A Mathematical View of Our World

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A Mathematical View of Our World

• 1st ed.
• Parks, Musser, Trimpe, Maurer, and Maurer

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Chapter 3

• Voting and Elections

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Section 3.1Voting Systems

• Goals
• Study voting systems
• Plurality method
• Borda count method
• Plurality with elimination method
• Pairwise comparison method
• Discuss tie-breaking methods

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3.1 Initial Problem

• The city council must select among 3 locations
  for a new sewage treatment plant.
• A majority of city councilors say they prefer
  site A to site B.
• A majority of city councilors say they prefer
  site A to site C.
• In the vote site B is selected.
• Did the councilors necessarily lie about their
  preferences before the election?
• The solution will be given at the end of the
  section.

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Voting Systems

• The following voting methods will be discussed
• Plurality method
• Borda count method
• Plurality with elimination method
• Pairwise comparison method

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Plurality Method

• When a candidate receives more than half of the
  votes in an election, we say the candidate has
  received a majority of the votes.
• When a candidate receives the greatest number of
  votes in an election, but not more than half, we
  say the candidate has received a plurality of the
  votes.

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Question

• Suppose in an election, the vote totals are as
  follows. Andy gets 4526 first-place votes. Lacy
  gets 1901 first-place votes. Peter gets 2265
  first-place votes.
• Choose the correct statement.
• a. Andy has a majority.
• b. Andy has a plurality only.

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Plurality Method, contd

• In the plurality method
• Voters vote for one candidate.
• The candidate receiving the most votes wins.
• This method has a couple advantages
• The voter chooses only one candidate.
• The winner is easily determined.

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Plurality Method, contd

• The plurality method is used
• In the United States to elect senators,
  representatives, governors, judges, and mayors.
• In the United Kingdom and Canada to elect members
  of parliament.

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Example 1

• Four persons are running for student body
  president. The vote totals are as follows
• Aaron 2359 votes
• Bonnie 2457 votes
• Charles 2554 votes
• Dion 2288 votes
• Under the plurality method, who won the election?

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Example 1, contd

• Solution With 2554 votes, Charles has a
  plurality and wins the election.
• Note that there were a total of 9658 votes cast.
  
• A majority of votes would be at least 4830 votes.
  Charles did not receive a majority of votes.

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Example 2

• Three candidates ran for Attorney General in
  Delaware in 2002. The vote totals were as
  follows
• Carl Schnee 103,913 votes
• Jane Brady 110,784
• Vivian Houghton 13,860
• What percent of the votes did each candidate
  receive and who won the election?

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Example 2, contd

• Solution A total of 228,557 votes were cast.
• Schnee received
• Brady received
• Houghton received
• Brady received a plurality and is the winner.

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Borda Count Method

• In the Borda count method
• Voters rank all of the m candidates.
• Votes are counted as follows
• A voters last choice gets 1 point.
• A voters next-to-last choice gets 2 points.
• A voters first choice gets m points.
• The candidate with the most points wins.

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Borda Count Method, contd

• The main advantage of the Borda count method is
  that it uses more information from the voters.
• A variation of the Borda count method is used to
  select the winner of the Heisman trophy.

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Example 3

• Four persons are running for student body
  president. Voters rank the candidates as shown
  in the table below.
• Under the Borda count method, who is elected?

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Example 3, contd

• Solution Convert the votes to points.

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Example 3, contd

• Solution Total the points for each person
• Aaron 9436 4104 5572 3145 22,257
• Bonnie 9828 10,497 4948 1228 26,501
• Charles 10,216 7101 3468 3003 23,788
• Dion 9152 7272 5328 2282 24,034
• Bonnie has the most points and is the winner.

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Example 3, contd

• Note that in this same election
• Charles won using the plurality method because he
  had more first place votes than any other
  candidate.
• Bonnie won using the Borda count method because
  her point total was highest, due to having many
  second-place votes.

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Plurality with Elimination Method

• In the plurality with elimination method
• Voters choose one candidate.
• The votes are counted.
• If one candidate receives a majority of the
  votes, that candidate is selected.
• If no candidate receives a majority, eliminate
  the candidate who received the fewest votes and
  do another round of voting.

