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 # of shares Ruth Smith 9 Ralph Smith 9 Albert Mansfield 7 Kathrine Ide 3 Gary Lambert 1 Marjorie Edwards 1 Total30

How do we Measure an individual’s power?  Critical Voter  Banzhaf Power Index  Developed by John F Banzhaf III  “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 ABCVotesOutcome YYY4Pass YNY3PassABCVotesOutcomeYYY4Pass NYY2Fail

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  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) Weight Winning Coalitions Extra Votes 3[A,B];[A,C]0 4[A,B,C]1 Weight Blocking Coalitions Extra Votes 2[A];[B,C]0 3[A,B];[A,C]1 4[A,B,C]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=10/24B=6/24C=6/24D=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  PermutationsWeights  Shapley-Shubik indexes:  A=4/6B=1/6C=1/6 ABC234 ACB234 BAC134 BCA124 CAB134 CBA124

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=10/24B=6/24C=6/24D=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)  We can see that the big shareholder is the pivotal voter in all permutations in groups 4002 through 5001  So, the big shareholder has a Shapley- Shubik index of 1000/9001  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 rule we can see that there is a  68% chance of heads  95% chance of heads  99.7% chance of 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