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.
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Shapley, L.S.: A Value forn-person Games, Annals of Mathematics Study No. 28, Princeton University Press, Princeton, N.J., 1953, pp. 307–317.
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.
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Notation: For a setS we denote by ¦S ¦ the number of elements thatS contains and frequently write it ass; similarlyt abbreviates ¦T ¦ for a setT, etc. 2S denotes the class of all subsets of the setS. Ø stands for the empty set.R, as usual, represents the real line and Z+ the set of positive integers. For a vectorv inR n, vi is the ith component ofv. The symboli is used both as a number and as the name of a player inN, but its meaning will be clear from the context.
This research was supported in part by the Office of Naval Research under contract N00014-67-A-0077-0014, task NR 047-094 and by the National Science Foundation under grant GP-32314 X in the Department of Operations Research in Cornell University.
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Dubey, P. On the uniqueness of the Shapley value. Int J Game Theory 4, 131–139 (1975). https://doi.org/10.1007/BF01780630
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DOI: https://doi.org/10.1007/BF01780630