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Plurality with Elimination, contd

• Contd
• This process is repeated until someone receives a
  majority of the votes and is declared the winner.
• The plurality with elimination method is used
• To select the location of the Olympic games.
• In France to elect the president.

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Plurality with Elimination, contd

• Rather than needing to potentially conduct
  multiple votes, the voters can be asked to rank
  all candidates during the first election.
• A preference table is used to display these
  rankings.

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Example 4

• Four persons are running for department
  chairperson. The 17 voters ranked the candidates
  1st through 4th.
• Under plurality with elimination, who is the
  winner?

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Example 4, contd

• Solution Some voters had the same preference
  ranking. Identical rating have been grouped to
  form the preference table below.
• The number at the top of each column indicates
  the number of voters who shared that ranking.

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Example 4, contd

• Solution, contd The first-place votes for each
  candidate are totaled
• Alice 6 Bob 4 Carlos 4 Donna 3
• No candidate received a majority, 9 votes.
• Donna, who has the fewest first-place votes, is
  eliminated.

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Example 4, contd

• Solution, contd A new preference table, without
  Donna, must be created.
• Donna is eliminated from each column.
• Any candidates ranked below Donna move up.

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Example 4, contd

• Solution, contd The first-place votes for each
  candidate are totaled
• Alice 7 Bob 4 Carlos 6
• No candidate received a majority.
• Bob, who has the fewest first-place votes, is
  eliminated.

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Example 4, contd

• Solution, contd A new preference table, without
  Bob, must be created.
• Bob is eliminated from each column.
• Any candidates ranked below Bob move up.

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Example 4, contd

• Solution, contd The first-place votes for each
  candidate are totaled
• Alice 9 Carlos 8
• Alice received a majority and is the winner.

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Pairwise Comparison Method

• In the pairwise comparison method
• Voters rank all of the candidates.
• For each pair of candidates X and Y, determine
  how many voters prefer X to Y and vice versa.
• If X is preferred to Y more often, X gets 1
  point.
• If Y is preferred to X more often, Y gets 1
  point.
• If the candidates tie, each gets ½ a point.
• The candidate with the most points wins.

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Pairwise Comparison, contd

• The pairwise comparison method is also called the
  Condorcet method.

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Example 5

• Three persons are running for department chair.
  The 17 voters rank all the candidates, as shown
  in the preference table below.
• Under the pairwise comparison method, who wins
  the election?

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Example 5, contd

• Solution There are 3 pairs of candidates to
  compare
• Alice vs. Bob
• Alice vs. Carlos
• Bob vs. Carlos
• For each pair of candidates, delete the third
  candidate from the preference table and consider
  only the two candidates in question.

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Example 5, contd

• Solution, contd

• Alice receives 10 first-place votes, while Bob
  only receives 7.
• We say Alice is preferred to Bob 10 to 7.
• Alice receives one point.

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Example 5, contd

• Solution, contd

• Alice is preferred to Carlos 9 to 8, so Alice
  receives another point.

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Example 5, contd

• Solution, contd

• Carlos is preferred to Bob 10 to 7, so Carlos
  receives one point.

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Example 5, contd

• Solution, contd The final point totals are
• Alice 2 points
• Bob 0 points
• Carlos 1 point
• Alice wins the election.

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Question

• Candidate B is the winner of an election with
  the following preference table .
• What voting method could have been used to
  determine the winner?
• a. Plurality method
• b. Borda count method
• c. Plurality with elimination method
• d. Pairwise comparison method

7 12 4 9 6
1st C B A C A
2nd A A B A B
3rd B C C B C
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Voting Methods, contd

• The four voting systems studied here can produce
  different winners even when the same voter
  preference table is used.
• Any of the four methods can also produce a tie
  between two or more candidates, which must be
  broken somehow.

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Tie Breaking

• A tie-breaking method should be chosen before the
  election.
• To break a tie caused by perfectly balanced voter
  support, election officials may
• Make an arbitrary choice.
• Flipping a coin
• Drawing straws
• Bring in another voter.
• The Vice President votes when the U. S. Senate is
  tied.

