The Mathematics of Voting

Chapter 1

Voting theory: application of methods that affect the outcome of an election.

Sec 1: Preference Ballots and Schedules

***Read Ex 1.1 on page 4, we will refer to it multiple times during the lesson.Preference Ballot- where voters RANK the candidates in order of their preference (how they like them). Each voter should have a ballot in the election process. Copy this table:

2020 1010 33

McDonalds 1 2 3

Zaxby's 2 3 2

Cook Out 3 4 4

Burger King

4 1 1

The table on the previous slide is actually a PREFERENCE SCHEDULE. It is a compilation of all the preference ballots into a compact table so we don't have to read every preference ballot cast. What the preference ballot tells us is that 20 people all ranked the restaurants in the following order: McDonalds first, Zaxbys second, Cook Out third, and Burger King last. Ten people ranked Burger King first, McDonalds second, Zaxbys third, and Cook Out fourth. Three people ranked Burger King, then Zaxbys, McDonalds, and Cook Out last.

If you add up the numbers at the top of each column, you'll see that 33 people voted in this survey. So with all the votes, which restaurant would satisfy the voters the best?

Sec 2: Plurality MethodPlurality method-the candidate with the MOST first place votes wins the election• With plurality, we don't care at all about anything other

than the number of first place votes a candidate receives.• If we apply the plurality method to the survey on favorite restaurants, who would win?

Easy, right? Well, even though it is a simple method, there is a reason not to use it too much. Take a look at EXAMPLE 1.2 on page 7 in your book. It is a preference schedule for a band that is voting on which bowl game to perform at. If you apply the plurality method, the band would be going to the Rose Bowl. But, if you look twice, 49 people voted for the Rose Bowl to be in first place, while 51 people voted for it to be their last choice! So, more people dislike it the most than people who like it the best. Would the Rose Bowl be where the band should go to satisfy the most people?

Take a look at the Hula Bowl in the table. It actually ranks in either first or second place for EVERYONE's vote! If you compare H to everyone else, H outranks them.Take a look: Hula vs Rose: Hula is beneath Rose in only the first column. This means Rose outranks Hula. But in the next two columns, Hula is closer to the top, and therefore outranks Rose. Hula outranks Rose by 51 (48 in the 2nd column+3 from the 3rd column) votes, while Rose outranks Hula by 49 votes. H outranks C 97 to 3H outranks O 100 to 0H outranks S 100 to 0

What's wrong with plurality continued....

** H actually ranks higher than everyone else when you look at the preference for the band as a whole.

What does all this mean? It means even though plurality seems like a good way to conduct an election, it actually has its flaws. It violates the Condorcet Criterion.

The Condorcet Criterion says if a candidate beats everyone in a head to head comparison (outranks everyone), that candidate should be the winner of the election.**Condorcet is pronounced like Con*door*say

2424 2323 33

1st A B C

2nd B D B

3rd C A D

4th D C A

Is there a Condorcet Candidate in the table? If so, who is it?

Insincere voting- when a voter changes his vote (or preferences) to influence or change the outcome of an election.For example, what if one of the four voters in the last column decided to change their first place vote from C to B? Then B would actually tie for first with A.

A C B C C C A C A A C A

B B D B B B B B B B B B

C D C A A D C A C C D C

D A A D D A D D D D A D

Page 281.

In these preference ballots, there are only 4 distinct ballots.

ABCD

CBDA

BDCA

CBAD**just place

above or below each box how many people voted that way. This creates the preference schedule.

ABCD

CBDA

BDCA

CBAD

5 3 1 3

5. Refers back to #3. Says to eliminate candidates with 20% or less of the first place votes if no one initially has a majority.

Exercise 3 has 21 voters so a majority would be 11 first place votes. Initially, A has 8, B has 3, C has 5, and D has 5. So no majority yet!Therefore, we have to find 20% of the votes and eliminate any candidate who has less than that number of first place votes.A) .20*21=4.2. So anyone with 4.2 or less first place votes will be eliminated. That would be candidates B and E. Now that B and E are eliminated, rewrite your preference schedule. Just bump up all candidates that are initially ranked below B and E. There is no 4th or 5th place now.

5 3 5 3 2 3

A A C D D A

C D D C C C

D C A A A D

This is the new schedule. As you can see, some are the

same, so condense it.

8 3 5 5

A A C D

C D D C

D C A A

B)

C) candidate A with 11 first place votes

Page 31 #13According to the directions, 120 of 150 votes have been tallied. A) If we want A to win by plurality, we have to figure out what the smallest number of remaining votes A needs to win. This number has to ensure that no matter who gets the remaining votes, it won't surpass A.If we give A 20 of the remaining votes, it would then have 46. We don't know who will get the remaining 10. What if all 10 go to C? Then C would have 52. That won't work. Give A a little more, maybe 24. This means A would have 50 and the most C would have is 48. This works, but since we're looking for the smallest number, lets see if we can use something smaller than 24. Try 23. This would mean A and C would both tie @ 49. So 23 would guarantee a tie, but 24 would guarantee A is the only winner.

Assignment: pages 28-31 2-4, 6-12, 13b-16.

Chapter 1 Sec 3: Borda Count Method

Another method for finding an election winner is using Borda Count. This method assigns points based on how many votes each candidate receives for each rank.

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1st A B C

2nd B D B

3rd C A D

4th D C A

Ex: In the table below from section 1, 24 people voted for A to be in first place, 0 people voted for A to be in 2nd place, 23 for 3rd place, and 3 for last place.

