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Finite Math A – Mrs. Leahy

Chapter 1: The Mathematics of Elections

Assignments to be completed in this chapter:

3 Textbook Assignments: HW 1.1/1.2, HW 1.3/1.4, HW 1.5

worth 10 points each – 30 total homework points,

graded on completion

due dates listed on schedule above

5 Minor Schoology Quizzes: Reading Quiz #1, #2, #3 ; Vocab Practice #1, #2

worth 2 points each – 10 total homework points must be completed by 3:00pm test day

1 Major Schoology Quiz: Final Vocab Quiz

worth 10 total homework points must be completed by 3:00pm test day

Pd 3 Pd 5 Pd 6 In Class Homework

Wed 7/27 A

Wed 7/27 A

Wed 7/27 A

Rules, Books, 1.1 Basic Elements of Elections

*Read: Pg 4 – 12 Schoology: Reading Quiz #1 *HW 1.1/1.2 Pg 28- 30 #1, 2, 3, 7, 8, 11, 12, 16, 17, 20 [10Q’s]

Fri 7/29 C

Fri 7/29 C

Thur 7/28 B

1.2 Plurality, 1.6 Fairness (Majority, Condorcet)

Mon 8/1

Mon 8/1

Mon 8/1

1.3 Borda Count HW 1.1/1.2 DUE TODAY

*Read: Pg 12 – 19 Schoology: Reading Quiz #2 *HW 1.3/1.4 Pg 31-32 #21, 22, 24, 31, 32, 34 [6Q’s]

Tues 8/2 A

Tues 8/2 A

Tues 8/2 A

1.4 Plurality with Elimination, 1.6 Fairness (Monotonicity)

*Schoology: Vocab Practice #1

Thurs 8/4 C

Thurs 8/4 C

Wed 8/3 B*

1.5 Pairwise Comparisons, 1.6 Fairness (I.I.A.) HW 1.3/1.4 DUE TODAY

*HW 1.5 Pg 32-33 #41, 42, 49, 50 [4 Q’s] *Read: Pg 25-26 “Conclusion” Schoology: Reading Quiz 3 *Schoology: Vocab Practice #2

Fri 8/5

Fri 8/5

Fri 8/5

HW 1.5 DUE TODAY Finish Schoology Quizzes if Needed. Work on Chapter 1 Review handout.

*Schoology: Final Vocab Quiz *Chapter 1 Review Handout – Not for a grade

Mon 8/8

Mon 8/8

Mon 8/8

SCHOOLOGY QUIZZES DUE TODAY

TEST CHAPTER 1

No Homework

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Chapter 1

Academic Standard: PS.ED.2 Use election theory techniques to analyze election data. Use

weighted voting techniques to decide voting power within a group.

Learning Outcomes A successful student will be able to

interpret and construct a preference schedule for an election involving

preference ballots.

implement the plurality, Borda count, plurality-with-elimination, and pairwise

comparisons vote counting methods.

rank candidates in a preference election.

identify fairness criteria as they pertain to preferential voting methods.

understand the significance of A rrows’ impossibility theorem.

Skills to Help Prepare You for the Next Exam At a minimum, be able to

interpret and construct preference schedules for elections involving preference

ballots (Exercises 1-10).

determine the winner and complete ranking of the candidates in an election

using preference ballots by applying the

- plurality method (Exercises 11-16).

- Borda count method (Exercises 21-26).

- plurality-with-elimination method (Exercises 31-36).

- method of pairwise comparisons (Exercises 41-48).

state the fairness criteria and identify when they are violated (Exercises 51-

55).

state Arrows’ impossibility theorem in your own words.

Study Tip

When it comes to implementing a particular algorithm, practice makes perfect.

Practice will also make you more efficient at solving routine problems on exam day

when time is short and your mind has other things to worry about .