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3.1 Initial Problem Solution

• A majority of city councilors said they preferred
  site A to site B and also site A to site C. If B
  won the election, did they necessarily lie?
• Solution
• The councilors would not have to lie in order for
  this to happen. This situation can occur with
  some voting methods.

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Initial Problem Solution, contd

• For example, this situation could occur if the
  voting method used was plurality with
  elimination.
• Suppose 11 councilors ranked the sites as shown
  in the table below.

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Initial Problem Solution, contd

• Notice that in this scenario
• Site A is preferred to site B 7 to 4.
• Site A is preferred to site C 7 to 4.
• However, in the vote count
• Site A, with the fewest first-place votes, is
  eliminated.
• In the second round of voting, site B wins.

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Section 3.2Flaws of the Voting Systems

• Goals
• Study fairness criteria
• The majority criterion
• Head-to-head criterion
• Montonicity criterion
• Irrelevant alternatives criterion
• Study fairness of voting methods
• Arrow impossibility theorem
• Approval voting

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3.2 Initial Problem

• The Compromise of 1850 averted civil war in the
  U.S. for 10 years.
• Henry Clay proposed the bill, but it was defeated
  in July 1850.
• A short time later, Stephen Douglas was able to
  get essentially the same proposals passed.
• How is this possible?
• The solution will be given at the end of the
  section.

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Flaws of Voting Systems

• We have seen that the choice of voting method can
  affect the outcome of an election.
• Each voting method studied can fail to satisfy
  certain criteria that make a voting method fair.

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Fairness Criteria

• The fairness criteria are properties that we
  expect a good voting system to satisfy.
• Four fairness criteria will be studied
• The majority criterion
• The head-to-head criterion
• The monotonicity criterion
• The irrelevant alternatives criterion

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The Majority Criterion

• If a candidate is the first choice of a majority
  of voters, then that candidate should be
  selected.

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Question

• Candidate A won an election with 3000 of the
  8500 votes. Was the majority criterion
  necessarily violated?
• a. yes
• b. no

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The Majority Criterion, contd

• For the majority criterion to be violated
• A candidate must have more than half of the
  votes.
• This same candidate must not win the election.
• Note
• This criterion does not say what should happen if
  no candidate receives a majority.
• This criterion does not say that the winner of an
  election must win by a majority.

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The Majority Criterion, contd

• If a candidate is the first choice of a majority
  of voters, then that candidate will always win
  using
• The plurality method.
• The plurality with elimination method.
• The pairwise comparison method.
• In both of these methods any candidate with more
  than half the vote will always win.

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The Majority Criterion, contd

• If a candidate is the first choice of a majority
  of voters, then that candidate might not win
  using
• The Borda count method.
• The candidate with the most points may not be the
  candidate with the most first-place votes.

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Example 1

• Four cities are being considered for an annual
  trade show. The preferences of the organizers
  are given in the table.

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Example 1, contd

• Which site has a majority of first-place votes?
• Which site wins using the Borda count method?

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Example 1, contd

• Solution There are 9 votes, so a majority would
  be 5 or more votes.
• The first place vote totals are
• Chicago 5 Seattle 3 Phoenix 1 Boston 0
• Chicago has a majority of first-place votes.

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Example 1, contd

• Solution, contd Find the point totals for Borda
  count.
• The points are calculated as follows
• Chicago 5(4) 0(3) 2(2) 2(1) 26
• Seattle 3(4) 4(3) 2(2) 0(1) 28
• Phoenix 1(4) 2(3) 2(2) 4(1) 18
• Boston 0(4) 3(3) 3(2) 3(1) 18
• Under the Borda count method, Seattle is the
  winner.

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Example 1, contd

• Note that Chicago had a majority of first-place
  votes, but under the Borda count method Seattle
  was the winner.
• This is an example of the Borda count method
  failing the majority criterion. In this case, we
  would say that the Borda count method was unfair.

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The Head-to-Head Criterion

• If a candidate is favored when compared
  separately with each of the other candidates,
  then the favored candidate should win the
  election.
• This is also called the Condorcet criterion.

