VOTING SYSTEMS

1
VOTING SYSTEMS

• Section 2.5

2
Introduction

• The rights and duties of a citizen are captured
  in a simple one word mantra VOTE!
• In this section we discuss different types of
  ways to count the votes.
• Counting the votes is the heart of democracy
• Is the counting fair? How well does the counting
  work? These are the questions of Voting Theory.
• Not to disappoint you but a mathematical
  economist by the name of Kenneth Arrow has a
  theorem called Arrows Impossibility Theorem.
• The Arrow Theorem states that a method for
  determining election results that is democratic
  and always fair is a mathematical impossibility.
• Donal Saari a Mathematican and author of Chaotic
  Elections! A Mathematician Looks at Voting, has
  said For a price I will serve as a consultant
  to your group for your next election. Tell me who
  you want to win and I will devise a 'democratic
  process' which ensures the election of your
  candidate.

3
Preference Ballots

• The voters of a club, or the UN Security Council
  or the IOC, use ballots to express the opinions
  of the constituency.
• A preference ballot is a ballot that ranks the
  choices or candidates by preference.

4
Preference Schedule

• When we tabulate the ballots, we put the results
  into a preference schedule. The schedule below is
  a vote for favorite USS Enterprise Captain on the
  Star Trek TV shows and movies.

5
Reading the preference schedule

• The total number of ballots cast are
  141084137.
• We see that Kirk received 14 first place votes.
• Archer received 11 first place votes.
• Pike received 8 first place votes, and Piccard
  got 4 first place votes.
• Since James T. Kirk received the most first place
  votes we can declare him the winner.
• This is an example of using the plurality method
  to determine the winner of an election.

6
Majority vs. Plurality

• Winning an election with a majority is when a
  candidate receives 501 first place votes or
  more.
• The plurality method says that the candidate with
  the most first place votes is the winner.
• In the preference schedule from before, we saw
  that there were a total of 37 votes.
• A candidate winning with a majority would need 19
  first place votes.
• None of the candidates received a majority.
• Kirk won the election with a plurality of 14
  votes.
• President Clinton never won a majority of the
  popular vote in his two elections. The only
  President not to do so.

7
The problems with Plurality

• The plurality method has many flaws and is
  considered a poor method for choosing a winner of
  an election.
• The Condorcet Criterion states that if there is a
  candidate in head-to-head comparisons is
  preferred by the voters over each of the other
  choices, then that choice should be the winner.
• The plurality method violates this criterion.
• In the previous example, 14 voters preferred
  Kirk, while 23 voters preferred someone other
  than Kirk.

8
Pairwise Comparison Method (Copelands Method)

• The method of pairwise comparison is like a
  round-robin tournament. Each candidate is matched
  head-to-head with every other candidate.
• If a candidate wins a head-to-head matchup, they
  receive 1 point. If the matchup is tied each
  candidate gets ½ a point.

9
How many matchups???? Who wins???

• To determine the correct number of matchups, use
  the combination formula .
• We have four candidates in the election and 2
  candidates in a matchup, so use 4 choose 2. Which
  is 6.
• Here are the matchups
• Kirk vs. Piccard 23 voters prefer Piccard, 14
  prefer Kirk thus Piccard gets the head-to-head
  victory. One point for Piccard.
• Kirk vs. Pike 23 voters prefer Pike, 14 prefer
  Kirk, thus Pike gets the victory and the point.
• Kirk vs. Archer 23 voters prefer Archer, 14
  prefer Kirk, Archer wins the matchup and gets a
  point.
• Piccard vs. Pike 28 voters prefer Piccard, 9
  prefer Pike, Piccard gets the point. Piccard has
  two points.
• Piccard vs. Archer 19 voters preferred Archer,
  only 18 for Piccard, thus Archer gets the point
  and she now has two.
• Pike vs. Archer 25 voters preferred Archer, only
  12 for Pike, Archer gets a point and he now has
  three points.
• Archer got 3 points, Piccard 2 points, Pike 1
  point, Kirk 0 points. Jonathan Archer is your
  winner.

10
Whats wrong with Pairwise Comparisons

• Pairwise Comparisons violate a criterion known
  as Independence-of-irrelevant-Alternatives.
• What this means is that if choice X wins an
  election, and one or more other choices are
  disqualified, then when the ballots are
  recounted, choice X should still win the election.