Since there are 4 places, give 4 points for first place votes, 3 points for 2nd place votes, 2 points for 3rd place votes, and 1 point for last place. Give last place 1 point and each higher rank gets one point more.

The complete Borda Count for this example would be

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1st A B C

2nd B D B

3rd C A D

4th D C A

A: 4(24)+3(0)+2(23)+1(3)=145B: 4(23)+3(27)+2(0)+1(0)=173C: 4(3)+3(0)+2(24)+1(23)=83D: 4(0)+3(23)+2(3)+1(24)=99

Since B has the most pints, it wins using Borda Count.

What's wrong with Borda Count? It violates the Majority Criterion. A candidate with majority doesn't necessarily win using Borda Count. If a candidate has majority, but another wins using Borda Count, it violates the majority criterion. It also can violate the Condorcet Criterion.

Page 34 #23If the election has 50 voters, then each candidate receives at least one point from 50 people and up to 4 points from 50 people. The best any candidate can do is to get all first place votes. That means every voter each gives that candidate 4 points. The worst any candidate can do is to receive a last place vote from all the candidates, which means all 50 voters give that candidate 1 point. A) 4x50=200

pointsB) 1x50=50

points

Assignment: page 32 17-20, 24-26.

Chapter 1 Sec 4: Plurality With Elimination

Also called Instant Runoff Voting, Plurality with Elimination requires a majority winner. If no candidate has a majority in the initial voting, eliminate candidates with the fewest first place votes one at a time until someone does have majority.1. Count the total number of votes. A majority would be MORE than half the total number of votes. If a candidate has a majority, that candidate is a winner. If not, move on to step 2.2. Eliminate a candidate with the fewest first place votes until someone has a majority. Once you eliminate a candidate, the candidates move up in each column. Check for a majority winner now. Repeat step #2 until you have a majority winner.

Ex: 2424 2323 33

1st A B C

2nd B D B

3rd C A D

4th D C A1. 24+23+3=50. 50/2=25. A majority winner would need 26 first place votes. Nobody does, so look for the candidate with the fewest first place votes. That would be D. It doesn't have any first place votes.2. Once you eliminate D, the new preference schedule would be:

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1st A B C

2nd B A B

3rd C C AUnfortunately, eliminating D doesn't change anything as far as first place votes. So, eliminate the remaining candidate with the fewest first place votes. That would be C.

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1st A B B

2nd B A A

After eliminating C, we now have a change in first place votes. The preference schedule above can now be condense because the last two columns are the same.

As you can see now, B has a majority of first place votes, so B is the winner using Plurality with Elimination.

2424 2626

1st A B

2nd B A

Page 34:27. There are 21 total votes, so a majority would be 11. A has 8, B has 3, C has 5, D has 5, and E has 0 first place votes. No majority winner yet. So, eliminate E.

55 33 55 55 33

1st A A C D B

2nd B D D C A

3rd C B A B C

4th D C B A D

Still no majority winner, so eliminate B:

55 33 55 55 33

1st A A C D A

2nd C D D C C

3rd D C A A DNow A has a majority, so it wins using PWE.

Assignment: page 34-36. 28-34

Chapter 1 Sec 5: Pairwise Comparisons

Pairwise comparisons pit each candidate against each other. The winner of each comparison would be the candidate that has the most votes that ranks it higher on the ballot. A candidate who wins a comparison gets a point. If two candidates tie, they each get 1/2 point.

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1st A B C

2nd B D B

3rd C A D

4th D C A

Ex:

We have to compare each candidate against all the other candidates. With N candidate, there will be pairwise comparisons.

Pairwise Comparisons for this example.

A vs B: A ranks higher than B in column 1 only which is 24 votes. B ranks higher than A in the second and third columns for a total of 26 votes. B wins in a Pairwise comparison against A. B gets one point.

2244

2233

33

1st

A B C

2nd

B D B

3rd

C A D

4th

D C A

A vs C: A outranks C in columns 1 & 2 for a total of 47 votes. C only outranks A for 3 votes in the last column. A wins and gets a point.

A vs D: A outranks D by 24 votes, D outranks A by 26 votes. D wins, gets a point.B vs C: 47 vs 3. B wins, gets a point.B vs D: 50 vs 0. B wins, gets a point.C vs D: 27 vs 23. C wins, gets a point.

PointsA: 1B: 3C: 1D: 1B

wins!!!

Page 36 # 35:A vs B: 13 vs 13....a tie! A and B both get 1/2 a point.A vs C: 21 vs 5... A wins, gets a point.A vs D: 14 vs 12... A wins, gets a point.B vs C: 16 vs 10...B wins, gets a point.B vs D: 8 vs 18...D wins, gets a point.C vs D: 13 vs 13...a tie! C and D get 1/2 a point.

PointsA: 2 1/2 pts

B: 1 1/2 pointsC: 1/2 pointD: 1 1/2 pointsA

wins!Chapter 1 Sec 6: Rankings

Extended Rankings require the actual ranking for all the candidates in the election, not just the winner.

Extended Pairwise comparisons for the problem we just did would be: A ranks 1st, B and D tie for 2nd, and C ranks 4th.

If we did extended plurality with elimination with #27 on page 34, this is how we would rank them:• Since E is eliminated first, E would rank last

(or 5th).

• Then we eliminated B, so it ranks 4th.

• C and D would tie

• A would rank first because it is the majority winner.

Assignment: page 37 41&42