## Weighted Voting and Power Indices

A voting arrangement in which voters may control unequal number of votes and decisions are made by forming *coalition*s with the total of votes equal or in access of an agreed upon *quota* is called a *weighted voting system*. The usual notation is _{1}, w_{2}, ..., w_{n}],_{i}, *players*.

(Bold numbers could be clicked upon. To increase the number, click to the right of its vertical center line. To decrease it click to the left of the line. Dragging the mouse near the center line will accomplish the same task, but faster.)

What if applet does not run? |

In the system [5: 3, 2, 1], all decisions could be made by the two principal players (3 and 2 votes, respectively), while that last player (the one with the single vote) has no influence whatsoever on the decision process. It is then clear that having a vote does not endow its owner with any real power in making decisions. (The last player in this example is known as a *dummy*.)

Banzhaf's is one possible indicator of the relevance of a particular player. Shapley-Shubik's is another. In both cases, the power wielded by a player is determined by the number of coalitions in which his or her role is important. However, the two indices formalize the notions of coalition and importance in different ways.

*Coalition* is any (non-empty) combination of the players. A coalition is *winning* provided the cumulative vote of its members is equal to or greater than the quota. A coalition is losing if it's not winning. A player is called *critical* to a winning coalition, if his or her removal from the coalition renders it losing. *Banzhaf's index* of a player p is the ratio of the number of winning coalitions to which p is critical to the total number of times the players are critical.

A coalition is just a set of its elements. No order is specified in which the players enter the coalition. Not so with the *sequential coalition*, used to define the Shapley-Shubik index. A sequential coalition of n players p_{1}, ..., p_{n} is any permutation p_{i1}, ..., p_{in} of that set. An element p_{ik} is said to be *pivotal* to a (sequential) coalition p_{i1}, ..., p_{in}, provided the (regular) coalition p_{i1}, ..., p_{ik-1} is losing, whereas the (regular) coalition p_{i1}, ..., p_{ik} is winning. Assuming that the quota does not exceed the total number of votes, every sequential coalition has a unique pivotal element. *Shapley-Shubik's index* of power of a player p is the ratio of the number of sequential coalitions for which p is pivotal to the total number of sequential coalitions, which is always n!.

|Contact| |Front page| |Contents| |Up|

Copyright © 1996-2018 Alexander Bogomolny