Weighted voting systems

Weighted voting systems are based on the idea that not all voters are equal. Instead, it can be desirable to recognize differences by giving voters different amounts of say concerning the outcome of an election. This type of voting system is used in everything from shareholder meetings to the United States Electoral College.

The Mathematics of Weighted Voting Systems

A weighted voting system is characterized by three things - the players, the weights and the quota. The voters are the players ("P₁" , "P₂" . . ."P_N" ). "N" denotes the total number of players. A player's weight ("w") is the number of votes he controls. The quota ("q") is the minimum number of votes required to pass a motion. Any integer is a possible choice for the quota as long as it is more than 50% of the total number of votes but is no more than 100% of the total number of votes. Each weighted voting system can be described using the generic form ["q": "w₁" , "w₂" . . ."w_N" ] . The weights are always listed in numerical order, starting with the highest.Tannenbaum, Peter. Excursions in Modern Mathematics. 6th ed. Upper Saddle River: Prentice Hall, 2006. 48-83.]

The Notion of Power

When considering motions, all reasonable voting methods will have the same outcome as majority rules. Thus, the mathematics of weighted voting systems looks at the notion of power: who has it and how much do they have?Bowen, Larry. "Weighted Voting Systems." Introduction to Comtemporary Mathematics. 1 Jan. 2001. Center for Teaching and Learning, University of Alabama. [http://www.ctl.ua.edu/math103/POWER/wtvoting.htm#BPI%20example] .] A player's power is defined as that player's ability to influence decisions. [Daubechies, Ingrid. "Weighted Voting Systems." Voting and Social Choice. 26 Jan. 2002. Math Alive, Princeton University. [http://www.math.princeton.edu/matalive/Voting/VotingLab2/Weighted.html] .]

Consider the voting system [6: 5, 3, 2] . Notice that a motion can only be passed with the support of "P₁". In this situation, "P₁" has veto power. A player is said to have veto power if a motion cannot pass without the support of that player. This does not mean a motion is guaranteed to pass with the support of that player.]

Now let's look at the weighted voting system [10: 11, 6, 3] . With 11 votes, "P₁" is called a dictator. A player is typically considered a dictator if his weight is equal to or greater than the quota. The difference between a dictator and a player with veto power is that a motion is guaranteed to pass if the dictator votes in favor of it. ]

A dummy is any player, regardless of his weight, who has no say in the outcome of the election. A player without any say in the outcome is a player without power. Dummies always appear in weighted voting systems that have a dictator but also occur in other weighted voting systems.]

Measuring a Player's Power

A player's weight is not always an accurate depiction of that player's power. Sometimes, a player with several votes can have little power. For example, consider the weighted voting system [20: 10, 10, 9] . Although "P₃" has almost as many votes as the other players, his votes will never affect the outcome. Conversely, a player with just a few votes may hold quite a bit of power. Take the weighted voting system [7: 4, 2, 1] for example. No motion can be passed without the unanimous support of all the players. Thus, "P₃" holds just as much power as "P₁".

It is more accurate to measure a player's power using either the Banzhaf Power Index or the Shapley-Shubik Power Index. The two power indexes often come up with different measures of power for each player yet neither one in necessarily a more accurate depiction. Thus, which method is best for measuring power is best for measuring power is based on which assumption best fits the situation.The Banzhaf measure of power is based on the idea that players are free to come and go from coalitions, negotiating their allegiance. The Shapley-Shubik measure centers on the assumption that a player makes a commitment to stay upon joining a coalition.

Notes