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Head-to-Head Criterion, contd

• For the head-to-head criterion to be violated
• A candidate must be preferred pairwise to every
  other candidate.
• This same candidate must not win the election.
• Note
• This criterion does not say what should happen if
  no candidate is preferred pairwise to every other
  candidate.

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Head-to-Head Criterion, contd

• If a candidate is favored pairwise to every other
  candidate, then that candidate will always win
  using
• The pairwise comparison method.
• This candidate will earn the most points from the
  pairwise comparisons.

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Head-to-Head Criterion, contd

• If a candidate is favored pairwise to every other
  candidate, then that candidate might not win
  using
• The plurality method.
• The plurality with elimination method.
• The Borda count method.

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Example 2

• Seven people are choosing an option for a
  retirement party catering, picnic, or
  restaurant. The preferences are shown in the
  table below.
• Which site is selected using the plurality
  method?
• Show that the head-to-head criterion is violated.

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Example 2, contd

• Solution
• The picnic has the most votes, 3, so it is the
  winning option under the plurality method.

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Example 2, contd

• Solution, contd
• The pairwise comparisons are made
• R is preferred to P 4 to 3.
• R is preferred to C 5 to 2.
• R is preferred separately to every other
  candidate, but R did not win the election. This
  is a violation of the head-to-head criterion.

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The Monotonicity Criterion

• Suppose a particular candidate, X, wins an
  election.
• If, hypothetically, this election were redone and
  the only changes were that some voters switched X
  with the candidate they had ranked one higher,
  then X should still win.
• This criterion is only used in special cases.

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Monotonicity Criterion, contd

• The monotonicity criterion is always satisfied
  by
• The plurality method.
• The Borda count method.
• The pairwise comparison method.

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Monotonicity Criterion, contd

• The monotonicity criterion is not always
  satisfied by
• The plurality with elimination method.

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Example 3

• Teachers are voting for a union president from
  the candidates Akst, Bailey, and Chung. The
  preferences are shown in the table below.

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Example 3, contd

• Who will win using the plurality with elimination
  method?
• Solution The first-place vote totals are
• Akst 14 Bailey 12 Chung 15.
• No candidate received a majority of at least 21
  votes, so Bailey is eliminated.

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Example 3, contd

• Solution, contd
• After Baileys elimination, a new preference
  table is created.
• Akst now has 26 first-place votes, and is the
  winner.

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Example 3, contd

• If 4 of the 5 teachers who ranked the candidates
  CAB changed to a ranking of ACB, would this
  affect the outcome of the election?
• Solution Akst won the first election and the
  only changes now are that 4 teachers moved him
  from 2nd to 1st place.

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Example 3, contd

• Solution, contd The new preference table is
  shown below.

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Example 3, contd

• Solution, contd The first-place vote totals
  are
• Akst 18 Bailey 12 Chung 11.
• No candidate received a majority of at least 21
  votes, so Chung is eliminated.

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Example 3, contd

• Solution, contd
• After Chungs elimination, a new preference table
  is created.
• Bailey now has 22 votes, and is the winner.

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Example 3, contd

• Solution, contd
• The only changes in the preference table were
  ones that favored Akst, who won the first
  election.
• However, Akst ended up losing the modified vote
  to Bailey.
• This is a violation of the monotonicity criterion.

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The Irrelevant Alternatives Criterion

• Suppose a candidate, X, is selected in an
  election.
• If, hypothetically, this election were redone
  with one or more of the unselected candidates
  removed from the vote, then X should still win.

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Irrelevant Alternatives Criterion, contd

• The irrelevant alternatives criterion is not
  always satisfied by any of the 4 voting methods
  studied.

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Example 4

• The 5 members of a book club are voting on what
  book to read next a mystery, a historical novel,
  or a science fiction fantasy. The preference
  table is shown below.

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Example 4, contd

• Which of the books is selected using the
  plurality with elimination method?
• Solution The first-place vote totals are
• M 2 H 1 S 2.
• No book has a majority, so H is eliminated.

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Example 4, contd

• Solution, contd After H is eliminated a new
  preference table is created.
• Book M receives 3 first-place votes, and is the
  winner. The book club will read the mystery.

