Discrete Math A Chapter 1, The Mathematics of Voting
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Discrete Mathematics Notes
Chapter 1: The Mathematics of Voting
Academic Standards: DM.5.1 Use election theory techniques to analyze election data
1.1 Preference Ballots and Preference Schedules
Voting on the surface seems really easy:
We hold an election.
We count the ballots.
We determine the outcome in a consistent and fair manner.
In this chapter we will learn:
Methods for collecting and recording votes
Methods for counting votes and ranking candidates
“Fairness Criterion” – what it means to be fair
Arrow’s Impossibility Theorem – discovery that proved there is no method of counting
votes in an election of three or more candidates that is ALWAYS fair.
Ballots:
Elections are expensive  Use a “linear preference ballot”
Example: What dessert do you want after dinner?
1st choice
2nd choice
3rd choice
4th choice
Transitivity: I ran out of cookies and cake.
What should I serve this guest?
Cookies
Pie
Cake
Candy
Holding Elections:
The Math Appreciation Society Officer Election (p.4)
Example:
The Math Appreciation Society is electing its president. The candidates are Alisha (A), Boris (B),
Carmen (C), and Dave. Each of the 37 members votes with a preference ballot. Who should be the
winner? Why?
Step 1: Everyone Votes
Step 2: Collect Ballots that are exactly the same
#of ballots:
Discrete Math A Chapter 1, The Mathematics of Voting
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Step 3: Organize the data neatly in a chart,
called a “preference schedule”
1. How many first place votes did each candidate
receive?
2. Did anyone get a majority of the first place votes?
Majority =
MORE than half of the total first-place votes
3. Who received the plurality of the first-place votes?
Plurality =
the MOST first-place votes, but not a majority
4. What if it came down to a vote between
B and C? Who would win?
1.2 The Plurality Method
Plurality Method:
The Candidate with the most first-place votes wins.
Extension of “majority rules” but no need to have more than 50%
Advantages:
Easy to determine
Easy to understand
Only need 1st choice-- not a ranked ballot
Most political elections in the USA are determined using the
plurality method.
A=
B=
C=
D=
Who wins by this method?
How would you RANK the candidates? (1st, 2nd, etc..)
Discrete Math A Chapter 1, The Mathematics of Voting
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FAIRNESS: Majority Rules
If candidate X has a majority of the first-place votes, then candidate X
should be the winner of the election.
The Majority Criterion:
Does the Math Appreciation Society election have a majority candidate?
So what’s the problem with Plurality?
How many first place votes did our winner get?
Where is our winner on EVERYONE ELSE’s BALLOT?
Example:
The band has the choice to perform at 5 different bowl games: the Rose
Bowl (R), the Hula Bowl (H), the Fiesta Bowl (F), the Orange Bowl (O),
and the Sugar Bowl (S).
Who is the PLURALITY winner?
Is there a better choice? Why?
FAIRNESS: Preference Matters
The Condorcet Criterion:
If candidate X is preferred by the voters over each of the other candidates in a
head-to-head comparison, then candidate X should be the winner of the
election.
One of the Plurality Method’s major flaws is that makes us choose a candidate over a much more reasonable
choice! However, just because the Plurality method MAY violate the Condorcet Criterion doesn’t mean that it
MUST!
Discrete Math A Chapter 1, The Mathematics of Voting
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Insincere Voting (Strategic Voting)
Another flaw of the Plurality Method is that election results can easily be manipulated by insincere voting.
What is insincere voting:
There is no way that our true preference is going to win.
We vote for a lesser choice that has a better chance of winning so we don’t “waste our vote.”
Practice Examples:
1a. How many people voted in this election?
1b. How many votes do you need for a
majority?
1c.
Use the plurality method to find the winner of the election.
1d.
Rank the candidates using the extended plurality method.
Discrete Math A Chapter 1, The Mathematics of Voting
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Exercise #13, pg 31
An election with 4 candidates and 150 voters is to be decided using the plurality method. After 120 ballots
have been recorded, A has 26 votes, B has 18 votes, C has 42 votes, and D has 34 votes.
a.
For A to win the election outright, at least how many of the remaining 30 votes need to have A in
first place? Explain.
b.
For C to win the election outright, at least how many of the remaining 30 votes need to have C in
first place? Explain.
# of voters
1st choice
2nd choice
3rd choice
3c.
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10
A
B
C
15
C
A
B
5
A
C
B
3
B
C
A
3a.
Use the plurality method to find the winner of the election.
3b.
Tie-breaking rule: If there is more than one alternate with a plurality of
the first-place votes, then the tie is broken by choosing the alternate with
the fewest last place votes. Who is the winner?
Tie-breaking rule: If there is more than one alternate with a plurality of
the first-place votes, then the tie is broken by a head-to-head comparison
between the two candidates. Who is the winner?