11
Borda Count Method

• Each voter ranks all of the candidates. If there
  are k candidates, each candidate receives k
  points for 1st choice, k 1 points for second
  choice, and so on. The candidate with the most
  points is declared the winner.
• The Borda Count method takes into account all the
  information from the ballot and it produces the
  best compromise candidate. The problem with Borda
  Count is that it can violate the majority
  criterion. That is a candidate could have the
  majority of the 1st place votes but lose with the
  Borda Count.

12
Borda Count Example

• Referring to the preference schedule slide, we
  see that there are 4 candidates in the election.
  First choice receives 4 points, second choice
  receives 3 points, third choice receives 2
  points, and fourth choice receives 1 point.
• Kirk received 14 first choice votes and 23 fourth
  choice votes for a Borda Count total of (14 x 4)
  (0 x 3) (0 x 2) (23 x 1) 79 points
• Archer received 11 first choice votes, 8 second
  choice votes and 18 third choice votes for a
  Borda Count total of (11 x 4) (8 x 3) (18 x
  2) (0 x 1) 44 24 36 104 points.
• Pikes Borda Count is (8 x 4) (5 x 3) (10 x
  2) (14 x 1) 32 15 20 14 81 points.
• Piccards Borda Count is (4 x 4) (24 x 3) (9
  x 2) 16 72 18 106 points.
• Jean-Luc Piccard wins the election in the Borda
  Count Method.

13
Borda Count Example 2

• This example shows how Borda Count violates the
  majority criterion.

14
Example 2 Continued

• Since there are 11 voters in the preference
  schedule, 6 votes are needed for a majority.
• Candidate Foster has received 6 votes. Under a
  majority rules, Foster should be declared the
  winner.
• However if Borda Count is used we see the
  following
• Foster gets (6 x 4) (5 x 1) 29 points
• Winkel gets (2 x 4) (9 x 3) 35 points
• Heerey gets (3 x 4) (2 x 3) (6 x 2) 30
  points
• LeVarge gets (3 x 3) (2 x 2) (6 x 1) 16
  points
• Under the Borda Count rule, Winkel would be
  declared the winner.

15
Ranked-Choice or Instant Run-Off Method

• Each voter ranks all of the candidates first,
  second, third, etc. If a candidate receives a
  majority of the first choice votes, that
  candidate is declared the winner.
• If no candidate receives a majority, then the
  candidate that received the fewest first choice
  votes is eliminated. Those votes are given to the
  next preferred candidate.
• If a candidate now has a majority, that candidate
  is declared the winner. If no candidate has a
  majority, then the process continues.

16
Instant Run-Off Example

• Referring back to our preference schedule, we see
  that Kirk has 14 first choice votes, Archer has
  11 first choice, Pike has 8 first choice votes,
  and Piccard got 4 first choice votes.
• Since Piccard got only 4 first choice votes he is
  now eliminated.
• Those candidates below Piccard now move up in the
  preference schedule.

17
Instant Run-Off Example

• Now Kirk has 14 first choice votes, Pike has 12
  first choice votes, and Archer has 11 first
  choice votes.
• Again no candidate has a majority so Archer is
  eliminated.
• The candidates below Archer move up in the
  preference schedule.

18
Instant Run-Off Example

• Now we see that Kirk only has 14 first choice
  votes and Pike has 23 first choice votes.
• Christopher Pike is declared the winner of this
  election by the instant run-off method.
• The problem with Instant run-off is that it
  violates the Monotonicity criterion.
• The criterion states that if choice X is the
  winner of an election, and in a reelection, the
  only changes in the ballots are changes that only
  favor X, then X should remain the winner.

19
Section 2.5 12

• Four candidates Harrison (H), Lennon (L),
  McCartney (M) and Starr (S) are running for
  regional manager. After the polls close the
  ranked ballots are tallied, the preference table
  is given to the right.

20
Questions

• How many votes were cast?
• Use the Plurality method to determine the winner.
• What percent of the votes did the winner in
  number 2 receive?
• Use instant run-off method to determine the
  winner.
• What percent of the votes did the winner in
  number 4 receive?
• Use Borda Count to determine the winner.
• How many points did the winner in number 6
  receive?
• Use pairwise comparison to determine the winner.
• How many points did the winner receive in number
  8?