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Finite Math A – Mrs. Leahy

Chapter 1 Notes: The Mathematics of Elections

1.1 The Basic Elements of Elections

*Powerpoint Intro and Sample Election*

Vocab to Know: single choice ballot, preference ballot, truncated preference ballot,

preference schedule, majority, plurality

Skills to Learn: Construct and interpret a preference schedule.

Preference Ballots:

Example: What dessert do you want after dinner? Transitivity: I ran out of cookies.

What should I serve this guest?

Example: The Math Appreciation Society Officer Election The Math Appreciation Society is electing its president. The candidates are Alisha (A), Boris (B), Carmen (C), and

Dave (D). 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:

1st choice Cookies

2nd choice Pie

3rd choice Cake

4th choice Candy

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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?

3. Who received the plurality of the first-place votes?

4. What if it came down to a vote between

B and C? Who would win?

Example: You collect election data by giving each voter a ballot with the candidates’ names and have

them number their choices. You group like ballots and make an alternate version of the preference

schedule. Use this to create a conventional preference schedule.

Ballot Alternate Preference Schedule Conventional Preference Schedule

10 15 3

Joe Smith (A) 4 1 2

John Citizen (B) 1 5 3

Jane Doe (C) 3 2 1

Fred Rubble (D) 5 3 4

Mary Hill (E) 2 4 5

Majority = MORE than half of the total first-place votes

Plurality = the MOST first-place votes, but not a majority

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1.2 The Plurality Method/1.6 Fairness Criteria

Vocab to Know: Plurality Method, Majority Criterion, Condorcet Candidate, Condorcet Criterion

Skills to Learn: Identify the winner and complete rankings using the plurality method.

Determine if the Majority Criterion or Condorcet Criterion has been violated.

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 a single-choice ballot. Math Appreciation Society Election

How many 1st place votes does each candidate have?

A = B =

C = D =

Who wins by this method?

How would you RANK the candidates? (1st, 2

nd, etc..)

Example 1: Given an election with four candidates, determine the winner and the complete rankings using the

plurality method.

This seems pretty easy… but there is a catch coming…

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Fairness Criteria

Fairness Criteria are a set of basic rules that define the formal requirements for fairness. We expect any

democratic election to satisfy these basic rules. If it violates one of these conditions, there is a potential

for unfair election results.

Go back and check… Does the Math Appreciation Society Election have a Majority Candidate?

If you think about it, it is ______________________ for an election counted using the Plurality Method to violate

the Majority Criterion. If you have more than half of the first place votes, you will automatically have the “most”

first place votes.

The real problem: Our Plurality Method winner

was “A” with 14 votes. Where is “A” on the rest of

the ballots? Does Candidate A “feel like” a good

choice to lead this group?

Example 2:

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?

Rule #1: The Majority Criterion

If a candidate has a majority of the first place votes, then that candidate should be declared the winner

of the election.

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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!

Example 3: Candidate “C” won this election using the Plurality Method. Is “C” a Condorcet

Candidate? If not, is there a Condorcet Candidate in this election?

Insincere Voting (Strategic Voting)

Another flaw of the Plurality Method is that election results can easily be manipulated by insincere voting, or

voting for a candidate you don’t really want to try to affect the outcome of the election.

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.”

Rule #2: The Condorcet Criterion

If a choice in an election is preferred over each of the other candidates in a head-to-head comparison,

then that choice should be declared the winner of the election.

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Fun Fact: Did you know

that the last time a third-party presidential

candidate WON a state

and earned any Electoral College Votes was

infamous Alabama

Governor George

Wallace in 1968?

Example 4: Use the plurality method to find the winner of the election.

a) 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?

b) 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?

Always assume that ties are allowed unless a tie-breaking procedure is specified. Mathematically unbreakable ties

must be broken using a game of chance like a coin toss or rolling a die for the high number.

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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

The Nintendo Game Mario Kart using a version of Borda Count to determine the winner of the GP race

circuit. After each race, racers earn points based on

how they placed. At the end of 4 races, the top three races (most total earned points) get a trophy and

participate in a parade.

In this “Modified Borda” 1st place received 15 points, 2nd place 12 points, and last earned 0 points.

