Abstract
The Banzhaf value (like the Shapley index) is a combinatorial vector function c(w) of a vector w. The difficulty of finding optimal w* satisfying the Court’s one-man one-vote mandate is mitigated by the observation that wc(w)≤l defines a bounded “semi-infinite” polyhedron. We describe some pertinent properties of c(w) relative to the polyhedron and an algorithm of successive LP’s which constructs a cone with vertex w* containing the polyhedron.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Banzhaf, J.F., III: Weighted Voting Doesn’t Work: A Mathematical Analysis. Rutgers Law Review 19 (1965).
Baker VS. Carr: 369 U.S. 186 (1962).
Dobish VS. Board of Supervisors of Wayne County: 279 NYS 2d 565 (1967).
Dubey, P. and L.S. Shapley: Mathematical Properties of the Banzhaf Power Index. Math, of Operations Research 4 (1979), 99–131.
Iannucci VS. Board of Supervisors of Washington County: 20 N.Y. 2d 244 (1967).
Imrie, R.W.: The Impact of the Weighted Vote on Representation in Municipal Governing Bodies of New York State. Annals of the N.Y. Academy of Sciences 219 (1973), 192–199.
Papayanopoulos, L.: Computerized Weighted Voting Reapportionment AFIPS Proceedings 50 (1981), 623–629.
Papayanopoulos, L.: Power Indices and Combinatorial Frequency Functions. Rutgers Graduate School of Management Report, 1981.
Papayanopoulos, L.: Quantitative Principles Underlying Apportionment Methods. Annals of the N.Y. Academy of Sciences 219 (1973), 181–191.
Reynolds VS. SIMS 377 U.S. 533 (1964).
Shapley, L.S. and M. Shubik: A Method of Evaluating the Distribution of Power in a Committee System. Annals of Political Sciences Review 48, (1954), 787.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1983 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Papayanopoulos, L. (1983). On the Partial Construction of the Semi-Infinite Banzhaf Polyhedron. In: Fiacco, A.V., Kortanek, K.O. (eds) Semi-Infinite Programming and Applications. Lecture Notes in Economics and Mathematical Systems, vol 215. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-46477-5_14
Download citation
DOI: https://doi.org/10.1007/978-3-642-46477-5_14
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-12304-0
Online ISBN: 978-3-642-46477-5
eBook Packages: Springer Book Archive