1.5 The Method of Pairwise Comparisons
Pairwise Comparisons:
A Head-To-Head comparison of candidates.
Whoever is “higher” on the ballot gets all the votes
for that ballot. The winner of a comparison gets 1 point, loser gets 0 points.
In a tied comparison, both candidates earn ½ point. (Ties are common)
A versus B
A versus C
A versus D
B versus C
A=
B=
C=
D=
Who wins by this method?
B versus D
C versus D
How would you RANK the candidates? (1st, 2nd, etc.)
Discrete Math A Chapter 1, The Mathematics of Voting
FAIRNESS:
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The method of pairwise comparisons-- cannot violate the majority criterion
cannot violate the Condorcet criterion
So what’s the problem with Pairwise comparisons?
Example:
The table shows the voter preference for who a new NFL
expansion team should make their number 1 draft pick.
If we check pairwise comparisons we find:
A = 3 points
B = 2 ½ points
C = 2 points
D = 1 ½ points
E=1
You can verify this on p. 18
of your textbook.
Sometime after this vote was taken, we find that candidate C will not be participating in the draft.
Since he was not our top choice, this should not affect our results, right?
If we recheck the pairwise comparisons we find:
A = 2 points
B = 2 ½ points
D = 1 ½ points
E = 0 points
You can verify this on p. 19
of your textbook.
FAIRNESS: A Drop-Out + Recount shouldn’t change your winner
The Independence-of-Irrelevant Alternatives Criterion:
If candidate X is the winner of an election and in a recount one of the non-winning candidates withdraws or
is disqualified, then X should still be a winner of the election.
Another Problem:
Concerning the Number of Possible Pairs:
It is important you don’t miss any!
2 candidates = 1 pair
3 candidates = 3 pairs
4 candidates = 6 pairs
5 candidates = 10 pairs
N candidates =
𝑛(𝑛 − 1)
2
AvB
AvB, AvC, BvC
AvB, AvC, AvD, BvC, BvD, CvD
AvB, AvC, AvD, AvE, BvC, BvD, BvE, CvD, CvE, DvE
pairs
How many pairwise comparisons would we have if there are 7 candidates?
Discrete Math A Chapter 1, The Mathematics of Voting
Examples:
1.
An election with 6 candidates is to be decided by pairwise comparisons.
How many pairings are possible?
How many pairwise comparisons do you need to win in order to be a Condorcet candidate?
2.
An election with 5 candidates is to be determined using pairwise comparisons.
Suppose that B loses two pairwise comparisons, C loses one, D loses one and ties one, and
E loses two and ties one.
How many points does each candidate get?
Who was the winner of the election?
Is anyone a Condorcet Candidate?
3.
For the election given by the preference schedule:
Find a winner using the method of pairwise
comparisons.
4.
For the election given by the preference schedule:
Rank the candidates using extended pairwise
comparisions.
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Discrete Math A Chapter 1, The Mathematics of Voting
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1.3 The Borda Count Method
Borda Count Method:
Point values assigned to place values.
Candidate earns 1pt for last place, 2pts second-to-last, etc.
Most points = winner
Advantages:
Easy to understand
Provides a good “compromise” candidate by
giving value to all ballot positions.
Variations of Borda Count:
Heisman Trophy winner
NBA Rookie of the Year
Academy Awards
Who wins the Math Appreciation Society Election by Borda Count?
First:
Assign 1pt to 4th place, 2pts to 3rd place, 3pts to 2nd place, and 4pts to 1st place.
Second:
Calculate the total points earned by each player.
Who is our winner?
How would you rank the candidates?
So what’s wrong with the Borda Count Method?
Use the Borda Count Method to determine a winner.
How many people voted in this election?
How many votes do you need for a majority?
Did anyone receive the majority of the first place votes?
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Is there a Condorcet Candidate in this election? (table repeated)
How can you check?
Although RARE, the Borda Count can violate the Majority Criterion and the Condorcet Criterion.
Practice Problems:
1.
Use the Borda Count Method to determine a
winner of the election shown in this preference
schedule:
2.
Rank the candidates using extended borda count.
Discrete Math A Chapter 1, The Mathematics of Voting
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An election is held with four candidates (A, B, C, D) and 110 voters. The winner of the election is
determined using the Borda count method.
a. What is the maximum number of points a candidate can receive?
b. What is the minimum number of points a candidate can receive?
c. How many points are given out by one ballot?
d. What is the total number of points given out to all five candidates?
e. If A gets 320 points, B gets 290 points, and C gets 180 points, how many points did D get?
