Weighted Voting Systems Brian Carrico What is

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, …, wn]

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 Total 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 Pass Y NY 3 Pass NY Y 2 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 [A, B]; [A, C] Extra Votes 0 4 1 [A, B, C] Weight Blocking Coalitions 2 [A]; [B, C] 3 [A, B]; [A, C] Extra Votes 0 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?

§ Permutations § § Example A B C 2 A C B 2 B A C 1 B C A 1 C A B 1 C B A 1 Shapley-Shubik indexes: A=4/6 B=1/6 C=1/6 Weights 3 4 3 4 2 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) § 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 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?

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