Banzhaf Power Index and Shapley-Shubik Power Indices

Brief Introduction (for a fuller explanation, see For All Practical Purposes, 7th Edition, New York, WH Freeman 2006, Chapter 11)

Banzhaf Power Index

A player whose desertion of a winning coalition turns it into a losing one is called a critical player.
A player’s power is proportional to the number of times the player is critical.

Shapley-Shubik Power Index
A permutation of voters is a listing of the voters according to their preference on a issue.
A pivotal voter in a permutation is the one who, combined with all voters before him or her in the permutation, has just enough votes to win.
There are N! permutations in a system of N voters
A voter's Shapley-Shubik index is the number of permutations in which that voter is pivotal, divided by N!.

Java Applet

The JAVA Applet will enable you to compute the Banzaf and Shapley-Shabik indices of several important voting systems, including the U.S. Electoral college, the council of ministers of the European Union, and the U.N. security council. You may also enter weights and a quota for a weighted voting system of your own design.

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