ALTERNATIVE FORMS OF THE SHAPLEY VALUE AND THE SHAPLEY-SHUBIK INDEX

We prove the validity of an alternative representation of the Shapley- Shubik (1954) index of voting power, based on the following model. Vot ing in an assembly consisting of n voters is conducted by roll-call. E very voter is assumed to vote ''yea'' or ''nay'' with equal probabilit y, and all n! possible orders in which the voters may be called are al so assumed to be equiprobable. Thus there are altogether 2(n)n! distin ct roll-call patterns. Given a simple voting game (a decision rule), t he pivotal voter in a roll-call is the one whose vote finally decides the outcome, so that the votes of all those called subsequently no lon ger make any difference. The main result, stated without proof by Mann and Shapley (1964), is that the Shapley-Shubik index of voter a in a simple voting game is equal to the probability of a being pivotal. We believe this representation of the index is much less artificial than the original one, which considers only the n! roll-calls in which all voters say ''yea'' (or all say ''nay''). The proof of this result proc eeds by generalizing the representation so that one obtains a value fo r each player in any coalitional game, which is easily seen to satisfy Shapley's (1953) three axioms. Thus the generalization turns out to b e an alternative representation of the Shapley value. This result impl ies a non-trivial combinatorial theorem.