Banzhaf Power Index

The Banzhaf index is a measure of how probable it is that someone's vote will change the outcome of an election in settings in which voters may have unequal numbers of votes. This Demonstration examines a scenario with six voters, each of whom have a user-selectable number of votes. You can also determine the percentage of total votes needed to win. From this data, the Demonstration shows all the winning coalitions that can form and all the members of each coalition ("the swing voters") who, if they had voted differently, would have changed the outcome of the election. The Banzhaf index, which ranges from 0 to 1, is the percentage of times each voter is a swing voter over all possible winning coalitions.



  • [Snapshot]
  • [Snapshot]
  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


The Banzhaf index is closely related to the Shapley value, often used in cooperative games.
Snapshot 1: the percentage of votes needed to win affects the power index
Snapshot 2: sometimes, parties with different amounts of raw voting power have the same power index
Snapshot 3: only two parties have any real power
Snapshot 4: a four party system in which the second party has somewhat more votes but has no more power when a two thirds majority is needed
Snapshot 5: a four party system in which the second party has somewhat more votes but has considerably more power
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+