Warm-Up

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Warm-Up

• Rank the following soft drinks according to your
  preference (1 being the soft drink you like best
  and 4 being the one you like least)
• Dr. Pepper
• Pepsi
• Mt. Dew
• Sprite

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Election Theory

• Basics of Election Theory

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How do we elect officials?

• Sometimes it is necessary to rank candidates
  instead of selecting a single candidate.
• We can summarize votes into a preference schedule.

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Preference Ballot a ballot in which voters are
asked to rank the candidates in order
There are 37 ballots, therefore 37 people voted
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Preference Schedule a table that organizes the
ballots
Number of voters 14 10 8 4 1
1st choice A C D B C
2nd choice B B C D D
3rd choice C D B C B
4th choice D A A A A
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The Methods

1. Plurality
2. Borda Count
3. Pairwise Comparison (Copeland)
4. Plurality with Elimination (Hare)
5. Approval
6. Sequential Pairwise

• You will work with a group to prepare a lesson on
  your method.
• Must include
• Explanation
• Example done for the class
• Example for the class to do that you will go
  over.
• What fairness criteria is broken?

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The Mathematics of Voting

• Majority
• The candidate with a more than half the votes
  should be the winner.
• Majority candidate
• The candidate with the majority of 1st place
  votes .

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The Plurality Method if X has the most
first-place votes, then X is the winner. X does
not have to have a majority of 1st place votes.
A is the winner with 14 votes
R is the winner with 49 votes
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Example
10 6 5 4 2
1st A B B C D
2nd C D C A C
3rd B C A D B
4th D A D B A

• How many candidates?
• 4
• 2) How many people voted?
• 27
• 3) Which candidate has the most first-place
  votes? Is it a majority or plurality?
• B, Plurality

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The Mathematics of Voting

• In the Borda Count Method each place on a ballot
  is assigned points. In an election with N
  candidates we give 1 point for last place, 2
  points for second from last place, and so on.

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The Mathematics of Voting

• Borda Count Method
• At the top of the ballot, a first-place vote is
  worth N points. The points are tallied for each
  candidate separately, and the candidate with the
  highest total is the winner. We call such a
  candidate the Borda winner.

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The Mathematics of Voting

• Borda Count Method

A gets 4(14)1(10)1(8)1(4)1(1) 56
10 8 4 1 81 points B gets
3(14)3(10)2(8)4(4)2(1) 42 30
16 16 2 106 pointsC gets
2(14)4(10)3(8)2(4)4(1) 28
40 24 8 4 104 pointsD gets
1(14)2(10)4(8)3(4)3(1) 14 20
32 12 3 81 points
B is the winner!!!
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The Plurality-with Elimination Method (Hare)

• Steps
• 1) Count the first place votes for each
  candidate. If a candidate has a majority of the
  first-place votes, that candidate is the winner.
  
• 2) If there isnt a candidate that has the
  majority of votes then, Cross out the candidate
  (or candidates if there is a tie) with the fewest
  first-place votes
• 3) Move other candidates up and count the number
  of the first-place votes again. If a candidate
  has a majority votes, that candidate is the
  winner. Otherwise, continue the process of
  crossing names and counting the first-place votes.

37 people voted so the majority would need 19
votes
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Example 1 37 VOTERS, need 19 votes for majority
winner
The Plurality-with Elimination Method
Number of voters 14 10 8 4 1
1st choice A C D B C
2nd choice B B C D D
3rd choice C D B C B
4th choice D A A A A
Step 1 No one receives 19 votes, so eliminate B
and rewrite the table
Number of voters 14 10 8 4 1
1st choice A C D D C
2nd choice C D C C D
3rd choice D A A A A
4th choice
Number of voters 14 10 8 4 1
1st choice A D D D D
2nd choice D A A A A
Step 3 D has 23 votes so D is the winner
Step 2 No one with 19 votes yet, so eliminate C
and re-write the table
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The Mathematics of Voting

• The Method of Pairwise Comparisons
• (Copeland)
• In a pairwise comparison between between X and Y
  every vote is assigned to either X or Y, the vote
  got in to whichever of the two candidates is
  listed higher on the ballot. The winner is the
  one with the most votes if the two candidates
  split the votes equally, it ends in a tie.

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The Mathematics of Voting

• The Method of Pairwise Comparisons
• The winner of the pairwise comparison gets 1
  point and the loser gets none in case of a tie
  each candidate gets ½ point. The winner of the
  election is the candidate with the most points
  after all the pairwise comparisons are tabulate.

