Social Choice: Do we all agree? Lesson

What generic name do you call a carbonated beverage?

pick one poll of soft drinks

How does your vote compare with the country? What regional differences do you notice?

A colorful US map survey of preferred generic names for carbonated beverages.

At a large university, a professor took a survey of the generic names for carbonated beverages.

The following table is the results:

Poll results of drinks

"Pop" is the most common!

Is that what you voted for?

Think about the results

Do you think the "winning" name represents the choice of the people?

The most common name was chosen based on the plurality method or the name with the most votes, even if it is less than 50% of all votes.

A survey or an election held in which you only vote for your first choice is a standard ballot. When you have more than two candidates or selection choices, it is possible for the least popular candidate to win. This results when the majority of votes between two or more candidates is split.

What percentage of votes is needed to win the common name?

Think about beating a 5 way tie! hint: 100 / 5 = 20

Answer: For 5 candidates, the plurality winner must have over 20 %

Suppose the students that selected Coke did not agree with the name "pop" being the most common. What changes, if they voted for the name "soda" instead?

Poll results: Pop, 22. Soda, 23., Coke 0., Soft Drink, 3. Other, 2.

Soda would be the new plurality winner!

Is the winner also a majority winner?

No, the majority winner would have the most votes and at least 50% of the votes. To be the majority winner, 26 votes would be needed.

Majority Rule has at least three desirable properties:

It should be anonymous with all voters treated equally. That is, the weight of each voter is the same. If any two voters switch ballots before submitting them, the outcome would be the same.
It should be neutral with all candidates treated equally. That is, if a new election was held and every voter reversed her or his vote, the outcome would be reversed as well.
It should be monotone. If a new election is held and a voter changes his ballot in favor of the winning candidate, that candidate must be the winner of the new election

Ranking the preference of a ballot is a way to create a an agenda of the votes. This type of ballot might tell you if one prefers "soda" over "pop."

Below is a preference schedule for generic names:

Preference Schedule for Generic Names
Name	Rank
Pop	2
Soda	1
Coke	4
Soft Drink	3
Other	4

In this case, this student likes the name "Soda" the most and "Other" the least.

How many other ways could you rank the names? Let's look at a simpler case.

How many other ways are possible for ranking 2 choices?

Ranking Two Choices
	Option 1	Option 2
1st	A	B
2nd	B	A

There are two ways to rank 2 choices (2! = 2x1=2).

How many other ways are possible for ranking 3 choices?

Ranking Three Choices
	Option 1	Option 2	Option 3	Option 4	Option 5	Option 6
1st	A	B	C	A	B	C
2nd	B	C	A	C	A	B
3rd	C	A	B	B	C	A

There are 6 ways to rank 3 choices (3! = 3 x 2 x 2 = 6).

How many ways are possible for making 4 choices?

There are 24 ways to rank 4 choices (4! = 4 x 3 x 2 x 1 = 24).

How many ways are possible for making 5 choices?

There are 24 ways to rank 4 choices (5! = 5 x 4 x 3 x 2 x 1 = 120).

How many other ways are possible for ranking N choices?

Answer N! the 1 stands for factorial meaning that numbers time each number below until you get one.

n! = n x (n-1) x (n-2) x (n-3)... x 1

A beverage ranking chart listing Coke, Sprite, Pepsi, and Mtn. Dew with checkboxes, and a key for ranking letters. A is for Coke, B is for Sprite, C is for Pepsi, and D is for Mountain Dew.

The results are summarized in the table below:

Preference Survey Results
Number of Students	18	7	6	11	1	7
1st	A	B	C	D	B	C
2nd	B	C	A	C	A	D
3rd	C	A	B	A	D	B
4th	D	D	D	B	C	A

By counting 1^st choices, we find A with 18 points (look at the first highlighted column in the table above) and C with 13 points (look at the table above 13 = 6 +7) We can now hold a run off election without holding another election. We create a new schedule with only the top two candidates.

Top Two Candidates
Number of Students	18	7	6	11	1	7
1st	A	C	C	C	A	C
2nd	C	A	A	A	C	A

Pepsi (Choice C) is the winner by the Run Off Method. The run off method eliminates all candidates except the top two candidates.

Sequential Run Off Method

Sequential Run Off Method is a variation of the run off method in which you repeatedly eliminate the candidate with fewest 1^st place votes until a winner emerges.

Sequential Run Off Method
Number of Students	18	7	6	11	1	7
1st	A	B	C	D	B	C
2nd	B	C	A	C	A	D
3rd	C	A	B	A	D	B
4th	D	D	D	B	C	A

Round 1: Eliminate B with 8 pts and transfer the points to the next ranked candidate as needed.

Eliminating B - Next Ranked Candidate
Number of Students	18	7	6	11	1	7
1st	A	B	C	D	B	C
2nd	B	C	A	C	A	D
3rd	C	A	B	A	D	B
4th	D	D	D	B	C	A

Round 2: Eliminate D with 11pts and transfer the points to the next ranked candidate as needed.

Eliminating D - next ranked candidate
Number of Students	18	7	6	11	1	7
1st	A	B	C	D	B	C
2nd	B	C	A	C	A	D
3rd	C	A	B	A	D	B
4th	D	D	D	B	C	A

Round 3: Eliminate A with 19pts and the winner is C with the remaining 31 pts.

Eliminate A
Number of Students	18	7	6	11	1	7
1st	A	C	C	C	A	C
2nd	C	A	A	A	C	A

Pairwise Comparison or Sequential Pairwise is another variation of the run off method. In pairwise comparison, an agenda is created to pit the first candidate against a second candidate in a one-on-one contest. The winner moves on to confront the third candidate and so on until a winner emerges.

Pairwise Comparison or Sequential Pairwise
Number of Students	18	7	6	11	1	7
1st	A	B	C	D	B	C
2nd	B	C	A	C	A	D
3rd	C	A	B	A	D	B
4th	D	D	D	B	C	A

Assume with have the following agenda.: A, B, C, D. In a contest between A vs. B, A is the winner.

A, B, C, D Agenda
Number of Students	18	7	6	11	1	7
1st	A ✅	B ✅	C	D	B ✅	C
2nd	B	C	A ✅	C	A	D
3rd	C	A	B	A ✅	D	B
4th	D	D	D	B	C	A ✅

In this scenario,

A = 18 + 6+ 11 =35

B = 7 + 1 + 7 = 15

Assume with have the following agenda.: A, B, C, D. In a contest between A vs. C, C is the winner.

A, B, C, D - Contest A vs. C
Number of Students	18	7	6	11	1	7
1st	A ✅	B	C ✅	D	B	C ✅
2nd	B	C ✅	A	C ