Weighted Voting SystemsWeighted Voting SystemsBrian CarricoBrian Carrico

What is a weighted voting system?What is a weighted voting system?

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

Examples:Examples: Shareholder electionsShareholder elections Some legislative bodiesSome legislative bodies Electoral CollegeElectoral College

Key Terms and NotationKey Terms and Notation

WeightWeight

QuotaQuota

Shorthand notation:Shorthand notation: [q: w[q: w11, w, w22, …, w, …, wnn]]

Coalition BuildingCoalition Building

Rarely will one voter have enough votes to Rarely will one voter have enough votes to meet the quota so coalitions are meet the quota so coalitions are necessary to pass any measurenecessary to pass any measure

Types of coalitionsTypes of coalitions Winning CoalitionWinning Coalition Losing CoalitionLosing Coalition Blocking CoalitionBlocking Coalition

Dummy votersDummy voters

Coalition IllustrationCoalition Illustration On the right is a table of On the right is a table of

the weights of the weights of shareholders of a shareholders of a company.company.

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

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

ShareholderShareholder # of shares# of shares

Ruth SmithRuth Smith 99

Ralph SmithRalph Smith 99

Albert MansfieldAlbert Mansfield 77

Kathrine IdeKathrine Ide 33

Gary LambertGary Lambert 11

Marjorie Marjorie EdwardsEdwards

11

TotalTotal 3030

How do we Measure How do we Measure an individual’s power?an individual’s power?

Critical VoterCritical Voter

Banzhaf Power IndexBanzhaf Power Index Developed by John F Banzhaf IIIDeveloped by John F Banzhaf III

1965- “Weighted Voting Doesn’t Work”1965- “Weighted Voting Doesn’t Work”

The number of winning or blocking coalitions The number of winning or blocking coalitions in which a participant is the critical voterin which a participant is the critical voter

Critical Voter IllustrationCritical Voter Illustration

Consider a committee of three membersConsider a committee of three members The voting system follows this pattern:The voting system follows this pattern:

[3: 2, 1, 1][3: 2, 1, 1] For ease, we’ll refer to the members as A, B, and CFor ease, we’ll refer to the members as A, B, and C

AA BB CC VotesVotes OutcomeOutcome

YY YY YY 44 PassPass

YY NN YY 33 PassPass

AA BB CC VotesVotes OutcomeOutcome

YY YY YY 44 PassPass

NN YY YY 22 FailFail

Extra VotesExtra Votes

A helpful concept in calculating Banzhaf A helpful concept in calculating Banzhaf Power IndexPower Index

A winning coalition with w votes has w-q A winning coalition with w votes has w-q extra votesextra votes

Any voter with more votes than the extra Any voter with more votes than the extra votes in the coalition is a critical votervotes in the coalition is a critical voter

Calculating Banzhaf IndexCalculating Banzhaf Index

In Winning Coalitions; A is a critical voter three In Winning Coalitions; A is a critical voter three times, B and C are critical voters oncetimes, B and C are critical voters once

In Blocking Coaltions; A is a critical voter three In Blocking Coaltions; A is a critical voter three times, B and C are critical voters oncetimes, B and C are critical voters once

Banzhaf Index of this system: (6,2,2)Banzhaf Index of this system: (6,2,2)

WeightWeight Winning Winning CoalitionsCoalitions

Extra Extra VotesVotes

33 [A,B];[A,C][A,B];[A,C] 00

44 [A,B,C][A,B,C] 11

WeightWeight Blocking Blocking CoalitionsCoalitions

Extra Extra VotesVotes

22 [A];[B,C][A];[B,C] 00

33 [A,B];[A,C][A,B];[A,C] 11

44 [A,B,C][A,B,C] 22

Notice a Pattern there?Notice a Pattern there?

Each voter is a critical voter in the same Each voter is a critical voter in the same number of winning coalitions as blocking number of winning coalitions as blocking coalitioncoalition

When a voter defects from a winning When a voter defects from a winning coalition they become the critical voter in a coalition they become the critical voter in a corresponding blocking coalitioncorresponding blocking coalition [A, B, C]=>[A][A, B, C]=>[A] [A, B]=>[A, C][A, B]=>[A, C] [A, C]=>[A, B][A, C]=>[A, B]

How does this help?How does this help?

Because these numbers are identical, we Because these numbers are identical, we can calculate the Banzhaf Power Index by can calculate the Banzhaf Power Index by finding the number of winning coalitions in finding the number of winning coalitions in which a voter is the critical voter and which a voter is the critical voter and double itdouble it

Can make computations easier in systems Can make computations easier in systems with many voterswith many voters

[51: 40, 30, 20, 10][51: 40, 30, 20, 10]

Banzhaf IndexBanzhaf Index

From the table above we can see that in From the table above we can see that in winning coalitions,winning coalitions, A is a critical vote 5 timesA is a critical vote 5 times B and C are critical votes 3 times eachB and C are critical votes 3 times each D is a critical vote onceD is a critical vote once

So, their Banzhaf Index is twice that,So, their Banzhaf Index is twice that, A=10, B=6, C=6, and D=2A=10, B=6, C=6, and D=2

Their voting power isTheir voting power is A=10/24A=10/24 B=6/24B=6/24 C=6/24C=6/24 D=2/24D=2/24

The Electoral CollegeThe Electoral College

Shapley-Shubik Power IndexShapley-Shubik Power Index

For coalitions built one voter at a timeFor coalitions built one voter at a time The voter whose vote turns a losing The voter whose vote turns a losing

coalition into a winning coalition is the coalition into a winning coalition is the most important votermost important voter

Shapley-Shubik uses permutations to Shapley-Shubik uses permutations to calculate how often a voter serves as the calculate how often a voter serves as the pivotal voterpivotal voter

This index takes into account commitment This index takes into account commitment to an issueto an issue

How do we find the pivotal voter?How do we find the pivotal voter?

The first voter in a permutation of voters The first voter in a permutation of voters whose vote would make a the coalition a whose vote would make a the coalition a winning coalition is the pivotal voterwinning coalition is the pivotal voter

The Shapley-Shubik power index is the The Shapley-Shubik power index is the fraction of the permutations in which that fraction of the permutations in which that voter is pivotalvoter is pivotal

Formula:Formula: (number times the voter is pivotal)(number times the voter is pivotal) (number of permutations of voters)(number of permutations of voters)

What does this overlook?What does this overlook?

ExampleExample PermutationsPermutations WeightsWeights

Shapley-Shubik indexes:Shapley-Shubik indexes: A=4/6A=4/6 B=1/6B=1/6 C=1/6C=1/6

AA BB CC 22 33 44

AA CC BB 22 33 44

BB AA CC 11 33 44

BB CC AA 11 22 44

CC AA BB 11 33 44

CC BB AA 11 22 44

For a larger corporationFor a larger corporation

Larger Corporation (cont)Larger Corporation (cont)

This is the same corporation we looked at This is the same corporation we looked at earlier distributed as [51: 40, 30, 20, 10]earlier distributed as [51: 40, 30, 20, 10]

The Shapley-Shubik Index for the four The Shapley-Shubik Index for the four people in the corporation is:people in the corporation is: A=10/24A=10/24 B=6/24B=6/24 C=6/24C=6/24 D=2/24D=2/24

So here, the Banzhaf and Shapley Shubik So here, the Banzhaf and Shapley Shubik indexes agree, but is this always true?indexes agree, but is this always true?

Comparing the IndexesComparing the Indexes

The Banzhaf index assumes all votes are The Banzhaf index assumes all votes are cast with the same probabilitycast with the same probability

Shapley-Shubik index allows for a wide Shapley-Shubik index allows for a wide spectrum of opinions on an issuespectrum of opinions on an issue

Shapley-Shubik index takes commitment Shapley-Shubik index takes commitment to an issue into accountto an issue into account

An illustration of the differenceAn illustration of the difference

Consider a corporation of 9001 Consider a corporation of 9001 shareholdersshareholders

Such a large corporation can only be Such a large corporation can only be analyzed if nearly all of the voters have the analyzed if nearly all of the voters have the same powersame power

So, we will consider a corporation with 1 So, we will consider a corporation with 1 shareholder owning 1000 shares and 9000 shareholder owning 1000 shares and 9000 shareholders each owning one share, and shareholders each owning one share, and assume a simple majorityassume a simple majority

Under Shapley-ShubikUnder Shapley-Shubik

The big voter will be the critical voter in The big voter will be the critical voter in any permutation that positions at least any permutation that positions at least 4001 of the small voters before him, but no 4001 of the small voters before him, but no more than 5000more than 5000

We can group the permutations into 9001 We can group the permutations into 9001 equal groups based on the location of the equal groups based on the location of the big shareholderbig shareholder

Shapley-Shubik (cont)Shapley-Shubik (cont)

We can see that the big shareholder is the We can see that the big shareholder is the pivotal voter in all permutations in groups pivotal voter in all permutations in groups 4002 through 50014002 through 5001

So, the big shareholder has a Shapley-So, the big shareholder has a Shapley-Shubik index of 1000/9001Shubik index of 1000/9001

The remaining 8001/9001 power goes The remaining 8001/9001 power goes equally to the 9000 small votersequally to the 9000 small voters

Under BanzhafUnder Banzhaf

We can estimate the big shareholder’s We can estimate the big shareholder’s Banzhaf Power Index can be estimated Banzhaf Power Index can be estimated assuming a each small shareholder assuming a each small shareholder decides his vote by a coin tossdecides his vote by a coin toss

The big shareholder will be a critical voter The big shareholder will be a critical voter unless his coalition is joined by fewer than unless his coalition is joined by fewer than 4001 small shareholders or at least 5001 4001 small shareholders or at least 5001 small shareholderssmall shareholders

Banzhaf (cont)Banzhaf (cont)

When the 9000 small shareholders toss their When the 9000 small shareholders toss their coins, the expected number of heads is ½ * coins, the expected number of heads is ½ * 9000 = 45009000 = 4500

The standard deviation is roughly 50The standard deviation is roughly 50 By the 68-95-99.7 rule we can see that there is aBy the 68-95-99.7 rule we can see that there is a

68% chance of 4450-4550 heads68% chance of 4450-4550 heads 95% chance of 4400-4600 heads95% chance of 4400-4600 heads 99.7% chance of 4350-4650 heads99.7% chance of 4350-4650 heads

You can see that the big shareholder’s Banzhaf You can see that the big shareholder’s Banzhaf Index is nearly 100%Index is nearly 100%

Which seems fairer?Which seems fairer?

The Shapley-Shubik Index gave the big The Shapley-Shubik Index gave the big shareholder roughly 11% of the power shareholder roughly 11% of the power while the Banzhaf Index gave him nearly while the Banzhaf Index gave him nearly 100% of the power100% of the power

The big shareholder has roughly 11% of The big shareholder has roughly 11% of the votesthe votes

Which index seems more realistic?Which index seems more realistic? Why are the indexes so different when Why are the indexes so different when

earlier they came out the same?earlier they came out the same?

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