In a traditional Borda Count, 1st would have received 12 points, 2nd 11 points, and last 1 point.

Why do you think the points are distributed this

way after each race in the game?

Example: Who wins the Math Appreciation Society

Election by Borda Count?

Who is our winner? How would you rank the candidates?

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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?

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. Then rank the candidates.

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2. An election is held with six candidate and 150 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 six candidates?

e. If A gets 650 points, B gets 290 points, C gets 380 points, D gets 730 points, and E gets 400 points, how

many points does candidate F earn ? Who wins the election?

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.

Your true preferences: Your top choice BEFORE some Your top choice AFTER some

candidates are eliminated: candidates are eliminated.

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Example: 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!

FIRST QUESTION:

How many votes is a MAJORITY?

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?

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Plurality-with-Elimination can violate the Condorcet Criterion. (and does in this election)

In other words, doing better before a RECOUNT should help you, not hurt you.

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!

Rule #3: 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.

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Examples: Determine the winner using the plurality-with-elimination method.

1.

2.

3.

Number of Voters 16 4 5 10 15

1st choice A B B E C

2nd choice B E D A B

3rd choice C A A D D

4th choice D C C B E

5th choice E D E C A

Number of Voters 40 16 36 8 2

1st choice A E E B D

2nd choice B A B A C

3rd choice C B D E E

4th choice D C C C B

5th choice E D A D A

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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 B versus D C versus D

A = B = C = D =

Who wins by this method? How would you RANK the candidates?

Is there a Condorcet Candidate in this election?

Concerning the Number of Possible Pairs: It is important you don’t miss any!

2 candidates = 1 pair AvB

3 candidates = 3 pairs AvB, AvC, BvC 4 candidates = 6 pairs AvB, AvC, AvD, BvC, BvD, CvD

5 candidates = 10 pairs AvB, AvC, AvD, AvE, BvC, BvD, BvE, CvD, CvE, DvE

N candidates = pairs

How many pairwise comparisons would we have if there are 7 candidates?

In an election with 10 candidates, how many pairings would you have to win to be a Condorcet Candidate?

𝑁(𝑁 − 1)

2

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Example:

For the election given by the preference schedule: Determine the winner and rank the candidates using the

method of pairwise comparisons.

FAIRNESS: 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:

Sometime after this vote was taken, we find that

candidate C will not be participating in the draft.

Candidate C drops out of the election. Since he was not our top choice, this should not affect our results, right?

If we recheck the pairwise comparisons we find:

Rule #3: The Independence of Irrelevant Alternatives Criterion

If candidate X is the winner of an election and one of the non-winning candidates withdraws of is

disqualified, then after a recount, candidate X should still be the winner of an election.

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The way you count votes matters.

Math Appreciation Society Election:

Plurality Method winner:

Borda Count winner:

Plurality with Elimination winner:

Pairwise Comparisons winner:

Changing the method of counting votes might change the outcome of the election. Sometimes, all four

methods may produce the same winner and other times they won’t.

IS THE ELECTION FAIR? Is there a candidate with the majority of the first place votes?

YES: The candidate with the majority of the first place votes should be the winner of the election. If they are NOT the winner of the election, then the Majority Criterion has been violated. The election

results are unfair.

Is there a Condorcet Candidate? (a candidate who wins every head to head pairing against every other candidate)

YES: The Condorcet Candidate should be the winner of the election. If they are NOT the winner of the election, then the Condorcet Criterion has been violated.

Was there at any point a RECOUNT or a RE-ELECTION process?

YES: Did the winner change? That is, is the winner of the or original count DIFFERENT from the

winner after the recount?

YES: In between the original election and the recount, what happened?

A Non-winning candidate dropped out or was disqualified: The Independence of

Irrelevant Alternatives Criterion has been violated.

Changes were made to the ballot that ONLY favored the original winner: The

Monotonicity Criterion has been violated.

ARROW’S IMPOSSIBILITY THEOREM: