Weighted Voting Systems

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Weighted Voting Systems

• Brian Carrico

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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

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Key Terms and Notation

• Weight
• Quota
• Shorthand notation
• q w1, w2, , wn

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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

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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
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How do we Measure an individuals power?

• Critical Voter
• Banzhaf Power Index
• Developed by John F Banzhaf III
• 1965- Weighted Voting Doesnt Work
• The number of winning or blocking coalitions in
  which a participant is the critical voter

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Critical Voter Illustration

• Consider a committee of three members
• The voting system follows this pattern
• 3 2, 1, 1
• For ease, well refer to the members as A, B, and
  C

A B C Votes Outcome
Y Y Y 4 Pass
Y N Y 3 Pass
A B C Votes Outcome
Y Y Y 4 Pass
N Y Y 2 Fail
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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

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Calculating Banzhaf Index
Weight Blocking Coalitions Extra Votes
2 AB,C 0
3 A,BA,C 1
4 A,B,C 2
Weight Winning Coalitions Extra Votes
3 A,BA,C 0
4 A,B,C 1

• 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)

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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, CgtA
• A, BgtA, C
• A, CgtA, B

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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

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51 40, 30, 20, 10
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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,
• A10, B6, C6, and D2
• Their voting power is
• A10/24 B6/24 C6/24 D2/24

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The Electoral College
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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

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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)

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What does this overlook?
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Example

• Permutations Weights
• Shapley-Shubik indexes
• A4/6 B1/6 C1/6

A B C 2 3 4
A C B 2 3 4
B A C 1 3 4
B C A 1 2 4
C A B 1 3 4
C B A 1 2 4
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For a larger corporation
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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
• A10/24 B6/24 C6/24 D2/24
• So here, the Banzhaf and Shapley Shubik indexes
  agree, but is this always true?

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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

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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

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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

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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

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Under Banzhaf

• We can estimate the big shareholders 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

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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 shareholders Banzhaf
  Index is nearly 100

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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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Homework