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The Mathematics of Voting

• The Method of Pairwise Comparisons
• There are 10 possible pairwise comparisons
• A vs. B, A vs. C, A vs. D, A vs. E, B vs. C,
• B vs. D, B vs. E, C vs. D, C vs. E, D vs. E

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The Mathematics of Voting

• The Method of Pairwise Comparisons
• A vs. B B wins 15-7. B gets 1 point. A vs.
  C A wins 16-6. A gets 1 point. etc.
• Final Tally A-3, B-2.5, C-2, D-1.5, E-1. A wins.

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• Sequential Pairwise Voting
• Sequential pairwise voting starts with an agenda
  and pits the first candidate against the second
  in a one-on-one contest.
• The loser is deleted and the winner then moves on
  to confront the third candidate in the list, one
  on one.
• This process continues throughout the entire
  agenda, and the one remaining at the end wins.
• Example Who would be the winner using the
  agenda A, B, C, D for the following preference
  list ballots of three voters?

Using the agenda A, B, C, D, start with A vs. B
and record (with tally marks) who is preferred
for each ballot list (column).
Rank Number of Voters (3) Number of Voters (3) Number of Voters (3)
First A C B
Second B A D
Third D B C
Fourth C D A
A vs. B II I
A vs. C I II
C vs. D I II
Candidate D wins for this agenda.
A wins B is deleted.
C wins A is deleted.
D wins C is deleted.
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• Approval Voting
• Under approval voting, each voter is allowed to
  give one vote to as many of the candidates as he
  or she finds acceptable.
• No limit is set on the number of candidates for
  whom an individual can vote however, preferences
  cannot be expressed.
• Voters show disapproval of other candidates
  simply by not voting for them.

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• Approval Voting (cont)
• The winner under approval voting is the candidate
  who receives the largest number of approval
  votes.
• This approach is also appropriate in situations
  where more than one candidate can win,
• EX in electing new members to an exclusive
  society such as the National Academy of Sciences
  or the Baseball Hall of Fame.
• Approval voting is also used to elect the
  secretary general of the United Nations.
• Approval voting was proposed independently by
  several analysts in 1970s.

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First basic fairness criterion

• The Majority Criterion if X has the majority of
  the first-place votes (more than half), then X is
  the winner.
• The plurality method satisfies the majority
  criterion.

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Second basic fairness criterion

• The Condorcet Criterion was introduced in 1785 by
  the French mathematician Le Marquis de Condorcet
• If candidate X is preferred over other candidates
  in a head-to-head comparison, then X is the
  winner
• If X is the winner under the Majority Criterion,
  then X is also the Condorcet winner.

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Third basic fairness criterion

• If a candidate is winning votes are changed in
  FAVOR of the winner
• Monotonicity Criterion If votes are changed in
  favor of the winning candidate, the winner should
  not change.

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• Monotonicity (The Hare system fails
  monotonicity.)
• Monotonicity says that if a candidate is a winner
  and a new election is held in which the only
  ballot change made is for some voter to move the
  former winning candidate higher on his or her
  ballot, then the original winner should remain a
  winner.
• In a new election, if a voter moves a winner
  higher up on his preference list, the outcome
  should still have the same winner.

Number of Voters (13) Number of Voters (13) Number of Voters (13) Number of Voters (13)
Rank 5 4 3 1
First A C B A
Second B B C B
Third C A A C
In this example, A won because A has the
most 1st place votes. Round 1 B is deleted
with Hare method because B has the fewest 1st
place votes. Round 2 C moves up to replace B
on the third column. However, C wins because
now has the most 1st place votesthis is a
glaring defect!
Number of Voters (13) Number of Voters (13) Number of Voters (13) Number of Voters (13)
Rank 5 4 3 1
First A C C A
Second C A A C
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Fourth basic fairness criterion

• What if a non-winning candidate drops out?
• Independence-of-Irrelevant Alternatives
  Criterion If a non-winning candidate drops out,
  or is disqualified, the winner should not change.

Original Borda Score A11, B10, C9
Rank Number of Voters (5) Number of Voters (5) Number of Voters (5) Number of Voters (5) Number of Voters (5)
First (3pts) A A A C C
Second (2 pts) B B B B B
Third (1 pt) C C C A A
New Borda Score A 11, B12, C8
Rank Number of Voters (5) Number of Voters (5) Number of Voters (5) Number of Voters (5) Number of Voters (5)
First (3 pts) A A A B B
Second (2pts) B B B C C
Third (1 pt) C C C A A
Suppose the last two voters change their
ballots (reverse the order of just the losers).
This should not change the winner.
B went from loser to winner and did not
switch with A!
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Summary
Majority Condercet Monotonicity Independence
Plurality Yes No Yes No
Borda Count No No Yes No
Plurality Elimination Yes No No No
Pairwise Comparison Yes Yes Yes No