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Example 4, contd

• If the science fiction book is removed from the
  list, is the irrelevant alternatives criterion
  violated in the new election?
• Solution A new preference table, without S, is
  created.

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Example 4, contd

• Solution, contd
• In the new table, M has 2 votes and H has 3.
• Book H is the new winner, violating the
  irrelevant alternatives criterion.

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Fairness Criteria, contd
84
Arrow Impossibility Theorem

• The Arrow Impossibility Theorem states that no
  system of voting will always satisfy all of the 4
  fairness criteria.
• This fact was proved by Kenneth Arrow in 1951.

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Question

• The results of an election using the plurality
  method were analyzed. It was found that the
  election did not violate any of the four fairness
  criteria. Does this contradict the Arrow
  Impossibility Theorem?
• a. yes
• b. no

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Approval Voting

• No voting system is always fair, but we can
  explore systems that are unfair less often than
  others. One such system is called approval
  voting.
• In approval voting
• Each voter votes for all candidates he/she
  considers acceptable.
• The candidate with the most votes is selected.

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Example 5

• Three candidates are running for two positions.
  There are 9 voters and the votes are shown in the
  table below.
• Who is the winner under approval voting?

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Example 5, contd

• Solution The vote totals are
• Ammee 6
• Bonnie 7
• Celeste 5
• Bonnie and Ammee are selected for the two
  positions.

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3.2 Initial Problem Solution

• Henry Clay presented the Compromise of 1850 as
  one bill containing all the proposals.
• Of the 60 senators, a majority would not approve
  the bill because they disagreed on individual
  issues within the bill.

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Initial Problem Solution, contd

• Stephen Douglas presented each proposal of the
  Compromise in a separate bill.
• A (different) majority of the senators passed
  each proposal and the Compromise of 1850 went
  into effect, although a majority never approved
  the measures as a whole.

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Section 3.3Weighted Voting Systems

• Goals
• Study weighted voting systems
• Coalitions
• Dummies and dictators
• Veto power
• Study the Banzhaf power index

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3.3 Initial Problem

• A stockholder owns 17 of the shares of a
  company.
• Among the other 3 stockholders, no one owns more
  than 32 of the shares.
• Why will no one listen to the stockholder with
  17?
• The solution will be given at the end of the
  section.

93
Weighted Voting Systems

• In a weighted voting system, an individual voter
  may have more than one vote.
• The number of votes that a voter controls is
  called the weight of the voter.
• An example of a weighted voting system is the
  election of the U.S. President by the Electoral
  College.

94
Weighted Voting Systems, contd

• The weights of the voters are usually listed as a
  sequence of numbers between square brackets.
• For example, the voting system in which Angie has
  a weight of 9, Roberta has a weight of 12, Carlos
  has a weight of 8, and Darrell has a weight of 11
  is represented as 12, 11, 9, 8.

95
Weighted Voting Systems, contd

• The voter with the largest weight is called the
  first voter, written P1.
• The weight of the first voter is represented by
  W1.
• The remaining voters and their weights are
  represented similarly, in order of decreasing
  weights.

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Example 1

• The voting system in which Angie has a weight of
  9, Roberta has a weight of 12, Carlos has a
  weight of 8, and Darrell has a weight of 11 was
  represented as 12, 11, 9, 8.
• In this case, P1 Roberta, P2 Darrell, P3
  Angie, and P4 Carlos.
• Also, W1 12, W2 11, W3 9, and W4 8.

97
Weighted Voting Systems, contd

• Yes or no questions are commonly called motions.
• A final decision of No defeats the motion and
  leaves the status quo unchanged.
• A final decision of Yes passes the motion and
  changes the status quo.

98
Weighted Voting Systems, contd

• A simple majority requirement means that a motion
  must receive more than half of the votes to pass.
• A supermajority requirement means that the
  minimum number of votes required to pass a motion
  is set higher than half of the total weight.
• A common supermajority is two-thirds of the total
  weight.

99
Weighted Voting Systems, contd

• The weight required to pass a motion is called
  the quota.
• Example A simple majority quota for the weighted
  voting system 12, 11, 9, 8 would be 21.
• Half of the total weight is (12 11 9 8)/2
  40/2 20. More than half of the weight would be
  at least 21 Yes votes.

