV, V. RM-5468, RM-5842, RM-6216, RM-6260, R-843. The Rand Corporation, Santa Monica, California, 1968, 1969, 1970, 1971.Google Scholar
  2. Bondareva, O. N.: Some Applications of the Methods of Linear Programming to the Theory of Cooperative Games. (Russian), Problemy Kibernetiki10, 1963, 119–139; esp. § 4.Google Scholar
  3. Choquet, G.: Theory of Capacities. Annals de l'Institut Fourier5, 1955, 131–295.Google Scholar
  4. Crapo, H. H., andG.-C. Rota: Combinatorial Geometries. Massachusetts Institute of Technology, Cambridge, Massachusetts, 1968.Google Scholar
  5. Edmonds, J.: Submodular Functions, Matroids, and Certain Polyhedra. Combinatorial Structures and Their Applications (proceedings of a conference at the University of Calgary, June 1969), Gordon and Breach, New York, 1970, 69–87.Google Scholar
  6. Edmonds, J., andG.-C. Rota: Submodular Set Functions (abstract, Waterloo Combinatorics Conference). University of Waterloo, Waterloo, Ontario, 1966.Google Scholar
  7. Gillies, D. B.: Some Theorems onn-Person Games (dissertation). Department of Mathematics, Princeton University, 1953.Google Scholar
  8. —: Solutions to General Non-Zero-Sum Games. Annals of Mathematics Study40, 1959, 47–85; esp. 77–81.Google Scholar
  9. Lucas, W. F.: A Counterexample in Game Theory. Management Science13, 1967, 766–767.Google Scholar
  10. —: A Game with no Solution, Bull. Am. Math. Soc.74, 1968, 237–239.Google Scholar
  11. Luce, R. D., andH. Raiffa: Games and Decisions. Wiley and Sons, New York, 1957.Google Scholar
  12. Meyer, P. A.: Probabilities and Potentials. Blaisdell, Waltham, Massachusetts, 1966, esp. 40 ff.Google Scholar
  13. Maschler, M., B. Peleg andL. S. Shapley: The Kernel and Bargaining Set for Convex Games. RM-5372, The Rand Corporation, Santa Monica, California, 1967, esp. 10f.Google Scholar
  14. -: The Kernel and Bargaining Set for Convex Games, (to appear).Google Scholar
  15. Peleg, B.: Composition of General Sum Games. RM-74, Econometric Research Program, Princeton University, 1965.Google Scholar
  16. -: Composition of Kernels of Characteristic Function Games. RM-15, Department of Mathematics, The Hebrew University of Jerusalem, 1965.Google Scholar
  17. Rosenmüller, J.: Some Properties of Convex Set Functions (duplicated). Mathematisches Institut der UniversitÄt Erlangen-Nürnberg, 852 Erlangen, Germany, 1970.Google Scholar
  18. Schmeidler, D.: Cores of Exact Games, I. CP-329, Center for Research in Management Science, University of California, Berkeley, 1971.Google Scholar
  19. Shapley, L. S.: Notes on then-Person Game III: Some Variants of the Von Neumann-Morgenstern Definition of Solution. RM-670, The Rand Corporation, Santa Monica, California, 1951.Google Scholar
  20. -: Open Questions. Theory ofn-Person Games (report of an informal conference), Department of Mathematics, Princeton University, 1953; 15.Google Scholar
  21. —: A Value forn-Person Games. Annals of Mathematics Study28, 1953, 307–317.Google Scholar
  22. —: Notes onn-Person Games VII: Cores of Convex Games. RM-4571, The Rand Corporation, Santa Monica, California, 1965.Google Scholar
  23. Von Neumann, J., and O.Morgenstern: Theory of Games and Economic Behavior. Princeton University Press, 1944.Google Scholar
  24. Whitney, H.: The Abstract Properties of Linear Dependence. Am. J. Math.57, 1935, 509–533; esp. 511.Google Scholar

Copyright information

© Physica-Verlag 1971

Authors and Affiliations

  • Lloyd S. Shapley
    • 1
  1. 1.The Rand CorporationSanta Monica

Personalised recommendations