EXCURSIONS IN MODERN MATHEMATICS SIXTH EDITION Peter Tannenbaum 1

CHAPTER 2 WEIGHTED VOTING SYSTEMS The Power Game 2

Weighted Voting Systems Outline/learning Objectives 3  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 2.1 Weighted Voting Systems 4

Weighted Voting Systems 5  Weighted Voting Systems are any voting arrangement in which the voters are not necessarily equal in terms of the number of votes they control.  A motion is a vote with only 2 choices. (usually “yes” and “no”)

Weighted Voting Systems 6  The Players are the voters in a weighted voting system.  Characterized by a “P” with a subscript. P 1, P 2, P 3, … P N

Weighted Voting Systems 7  The Weights are the number of votes controlled by each player.  Characterized by a “w” with a subscript which denotes the player number.  Adding the weights gives the total number of votes in the system. w 1 is the number of votes for P 1 V = w 1 + w 2 + w 3 + … w N

Weighted Voting Systems 8  The Quota is the minimum number of votes needed to pass the motion.  Characterized by a “q” and is usually greater than 50% and less than the total number of votes. V >= q > V/2

Weighted Voting Systems 9  A weighted voted system takes the form:  Weights are ordered highest to lowest. [q: w 1, w 2, w 3, … w N ] w 1 > w 2 > w 3 > … > w N

Weighted Voting Systems 10  Quota is two-thirds of the total number of votes. [14: 8, 7, 3, 2]  Quota is too low (anarchy). [10: 8, 7, 3, 2]  Quota is too high (gridlock). [21: 8, 7, 3, 2]  One Partner – One Vote (unanimous). [19: 8, 7, 3, 2][4: 1, 1, 1, 1]

Weighted Voting Systems 11  A Dictator is a player who’s weight is bigger than or equal to the quota.  P 1 has enough weight (votes) to carry a motion single handedly. [11: 12, 5, 4]

Weighted Voting Systems 12  A Dummy is a player who has no power.  There is never a time when a dummy makes a difference in the outcome. [30: 10, 10, 10, 9]

Weighted Voting Systems 13  Veto Power occurs if a motion cannot pass unless a specific player votes in favor of the motion. [12: 9, 5, 4, 2] w < q and V – w < q

Weighted Voting Systems  [ 7: 5, 3, 2 ]  [ 4: 3, 2, 2 ]  [ 37: 8, 6, 5, 3, 3, 3, 2, 2, 1, 1, 1, 1 ] 14

Weighted Voting Systems  [ 15: 5, 4, 3, 2, 1 ]  [ 5: 1, 1, 1, 1, 1 ]  [ 12: 13, 2, 7, 1, 1 ] 15

WEIGHTED VOTING SYSTEMS 2.2 The Banzhaf Power Index 16

The Banzhaf Power Index 17  The Banzhaf Power Index is method to determine the probability of changing the outcome of a vote when power is not equally divided. [101: 99, 98, 3]  Requires two players to pass a motion.

The Banzhaf Power Index 18  A Coalition is any set of players that join forces and vote the same way. [10: 8, 7, 3, 1] {P 1, P 2 } W = w 1 + w 2 W =  To determine the number of coalitions: 2 N - 1

The Banzhaf Power Index 19  A grand coalition is a coalition consisting of all players. [101: 99, 96, 3] {P 1, P 2, P 3 }  Some coalitions have enough to win and some don’t. We call the former a winning coalition and the later a losing coalition. [101: 99, 96, 3]{P 1, P 2 } W = 195 {P 2, P 3 } W = 99

The Banzhaf Power Index 20  A critical player is a player who is required in a coalition to win. [101: 99, 98, 3] W – w < q

The Banzhaf Power Index 21  Banzhaf: “A player’s power should be measured by how often the player is a critical player.”  Each player is a critical player 2 times out of 6.

The Banzhaf Power Distribution 22 Computing a Banzhaf Power Distribution  Step 1. Make a list of all possible winning coalitions.