100
Question

• Given the weighted voting system 10, 9, 8, 8,
  5, find the quota for a supermajority
  requirement of two-thirds of the total weight.
• a. 27
• b. 21
• c. 26
• d. 20

101
Weighted Voting Systems, contd

• The quota for a weighted voting system is usually
  added to the list of weights.
• Example For the weighted voting system 12, 11,
  9, 8 with a quota of 21 the complete notation is
  21 12, 11, 9, 8.

102
Example 2

• Given the weighted voting system
• 21 10, 8, 7, 7, 4, 4, suppose P1, P3, and P5
  vote Yes on a motion.
• Is the motion passed or defeated?

103
Example 2, contd

• Solution
• The given voters have a combined weight of 10 7
  4 21.
• The quota is met, so the motion passes.

104
Example 3

• Given the weighted voting system
• 21 10, 8, 7, 7, 4, 4, suppose P1, P5, and P6
  vote Yes on a motion.
• Is the motion passed or defeated?

105
Example 3, contd

• Solution
• The given voters have a combined weight of 10 4
  4 18.
• The quota is not met, so the motion is defeated.

106
Coalitions

• Any nonempty subset of the voters in a weighted
  voting system is called a coalition.
• If the total weight of the voters in a coalition
  is greater than or equal to the quota, it is
  called a winning coalition.
• If the total weight of the voters in a coalition
  is less than the quota, it is called a losing
  coalition.

107
Question

• Given the weighted voting system 27 10, 9, 8,
  8, 5, is the coalition P1, P4, P5 a winning
  coalition or a losing coalition?
• a. winning
• b. losing

108
Example 4

• For the weighted voting system
• 8 6, 5, 4, list all possible coalitions and
  determine whether each is a winning or losing
  coalition.

109
Example 4, contd

• Solution Each coalition and its status is listed
  in the table below.

110
Coalitions, contd

• In a weighted voting system with n voters,
  exactly 2n - 1 coalitions are possible.
• Example
• How many coalitions are possible in a weighted
  voting system with 7 voters?
• Solution The formula tells us there are
• 27 - 1 128 1 127 coalitions.

111
Example 5

• The voting weights of EU members in a council in
  2003 are shown in the table.

112
Example 5, contd

• If resolutions must receive 71 of the votes to
  pass, what is the quota?
• How many coalitions are possible?

113
Example 5, contd

• Solution
• There are 87 votes total. So the quota is 71 of
  87, or approximately 62 votes.
• There are n 15 members, so there are 215 1
  32,767 coalitions possible.

114
Dictators and Dummies

• A voter whose presence or absence in any
  coalition makes no difference in the outcome is
  called a dummy.
• A voter whose presence or absence in any
  coalition completely determines the outcome is
  called a dictator.
• When a weighted voting system has a dictator, the
  other voters in the system are automatically
  dummies.

115
Veto Power

• In between the complete power of a dictator and
  the zero power of a dummy is a level of power
  called veto power.
• A voter with veto power can defeat a motion by
  voting No but cannot necessarily pass a motion
  by voting Yes.
• Any dictator has veto power, but a voter with
  veto power is not necessarily a dictator.

116
Example 6

• Consider the weighted voting system
• 12 7, 6, 4.
• List all the coalitions and determine whether
  each is a winning or losing coalition.
• Are there any dummies or dictators?
• Are there any voters with veto power?

117
Example 6, contd

• Solution
• Each coalition and its status is listed in the
  table below.

118
Example 6, contd

• Solution, contd
• Removing the third voter from any coalition does
  not change the status of the coalition. P3 is a
  dummy.

119
Example 6, contd

• Solution, contd
• No voter has complete power to pass or defeat a
  motion. There is no dictator.

120
Example 6, contd

• Solution, contd
• If P1 is not in a coalition, then it is a losing
  coalition. P1 has veto power.

121
Question

• In the weighted voting system
• 27 10, 9, 8, 8, 5, is P1 a
• a. dictator
• b. dummy
• c. voter with veto power
• d. none of the above

122
Example 7

• Consider the weighted voting system 10 10, 5,
  4.
• Are there any dummies, dictators, or voters with
  veto power?

