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International Journal of Game Theory

, Volume 4, Issue 3, pp 131–139 | Cite as

On the uniqueness of the Shapley value

  • P. Dubey
Papers

Abstract

L.S. Shapley [1953] showed that there is a unique value defined on the classD of all superadditive cooperative games in characteristic function form (over a finite player setN) which satisfies certain intuitively plausible axioms. Moreover, he raised the question whether an axiomatic foundation could be obtained for a value (not necessarily theShapley value) in the context of the subclassC (respectivelyC′, C″) of simple (respectively simple monotonic, simple superadditive) gamesalone. This paper shows that it is possible to do this.

Theorem I gives a new simple proof ofShapley's theorem for the classG ofall games (not necessarily superadditive) overN. The proof contains a procedure for showing that the axioms also uniquely specify theShapley value when they are restricted to certain subclasses ofG, e.g.,C. In addition it provides insight intoShapley's theorem forD itself.

Restricted toC′ orC″, Shapley's axioms donot specify a unique value. However it is shown in theorem II that, with a reasonable variant of one of his axioms, a unique value is obtained and, fortunately, it is just theShapley value again.

Keywords

Characteristic Function Economic Theory Game Theory Function Form Simple Proof 
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.

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References

  1. Lucas, W.F.: Measuring Power in Weighted Voting Systems, Cornell University, Ithaca, N.Y. 14850, October 1973, (Draft).Google Scholar
  2. Shapley, L.S.: A Value forn-person Games, Annals of Mathematics Study No. 28, Princeton University Press, Princeton, N.J., 1953, pp. 307–317.Google Scholar
  3. Spinneto, R.D.: Solution Concepts ofn-person Cooperative Games as Points in the Game Space, Technical Report No.138, Department of Operations Research, College of Engineering, Cornell University, Ithaca, N.Y. 14850, 1971.Google Scholar

Copyright information

© Physica-Verlag Rudolf Liebing KG 1975

Authors and Affiliations

  • P. Dubey
    • 1
  1. 1.Cornell UniversityIthaca

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