Excursions in Modern Mathematics Sixth Edition Peter Tannenbaum

Chapter 2 Weighted Voting Systems The Power Game

Weighted Voting Systems Outline/learning Objectives Represent a weighted voting system using a mathematical model. Use the Banzhaf and Shapley-Shubik indices to calculate the distribution of power in a weighted voting system.

Weighted Voting Systems

Weighted Voting Systems The Players The voters in a weighted voting system. The Weights That each player controls a certain number of votes. The Quota The minimum number of votes needed to pass a motion (yes-no votes)

Weighted Voting Systems Dictator The player’s weight is bigger than or equal to the quota. Consider [11:12, 5, 4] owns enough votes to carry a motion single handedly.

Weighted Voting Systems Dummy A player with no power. Consider [30: 10, 10, 10, 9] turns out to be a dummy! There is never going to be a time when is going to make a difference in the outcome of the voting.

Weighted Voting Systems Veto Power If a motion cannot pass unless player votes in favor of the motion. Consider [12: 9, 5, 4, 2] has the power to obstruct by preventing any motion from passing.

Weighted Voting Systems 2.2 The Banzhaf Power Index

Weighted Voting Systems Coalitions Any set of players that might join forces and vote the same way. The coalition consisting of all the players is called a grand coalition.

Weighted Voting Systems Winning Coalitions Some coalitions have enough votes to win and some don’t. We call the former winning coalitions and the latter losing coalitions.

Weighted Voting Systems Critical players In a winning coalition, a player is said to be a critical player for the coalition if the coalition must have that player’s votes to win. If and only if

Weighted Voting Systems Computing a Banzhaf Power Distribution Step 1. Make a list of all possible winning coalitions.

Weighted Voting Systems Computing a Banzhaf Power Distribution Step 2. Within each winning coalition determine which are the critical players. (To determine if a given player is critical or not in a given winning coalition, we subtract the player’s weight from the total number of votes in the coalition- if the difference drops below the quota q, then that player is critical. Otherwise, that player is not critical.

Weighted Voting Systems Computing a Banzhaf Power Distribution Step 3.Count the number of times that is critical. Call this number Repeat for each of the other players to find

Weighted Voting Systems Computing a Banzhaf Power Distribution Step 4. Find the total number of times all players are critical. This total is given by

Weighted Voting Systems Computing a Banzhaf Power Distribution Step 5. Find the ratio 1 . This gives the Banzhaf power index of . Repeat for each of the other players to find 2, 3, …, N . The complete list of ’s gives the Banzhaf power distribution of the weighted voting system.  

Weighted Voting Systems 2.3 Applications of Banzhaf Power

Weighted Voting Systems Applications of Banzhaf Power The Nassau County Board of Supervisors John Banzhaf first introduced the concept The United Nations Security Council Classic example of a weighted voting system The European Union (EU) Relative Weight vs Banzhaf Power Index

Weighted Voting Systems 2.4 The Shapley-Shubik Power Index

Weighted Voting Systems Three-Player Sequential Coalitions

Weighted Voting Systems Shapley-Shubik- Pivotal Player The player that contributes the votes that turn what was a losing coalition into a winning coalition.

Weighted Voting Systems Computing a Shapley-Shubik Power Distribution Step 1. Make a list of all possible sequential coalitions of the N players. Let T be the number of such coalitions.

Weighted Voting Systems Computing a Shapley-Shubik Power Distribution Step 2. In each sequential coalition determine the pivotal player.

Weighted Voting Systems Computing a Shapley-Shubik Power Distribution Step 3.Count the number of times that is pivotal. Call this number . Repeat for each of the other players to find

Weighted Voting Systems Computing a Shapley-Shubik Power Distribution Step 4. Find the ratio 1 This gives the Shapley Shubik power index of . Repeat for each of the other players to find  2,  3, …,  N . The complete list of  ’s gives the Shapley-Shubik power distribution of the weighted voting system.  

Weighted Voting Systems The Multiplication Rule If there are m different ways to do X, and n different ways to do Y, then X and Y together can be done in m x n different ways.

Weighted Voting Systems The Number of Sequential Coalitions The number of sequential coalitions with N players is N! = N x (N-1) x…x 3 x 2 x 1.

Weighted Voting Systems Applications of Shapley-Shubik Power The Electoral College There are 51! Sequential coalitions The United Nations Security Council Enormous difference between permanent and nonpermanent members The European Union (EU) Relative Weight vs Shapley-Shubik Power Index

Weighted Voting Systems Conclusion The notion of power as it applies to weighted voting systems How mathematical methods allow us to measure the power of an individual or group by means of an index. We looked at two different kinds of power indexes– Banzhaf and Shapley-Shubik