123
Example 7, contd

• Solution
• P1 has enough weight to pass a motion by voting
  Yes no matter how anyone else votes.
• If P1 votes No, the motion will not pass no
  matter how anyone else votes.
• P1 is a dictator and thus all other voters are
  dummies.

124
Critical Voters

• If a voters weight is large enough so that the
  voter can change a particular winning coalition
  to a losing coalition by leaving the coalition,
  then that voter is called a critical voter in
  that winning coalition.

125
Question

• Given the weighted voting system 27 10, 9, 8,
  8, 5, is the voter P4 a critical voter in the
  winning coalition P1, P2, P4, P5?
• a. yes
• b. no

126
Example 8

• Consider the weighted voting system 21 10, 8,
  7, 7, 4, 4.
• Which voters in the coalition P2, P3, P4, P5
  are critical voters in that coalition?

127
Example 8, contd

• Solution The weight in the winning coalition is
  26.
• If P2 leaves, the weight goes down to
• 26 8 18 lt quota.
• If P3 leaves, the weight goes down to
• 26 7 19 lt quota.

128
Example 8, contd

• Solution contd
• If P4 leaves, the weight goes down to
• 26 7 19 lt quota.
• If P5 leaves, the weight goes down to
• 26 4 22 gt quota.
• The critical voters in this coalition are P2, P3,
  and P4.

129
The Banzhaf Power Index

• The more times a voter is a critical voter in a
  coalition, the more power that voter has in the
  system.
• The Banzhaf power of a voter is the number of
  winning coalitions in which that voter is
  critical.

130
Banzhaf Power Index, contd

• The sum of the Banzhaf powers of all voters is
  called the total Banzhaf power in the weighted
  voting system.
• An individual voters Banzhaf power index is the
  ratio of the voters Banzhaf power to the total
  Banzhaf power in the system.
• The sum of the Banzhaf power indices of all
  voters is 100.

131
Banzhaf Power Index, contd

• An individual voters Banzhaf power index is
  calculated using the following process
• Find all winning coalitions for the system.
• Determine the critical voters for each winning
  coalition.
• Calculate each voters Banzhaf power.
• Find the total Banzhaf power in the system.
• Divide each voters Banzhaf power by the total
  Banzhaf power.

132
Example 9

• For the weighted voting system 18 12, 7,
  6, 5, determine
• The total Banzhaf power in the system.
• The Banzhaf power index of each voter.

133
Example 9, contd

• Solution Step 1 Find all the winning coalitions.

134
Example 9, contd

• Solution Step 2 Determine the critical voters
  for each winning coalition.
• Remove each voter one at a time and check to see
  whether the resulting coalition is still a
  winning coalition.
• This work is shown in the next slides.

135
Example 9, contd
136
Example 9, contd
137
Example 9, contd

• Solution Step 3 Count the number of times each
  voter is a critical voter
• P1 5 times
• P2 3 times
• P3 3 times
• P4 1 time
• Step 4 The total Banzhaf power in the system is
  5 3 3 1 12

138
Example 9, contd

• Solution Step 5 Divide each voters Banzhaf
  power by the total Banzhaf power to find the
  Banzhaf power indices.

139
3.3 Initial Problem Solution

• One of the 4 stockholders owns 17 of the shares
  of a company. Among the other 3, no one owns
  more than 32. Why will no one listen to the
  stockholder with 17?

140
Initial Problem Solution, contd

• We know one stockholder owns 32 and one owns
  17, so the other 51 is split between the
  remaining two stockholders.
• Suppose the other percents are 26 and 25.

141
Initial Problem Solution, contd

• The winning coalitions in this case are
• 32, 26, 25, 17
• 32, 26, 25
• 32, 26, 17
• 32, 25, 17
• 26, 25, 17
• 32, 26
• 32, 25
• 26, 25

142
Initial Problem Solution, contd

• The voter with 17 is in several winning
  coalitions, but removing that voter does not
  cause any of them to become losing coalitions.
• The voter with 17 is not a critical voter in any
  winning coalition.
• The reason no one will listen to the voter is
  that he or she is a dummy.