The Banzhaf power index for political games

doi:10.1016/0165-4896(82)91084-8

Mathematical Social Sciences

Volume 2, Issue 3, April 1982, Pages 299-315

https://doi.org/10.1016/0165-4896(82)91084-8 Get rights and content

Abstract

The Banzhaf index of a voting game is a measure of a priori power of the voters. The model on which the index is based treats the voters symmetrically, i.e. the ideology, outlook, etc., of the voters influencing their voting behavior is ignored. Here we present a nonsymmetric generalization of the Banzhaf index in which the ideology of the voters affecting their voting behavior is taken into account. A model of ideologies and issues is presented. The conditions under which our model gives the Shapley-Shubik index (another index of a priori power of the voters) are given. Finally several examples are presented and some qualitative results are given for straight majority and pure bargaining games.

References (12)

J.F. Banzhaf
Weighted voting doesn't work: A mathematical analysis
Rutgers Law Rev.
(1965)
J.F. Banzhaf
Multi-member electoral districts - Do they violate the ‘One Man, One Vote’ principle?
Yale Law J.
(1966)
J.F. Banzhaf
One man, 3.312 votes: A mathematical analysis of the electoral college
Villanova Law Rev.
(1968)
J.F. Banzhaf
One man,? votes: Mathematical analysis of political consequences and judicial choices
George Washington Law Rev.
(1968)
W.F. Lucas
Measuring power in weighted voting systems
J. Nagel
The Descriptive Analysis of Power
(1975)

There are more references available in the full text version of this article.

Cited by (12)

Probabilistic Owen-Shapley spatial power indices
2022, Games and Economic Behavior
In this paper we study probabilistic Owen-Shapley spatial power indices, which are generalizations of the Owen-Shapley spatial power index (1977). We provide an explicit formula for calculating these spatial indices for unanimity games and give an axiomatic characterization of the family of probabilistic Owen-Shapley spatial power indices. We employ an equal power change property, a spatial dummy property, anonymity, a positional invariance property, and a positional continuity property. Some examples are also given.
The Owen and Shapley spatial power indices: A comparison and a generalization
2017, Mathematical Social Sciences
Citation Excerpt :
When this is the case, we say that it is a spatial version of the basic index. Owen (1971) and Shapley (1977) propose spatial versions of the Shapley–Shubik power index, Shenoy (1982) proposes a spatial version of the Banzhaf power index, Rapoport and Golan (1985) give a spatial version of the Deegan–Packel power index. In this work, we are concerned with some spatial versions of the Shapley–Shubik power index.
Spatial games take into account the position of any voter in the space. In this class of games, two main indices of political power were defined. The first by Owen (1971) and the second, by Shapley (1977), later on extended in a two-dimensional space by Owen and Shapley (1989). We propose a generalization of Owen index. We show that the method proposed by this later in which players ordering is based on the distance between bliss and political issues points, yields the Shapley index if issues can be any point in the space.
The Banzhaf Index in Representative Systems with Multiple Political Parties
1999, Games and Economic Behavior
The proposed modification of the Banzhaf index is used to evaluate voter power in representative multiple-party (i.e., more than two) systems. The modified index shows that a voter's ability to affect the outcome of legislative decisions increases asymptotically as the inverse of the square root of the district's population. The square-root effect thus holds even in cases involving more than two parties. Journal of Economic Literature Classification Numbers: C71, D72.
The Owen–Shapley Spatial Power Index in Three-Dimensional Space
2021, Group Decision and Negotiation
An issue based power index
2021, International Journal of Game Theory
The Public Good Spatial Power Index in political games
2021, SSRN

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^☆: This research was supported in part by the University of Kansas General Research Fund and the School of Business Research Fund.

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The Banzhaf power index for political games☆

Abstract

Weighted voting doesn't work: A mathematical analysis

Rutgers Law Rev.

Multi-member electoral districts - Do they violate the ‘One Man, One Vote’ principle?

Yale Law J.

One man, 3.312 votes: A mathematical analysis of the electoral college

Villanova Law Rev.

One man,? votes: Mathematical analysis of political consequences and judicial choices

George Washington Law Rev.

Measuring power in weighted voting systems

The Descriptive Analysis of Power