On the uniqueness of the Shapley value

L.S. Shapley [1953] showed that there is a unique value defined on the class D of all superadditive cooperative games in characteristic function form (over a finite player set N) which satisfies certain intuitively plausible axioms. Moreover, he raised the question whether an axiomatic foundation could be obtained for a value (not necessarily the Shapley value) in the context of the subclass C (respectively C′, C″) of simple (respectively simple monotonic, simple superadditive) games alone. This paper shows that it is possible to do this. Theorem I gives a new simple proof of Shapley's theorem for the class G of all games (not necessarily superadditive) over N. The proof contains a procedure for showing that the axioms also uniquely specify the Shapley value when they are restricted to certain subclasses of G, e.g., C. In addition it provides insight into Shapley's theorem for D itself. Restricted to C′ or C″, Shapley's axioms do not 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 the Shapley value again.