Weighted Voting Systems

Brian Carrico

What is a weighted voting system?

 A weighted voting system is a decision making procedure in which the participants have varying numbers of votes.

 Examples:  Shareholder elections  Some legislative bodies  Electoral College

Key Terms and Notation

 Weight  Quota  Shorthand notation:  [q: w 1 , w 2 , …, w n ]

Coalition Building

 Rarely will one voter have enough votes to meet the quota so coalitions are necessary to pass any measure  Types of coalitions  Winning Coalition  Losing Coalition  Blocking Coalition  Dummy voters

Coalition Illustration

   On the right is a table of the weights of shareholders of a company.

A simple majority (16 votes) is needed for any measure.

Ide, Lambert, and Edwards are all Dummy Voters as any winning coalition including any subset of those three would be a winning coalition without them.

Shareholder Ruth Smith Ralph Smith Albert Mansfield 7 Kathrine Ide 3 Gary Lambert Marjorie Edwards Total 1 1 # of shares 9 9 30

How do we Measure an individual’s power?

 Critical Voter  Banzhaf Power Index  Developed by John F Banzhaf III  1965 “Weighted Voting Doesn’t Work”  The number of winning or blocking coalitions in which a participant is the critical voter

Critical Voter Illustration

  Consider a committee of three members The voting system follows this pattern:   [3: 2, 1, 1] For ease, we’ll refer to the members as A, B, and C A B C Votes Outcome Y Y Y 4 Y N Y 3 Pass Pass A B C Votes Outcome Y Y Y 4 N Y Y 2 Pass Fail

Extra Votes

 A helpful concept in calculating Banzhaf Power Index  A winning coalition with w votes has w-q extra votes  Any voter with more votes than the extra votes in the coalition is a critical voter

Calculating Banzhaf Index

Weight Winning Coalitions 3 Extra Votes [A,B];[A,C] 0 Weight Blocking Coalitions 2 [A];[B,C] Extra Votes 0 4 3 [A,B];[A,C] 1 [A,B,C] 1 4 [A,B,C] 2    In Winning Coalitions; A is a critical voter three times, B and C are critical voters once In Blocking Coaltions; A is a critical voter three times, B and C are critical voters once Banzhaf Index of this system: (6,2,2)

Notice a Pattern there?

 Each voter is a critical voter in the same number of winning coalitions as blocking coalition  When a voter defects from a winning coalition they become the critical voter in a corresponding blocking coalition    [A, B, C]=>[A] [A, B]=>[A, C] [A, C]=>[A, B]

How does this help?

 Because these numbers are identical, we can calculate the Banzhaf Power Index by finding the number of winning coalitions in which a voter is the critical voter and double it  Can make computations easier in systems with many voters

[51: 40, 30, 20, 10]

Banzhaf Index

 From the table above we can see that in winning coalitions,    A is a critical vote 5 times B and C are critical votes 3 times each D is a critical vote once  So, their Banzhaf Index is twice that,  A=10, B=6, C=6, and D=2  Their voting power is  A=10/24 B=6/24 C=6/24 D=2/24

The Electoral College

Shapley-Shubik Power Index

  For coalitions built one voter at a time The voter whose vote turns a losing coalition into a winning coalition is the most important voter  Shapley-Shubik uses permutations to calculate how often a voter serves as the pivotal voter  This index takes into account commitment to an issue

How do we find the pivotal voter?

 The first voter in a permutation of voters whose vote would make a the coalition a winning coalition is the pivotal voter  The Shapley-Shubik power index is the fraction of the permutations in which that voter is pivotal  Formula:  (number times the voter is pivotal)  (number of permutations of voters)

What does this overlook?

Example

 Permutations A A B C C B 2 2 B B C A C A C A B 1 1 1   C B A Shapley-Shubik indexes: 1 A=4/6 B=1/6 C=1/6 3 3 3 2 3 2 Weights 4 4 4 4 4 4

For a larger corporation

Larger Corporation (cont)

 This is the same corporation we looked at earlier distributed as [51: 40, 30, 20, 10]  The Shapley-Shubik Index for the four people in the corporation is:  A=10/24 B=6/24 C=6/24 D=2/24  So here, the Banzhaf and Shapley Shubik indexes agree, but is this always true?

Comparing the Indexes

 The Banzhaf index assumes all votes are cast with the same probability  Shapley-Shubik index allows for a wide spectrum of opinions on an issue  Shapley-Shubik index takes commitment to an issue into account

An illustration of the difference

 Consider a corporation of 9001 shareholders  Such a large corporation can only be analyzed if nearly all of the voters have the same power  So, we will consider a corporation with 1 shareholder owning 1000 shares and 9000 shareholders each owning one share, and assume a simple majority

Under Shapley-Shubik

 The big voter will be the critical voter in any permutation that positions at least 4001 of the small voters before him, but no more than 5000  We can group the permutations into 9001 equal groups based on the location of the big shareholder

Shapley-Shubik (cont)

  So, the big shareholder has a Shapley Shubik index of 1000/9001  We can see that the big shareholder is the pivotal voter in all permutations in groups 4002 through 5001 The remaining 8001/9001 power goes equally to the 9000 small voters

Under Banzhaf

 We can estimate the big shareholder’s Banzhaf Power Index can be estimated assuming a each small shareholder decides his vote by a coin toss  The big shareholder will be a critical voter unless his coalition is joined by fewer than 4001 small shareholders or at least 5001 small shareholders

Banzhaf (cont)

    When the 9000 small shareholders toss their coins, the expected number of heads is ½ * 9000 = 4500 The standard deviation is roughly 50 By the 68-95-99.7 rule we can see that there is a  68% chance of 4450-4550 heads   95% chance of 4400-4600 heads 99.7% chance of 4350-4650 heads You can see that the big shareholder’s Banzhaf Index is nearly 100%

Which seems fairer?

 The Shapley-Shubik Index gave the big shareholder roughly 11% of the power while the Banzhaf Index gave him nearly 100% of the power  The big shareholder has roughly 11% of the votes   Which index seems more realistic?

Why are the indexes so different when earlier they came out the same?

Homework