caa a22 4500 477078567 CHVBK 20180405111444.0 cr unu---uuuuu 170330e19960601xx s 000 0 eng 10.1007/BF00118651 doi (NATIONALLICENCE)springer-10.1007/BF00118651 Alternative forms of the Shapley value and the Shapley-Shubik index [Elektronische Daten] [Dan Felsenthal, Moshé Machover] We prove the validity of an alternative representation of the Shapley-Shubik (1954) index of voting power, based on the following model. Voting in an assembly consisting of n voters is conducted by roll-call. Every voter is assumed to vote "yea" or "nay" with equal probability, and all n! possible orders in which the voters may be called are also assumed to be equiprobable. Thus there are altogether 2 n n! distinct roll-call patterns. Given a simple voting game (a decision rule), the 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 longer 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 proceeds by generalizing the representation so that one obtains a value for each player in any coalitional game, which is easily seen to satisfy Shapley's (1953) three axioms. Thus the generalization turns out to be an alternative representation of the Shapley value. This result implies a non-trivial combinatorial theorem. Kluwer Academic Publishers, 1996 Felsenthal Dan Department of Political Science, University of Haifa, 31905, Haifa, Israel aut Machover Moshé Department of Philosophy, King's College London, Strand, WC2R 2LS, London, U.K. aut Public Choice Kluwer Academic Publishers; http://www.wkap.nl 87/3-4(1996-06-01), 315-318 0048-5829 87:3-4<315 1996 87 11127 https://doi.org/10.1007/BF00118651 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/BF00118651 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Felsenthal Dan Department of Political Science, University of Haifa, 31905, Haifa, Israel aut NATIONALLICENCE 700 1- Machover Moshé Department of Philosophy, King's College London, Strand, WC2R 2LS, London, U.K aut NATIONALLICENCE 773 0- Public Choice Kluwer Academic Publishers; http://www.wkap.nl 87/3-4(1996-06-01), 315-318 0048-5829 87:3-4<315 1996 87 11127 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer