9887

Power in Weighted Voting Systems

A weighted voting system consists of a number of players, each with a certain number of votes, and a quota of votes needed to pass a measure. There are various methods of measuring the "power" that an individual in a weighted voting system has. This Demonstration lets you compare the proportion of votes a player has versus that player's power as measured by the Shapley–Shubik and Banzhaf power indices. The thumbnail shows the famous example [51: 50, 49, 1] of a system with three players having 50, 49, and 1 votes, respectively, and with the quota set at 51 votes. Note that the distribution of power of the voters is very different than the distribution of votes.
  • Contributed by: Marc Brodie (Wheeling Jesuit University)

THINGS TO TRY

SNAPSHOTS

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

DETAILS

The Shapley–Shubik index of power of a player is the proportion of orderings of the players in which the given player is "pivotal". The pivotal player in a given ordering is the player whose vote(s), when added to the total of the votes of the previous players, result in enough votes to reach the quota and pass a measure.
The total Banzhaf power of a player is the number of winning coalitions (subsets of players with enough total votes to reach the quota and pass a measure) in which the given player is "critical". The Banzhaf index of power of a player is that player's total Banzhaf power divided by the sum of all players' total Banzhaf power. A critical player in a winning coalition is a player whose removal from the coalition would cause enough votes to be lost so that the remaining players do not have enough votes for the quota.
See A. D. Taylor, Mathematics and Politics—Strategy, Voting, Power and Proof, New York: Springer-Verlag, 1995.
The "show details" checkbox lets you see the orderings and pivotal players (in the case of Shapley–Shubik) and the winning coalitions and critical players (in the case of Banzhaf) for systems in which there are three or four players.
The quota cannot exceed the total number of votes nor can it be less than or equal to half the total number of votes.
    • 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.









 
RELATED RESOURCES
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 © 2014 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+