1.4 Plurality-with-Elimination (Instant Runoff Voting- IRV)
Plurality-with-Elimination:
Goal = Find a candidate with a majority of 1st place votes
Method =
Eliminate “Least Fit” candidates with the fewest 1st place votes, one at a time, until
someone gets a majority. Each time a candidate is eliminated, the next candidate on the
ballot receives those first-place votes.
This method has become increasingly popular for local municipal elections.
FIRST QUESTION:
How many votes is a MAJORITY?
Discrete Math A Chapter 1, The Mathematics of Voting
Find the winner of the Math Appreciation
Society Election by the method of
PLURALITY-with-ELIMINATION
On homework, you should always start
by carefully copying down the
preference schedule. Do not write in
your book!
A=
B=
C=
D=
How would you RANK the candidates?
FAIRNESS:
Plurality-with-Elimination CANNOT violate the Majority Criterion.
So what’s wrong with Plurality-with-Elimination?
Plurality-with-Elimination can violate the Condorcet Criterion. (and does in this election)
Plurality with Elimination can violate our final fairness criterion called “Monotonicity.”
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Discrete Math A Chapter 1, The Mathematics of Voting
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Example: Three cities, Athens (A), Barcelona (B), and
Calgary (C) are competing to host the summer Olympic
Games. The final decision is made by a secret vote of the
29 members of the Executive Council of the International
Olympic Committee, and the winner is to be chosen using
Plurality-with-Elimination. Two days before the actual
election, a straw poll is conducted, just to see how things
stand.
Although the results are supposed to be secret, word gets out
that Calgary is going to win. The four delegates represented
by the last column decide to jump on the Calgary bandwagon
and switch their votes to put Calgary first on the day of the
official vote.
How could this happen?
How could Calgary lose an election it was winning in the straw poll just because it got additional first-place votes
in the official election? While you will never convince the conspiracy theorists in Calgary that the election was not
rigged, double-checking the figures makes it clear that everything is on the up and up - Calgary is simply the victim
of a quirk in the plurality-with-elimination method: the possibility that you can actually do worse by doing better!
FAIRNESS: Doing BETTER in a RECOUNT should help you, not hurt you.
The Monotonicity Criterion:
If candidate X is the winner of an election and in a reelection, the only changes in the ballots are changes that favor
X (and only X), then X should remain a winner of the election.
Discrete Math A Chapter 1, The Mathematics of Voting
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Examples:
1.
Find the winner of the election
using the Plurality-withElimination method.
Then rank the candidates using
extended Plurality-withElimination.
2.
Find the winner of the election using
the Plurality-with-elimination method.
Then rank the candidates using
extended Plurality-with-Elimination.
A Note on TIES and TIE-Breaking:
It is ESSENTIAL that before anyone votes, everyone has agreed on a voting method and also a method of tiebreaking. If you wait until after the votes have been counted, people will favor the method of counting that favors
their candidate.
Common Tie Breakers: Most 1st place votes, Head to Head between tied, Fewest last place votes.
Chance must be used between unbreakable ties: Coin Toss, Rock-Paper-Scissors, Etc.
Assume ties are allowed unless the problem specifics state otherwise.
Did you know that ties are allowed for the Academy Awards?
Discrete Math A Chapter 1, The Mathematics of Voting
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IS THE ELECTION FAIR???
Fairness Criterion Review:
Has the Majority Criterion been violated?
If someone has a majority of the first place votes, then it is only fair that they win.
Is there a majority winner? (Determine by finding the number > 50% of the total vote)
If yes, they should have won the election using the stated method. If that choice didn’t win, then the
criterion has been violated.
Has the Condorcet Criterion been violated?
When compared against each other two at a time, a candidate is preferred to all other
candidates. It is only fair that that candidate wins the election.
Is there a Condorcet Candidate? (Determine by checking all of the head-to-head pairings) If yes, then that
choice should have won the election using the stated method. If that choice didn’t win, then the criterion
has been violated.
Has the Monotonicity Criterion been violated?
A candidate wins an election. In a recount, the candidate does better. It is only fair that the
candidate should still win the election.
Who won the original election? Did that choice win the reelection or vote recount? If they did not, the
criterion MAY have been violated. Check to see if insincere voting or a change in a preference ballot
occurred. Did receiving MORE 1st place votes cause someone to lose in the recount? If yes, then the
criterion has been violated.
Has the Independence-of-Irrelevant-Alternatives Criterion been violated?
A candidate wins an election. After the election, if a NONWINNING candidate drops out or
disqualified, it is only fair that the original winner should win again in the recount.
Who won the original election? Did that choice win the reelection or vote recount? If they did not, the
criterion MAY have been violated. Check to see if a choice was disqualified. If disqualifying a losing
candidate changes the outcome of the election, then the criterion has been violated.
ARROW’S IMPOSSIBILITY THEOREM: