Chapter 10: The Manipulability of Voting Systems Chapter 11: Weighted Voting Systems

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Chapter 10 The Manipulability of Voting
SystemsChapter 11 Weighted Voting Systems

• Presented by Katherine Goulde

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Chapter 10 Outline

• Introduction and example
• Majority Rule and Condorcets Method
• Voting Systems for 3 or more candidates
• Borda Count
• Sequential Pairwise Voting
• Plurality Voting
• Impossibility- The Gibbard- Satterthwaite Theorem
• The Chairs Paradox

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Manipulability the Borda Count

• The Borda count assigns point values to the
  candidates and the winner is the candidate with
  the most points
• Voter 1 Voter 2
• A B
• B C
• C A
• D D
• Candidate A has a score of 4
• Candidate B has a score of 5
• Candidate C has a score of 3
• Candidate D has a score of 0
• Therefore B wins!

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What if Voter 1 wants to manipulate the
election???

• Voter 1 Voter 2
• A B
• B C
• C A
• D D
• Voter 1 Voter 2
• A B
• D C
• C A
• B D

• Original Ballot where B wins the election
• However, Voter 1 wants A to win. How can Voter 1
  ensure that A wins?
• In this second ballot, A has a Borda count of 4,
  B has 3, C has 3, and D has 2. Therefore A is the
  winner.
• Is there any other way to obtain this result?

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• Unilateral Change- A change (in ballot) by a
  voter while every other voter keeps his or her
  ballot exactly as it was
• - single-voter manipulation
• A voting system is manipulable if
• there are two sequences of preference list
  ballots and a Voter so that
• Neither election results in a tie
• The only ballot change is by the Voter
• The Voter prefers the outcome of the second
  election to that of the first election.
• Take the two-candidate case with majority rule,
  and recall that it is monotone
• In this instance, nonmanipulability is the same
  thing as monotonicity

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Majority Rule and Condorcets Method

• Mays Theorem for Manipulability
• Among all two-candidate voting systems that never
  result in a tie, majority rule is the only one
  that treats all voters equally, treats both
  candidates equally, and is nonmanipulable
• The Nonmanipulability of Condorcets Method
• Condorcets method is nonmanipulable in the sense
  that a voter can never unilaterally change an
  election result from one candidate to another
  candidate that her or she prefers

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Cordorcets Voting Paradox and Manipulability

• Election 1
• In this example, C is the Condorcet winner

• Election 2
• In this example, there is no Condorcet winner at
  all

Voter 1 Voter 2 Voter 3 A B
C C C A
B A B
Voter 1 Voter 2 Voter 3 A B
C B C A
C A B
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Back to the Borda Count

• The Nonmanipulability of the Borda Count with
  exactly 3 candidates
• With exactly 3 candidates, the Borda count cannot
  be manipulated in the sense of a voter
  unilaterally changing an election outcome from
  one single winner to another single winner that
  he prefers
• Why?
• Imagine B is the Borda winner, but you prefer A.
  Consider 3 cases
• A gt B gt C
• C gt A gt B
• A gt C gt B

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• The Manipulability of the Borda Count with Four
  or More Candidates
• With four or more candidates and two or more
  voters, the Borda count can be manipulated in the
  sense that there exists an election in which a
  voter can unilaterally change the election
  outcome from one single winner to another single
  winner that he prefers
• Weve covered the example of 4 candidates and 2
  voters.
• 1) Any candidates in addition to the 4 can be
  placed below those on every ballot
• 2) The rest of the voters can be paired off with
  the members of each pair holding ballots that
  rank the candidates in exactly opposite orders

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Sequential Pairwise Voting

• Assume we are able to set the order.
• Choose the winner, and place the candidate last
• Look for the others that would beat that
  candidate one on one.
• Using this, we can arrange for any of the
  candidates to be the winner.

Voter 1 Voter 2 Voter 3 A C
B B A D D
B C C D A
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Plurality Voting and Group Manipulability

• Plurality voting cannot be manipulated by a
  single individual. However, it is group
  manipulable in the sense that there are elections
  in which a group of voters can change their
  ballots so that the new winner is preferred to
  the old winner by everyone in the group
• Real-world election third party candidate acts
  as a spoiler

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Impossibility the G-S Theorem

• Cordorcets theorm
• 1) Elections never result in ties
• 2) Satisfies the Pareto condition
• 3) Is nonmanipulable
• 4) Isnt a dictatorship
• Can we extend this so that there is always a
  winner??
• The Gibbard- Satterthwaite Theorem With three or
  more candidates and any number of voters, there
  doesnt exist a voting system that always
  produces a winner, never has ties, satisfies the
  Pareto condition, is nonmanipulable, and is not a
  dictatorship.
• for proof click here
• Weaker extension Any voting system for 3
  candidates that agrees with Condorcets Method
  whenever there is a winner is manipulable.

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The Chairs Paradox

• The fact that with three voters and three
  candidates, the voter with tie-breaking power
  (the chair) can, if all 3 voters act rationally
  in their own self-interest, end up with her or
  his least-preferred candidate as the election
  winner
• Each voter gets to vote for one of the
  candidates. If a candidate gets 2 or more, he or
  she wins. If each candidate receives one vote,
  then whichever person the chair voted for wins.
• Each voter will choose the best strategy given
  what the others might do.

Chair You Me A B
C B C A C A
B
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The Chairs Paradox
Chair You Me A B
C B C A C A
B

• The chair will vote for A. Me will vote for C.
  You will also vote for C.

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Chapter 11 Outline

• Introduction and definitions
• The Shapley- Shubik Power Index
• 3 voters, 4 voters, a committee
• The Banzhaf Power Index
• Critical voters, winning blocking, combinations
• Comparing Voting Systems
• 3 voters, using minimal winning coalitions

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Introduction and Definitions

• Weighted voting system a voting system in which
  each participant is assigned a voting weight . A
  quota is specified, and if the sum of the voting
  weights of the voters supporting a motion is at
  least the quota, the motion is approved
• Weight the number of votes assigned to a voter
• Quota the minimum number of votes necessary to
  pass a measure in a weighted voting system
• Notation for Weighted voting systems
• q W1, W2, , Wn where there are n voters, q
  quota, and voting weights W1, W2, , Wn

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Introduction and Definitions

• Dictator a participant who can pass or block any
  issue even if all other voters oppose it
• 10 7, 13
• Dummy Voter a participant who has no power, is
  never critical, and is never the pivotal voter
• 8 5, 3, 1
• Veto power had by a voter if no issue can pass
  without his vote. (a voter with veto power is a
  one-person blocking coalition)
• 6 5, 3, 1 or 8 5, 3, 1
• Power index a numerical measure of an individual
  voters ability to influence a decision the
  individuals voting power

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The Shapley-Shubik Power Index

• 1954- Lloyd Shapley and Martin Shubik
• This index is defined in terms of permutations (a
  permutation of voters in an ordering of all of
  the voters in a voting system)
• 1) Voters are ordered in accordance with their
  commitment to an issue (from most favorable to
  those most against)
• 2)The first voter in a permutation who, when
  joined by those coming before her, would have
  enough voting weight to win is the pivotal voter
  in that particular permutation.
• Examples animal rights, environmentalism

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The Shapley-Shubik Power Index

• This power index is computed by
• 1) counting the number of permutations in which
  that voter is pivotal
• 2) divide this number by the total number of
  possible permutations
• If there are n voters, the total number of
  possible permutations is n!
• Example 6 5, 3, 1.
• Result A 4/6, B,C 1/6

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How to compute the S-S Power Index

• If all the voters have the same voting weight,
  then each has the same share of power.
• If all but one of two voters have equal power, we
  can still easily calculate the S-S power index
• Example 7-person committee with the voting
  system 5 3, 1, 1, 1, 1, 1, 1
• CMMMMMM
• MCMMMMM
• MMCMMMM
• MMMCMMM
• MMMMCMM
• MMMMMCM
• MMMMMMC

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How to compute the S-S Power Index

• The chair is the pivotal voter 3 of the 7 times,
  so his S-S power index is 3/7.
• The remaining 4/7 is split among the six other
  voters (since all have the same weight), so each
  has (4/7)/6 2/21 as their S-S power index.

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The Banzhaf Power Index

• Based on the count of coalitions in which a voter
  is critical
• Coalition a set of voters who are prepared to
  vote for, or to oppose, a motion.
• Winning coalition favors the motion has enough
  votes to pass it
• Blocking coalition opposes the motion has
  enough votes to block it
• Losing coalition set of voters that does not
  have the votes to have its way
• Critical voter a member of the winning (or
  blocking) coalition whose vote is essential for
  the coalition to win (or block) a measure

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The Banzhaf Power Index

• To determine the B. Power index of voter A, count
  all the possible winning and blocking coalitions
  of which A is a member and casts a critical vote
• The weight of a winning coalition must be great
  than or equal to q (where q is the quota)
• The weight of a blocking coalition must be big
  enough to block the yes voters the q votes they
  need to win. So it must be at least n-q1 (where
  n is number of voters)
• Extra Votes Principle
• A winning coalition with total weight w has w-q
  extra votes. A blocking coalition with weight w
  has w-(n-q 1) extra votes. The critical voters
  are those whose weight is more than the
  coalitions extra votes. These are the voters the
  coalition cant afford to lose.

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Calculating the Banzhaf Index
Win. Coalit. Weight Extra votes A (c.v) B (c.v.) C (c.v)
A,B 3 0 1 1 0
A,C 3 0 1 0 1
A,B,C 4 1 1 0 0
Totals 3 1 1

• Take the voting system 32,1,1
• Winning coalition- have a weight of
• 3 or 4
• A has 3 critical votes,
• B and C both have1

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Calculating the Banzhaf Index
Block. Weight Extra A (c.v) B (c.v) C (c.v)
A 2 0 1 0 0
B,C 2 0 0 1 1
A,B 3 1 1 0 0
A,C 3 1 1 0 0
A,B,C 4 2 0 0 0
Totals 3 1 1
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Calculating the Banzhaf Index

• In the blocking coalitions, A is critical in 3
  and B and C are both critical in 1 each
• So, taking the blocking coalitions and winning
  coalitions together,
• A has an index of 6
• B has an index of 2
• C has an index of 2

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Comparing Voting Systems

• Two voting systems are equivalent if there is a
  way for all of the voters of the first system to
  exchange places with the voters of the second
  system and preserve all winning coalitions.
• 50 49, 1 and 4 3, 3 - unanimous support
• 2 2, 1 and 5 3, 6 dictator
• Every 2-voter system is equivalent to a system
  with a dictator or one that needs consensus
• Minimal winning coalition a winning coalition in
  which each voter is a critical voter

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Minimal Winning Coalitions

• Take the voting system 6 5, 3, 1 where the
  respective voters are A, B, C.
• The 3 winning coalitions are A,B, A,C and
  A,B,C.
• Which coalitions are minimal?
• Only A,B and A,C, but not A,B,C since only
  A is critical

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Minimal Winning Coalitions

• Instead of using weights and quotas to describe a
  voting system, one can describe it by using its
  minimal winning coalitions.
• The following conditions must be satisfied
• 1) The list cant be empty (otherwise there is no
  way to approve a motion)
• 2) There cant be one minimal coalition that
  contains another one
• 3)Every pair of coalitions in the list must
  overlap- otherwise two opposing motions could
  pass.

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3-Voter Systems Minimal Winning Coalitions

• Make a list of all voting systems with 3 voters
• The 3 voters are A, B, C
• 1) Suppose the M.W.C is A
• Dictatorship
• 2) Suppose the M.W.C is A,B,C
• Consensus rule
• 3) Suppose the M.W.C. is A,B
• A clique where C is the dummy voter
• 4) Suppose the M.W.C. are A,B and A,C
• A has veto power- the chair veto
• All 2-member coalitions are M.W.C
• Majority rule

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3-Voter Systems Minimal Winning Coalitions
System Min. W. Coaltions Weights Banzhaf Index
Dictator A 3 3, 1, 1 (8, 0, 0)
Clique A, B 4 2, 2, 1 (4, 4, 0)
Majority A,B A,C, B,C 2 1, 1, 1 (4, 4, 4)
Chair Veto A,B A, C 3 2, 1, 1 (6, 2, 2)
Consensus A, B, C 3 1, 1, 1 (2, 2, 2)
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Discussion

• Chapter 10
• Where do we see manipulation of voting systems?
• Are there any political elections that stand out
  in your mind?
• Chapter 11
• What are some applications of weighted voting
  systems?
• How would you describe a jury as a weighted
  voting system?
• What might advantages/disadvantages of certain
  types of weighted voting systems?
• Homework
• Chapter 10 pg 387 9
• Chapter 11 pg 425 7