The Banzhaf Power Distribution 23 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.

The Banzhaf Power Distribution 24 Computing a Banzhaf Power Distribution  Step 3.Count the number of times that P 1 is critical. Call this number B 1 Repeat for each of the other players to find B 2, B 3, … B N

The Banzhaf Power Distribution 25 Computing a Banzhaf Power Distribution  Step 4. Find the total number of times all players are critical. This total is given by T = B 1 + B 2 + B 3 + … B N

The Banzhaf Power Distribution 26 Computing a Banzhaf Power Distribution  Step 5. Find the ratio  1 = B 1 / T This gives the Banzhaf power index of P 1. 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.

The Banzhaf Power Index 27  β 1 is the Banzhaf power index for the specific player P 1.  The complete list of β s is the Banzhaf power distribution. β 1 =40% is the Banzhaf power index for P 1. β 1 =40%; β 2 =25%; β 3 =20%; β 4 =15% is the Banzhaf power distribution.

Weighted Voting Systems  [ 7: 5, 3, 2, 1 ] 28 2 N - 1

Weighted Voting Systems  [ 4: 3, 2, 2 ] 29 2 N - 1

WEIGHTED VOTING SYSTEMS 2.3 Applications of Banzhaf Power 30

Applications 31 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 32

The Shapley-Shubik Power Index 33  The Shapley-Shubik Power Index is an alternative way to computing voting power.  In situations like political alliances, the order that the players join is important.  When a motion is considered, the player that joins the coalition and allows it to reach quota can be considered most essential.

The Shapley-Shubik Power Index 34  Sequential coalitions are coalitions where the order of the players matter. [14: 10, 3, 2, 1] etc.

The Shapley-Shubik Power Index 35  The number of possible sequential coalitions is N factorial. N! = N x (N-1) x…x 3 x 2 x 1 [14: 10, 3, 2, 1] N = # of players N = 4 N! = 4 x 3 x 2 x 1 = 24 T = N!

The Shapley-Shubik Power Index 36  The pivotal player is the player who, in turn, causes the coalition to pass a motion. [14: 10, 3, 2, 1] 3, 10, 2, 1  If you consider all players voting “no” and then start switching the votes in order to “yes” = 15

The Shapley-Shubik Power Index 37 Three-Player Sequential Coalitions

The Shapley-Shubik Power Index 38 Shapley-Shubik Pivotal Player The player that contributes the votes that turn what was a losing coalition into a winning coalition.

The Shapley-Shubik Power Index 39 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. [14: 10, 3, 2, 1] etc. T = N! = 4 x 3 x 2 x 1 = 24

The Shapley-Shubik Power Index 40 Computing a Shapley-Shubik Power Distribution  Step 2. In each sequential coalition determine the pivotal player. [14: 10, 3, 2, 1] etc.

The Shapley-Shubik Power Index 41 Computing a Shapley-Shubik Power Distribution  Step 3.Count the number of times that P 1 is pivotal. Call this number SS 1. Repeat for each of the other players to find SS 2, SS 3, … SS N. [14: 10, 3, 2, 1] SS 1 = 11 SS 2 = 9 SS 3 = 2 SS 4 = 2

The Shapley-Shubik Power Index 42 Computing a Shapley-Shubik Power Distribution  Step 4. Find the ratio  1 = SS 1 /T. This gives the Shapley Shubik power index of P 1. Repeat for the other players to find  2,  3, …,  N. The complete list of  ’s gives the Shapley-Shubik power distribution of the weighted voting system.  1 = SS 1 /T = 11/24 = 45.8%  2 = SS 2 /T = 9/24 = 37.5%  3 = SS 3 /T = 2/24 = 8.3%  4 = SS 4 /T = 2/24 = 8.3%

Example- [4: 3, 2, 1] 43

The Shapley-Shubik Power Index 44 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

Summary 45  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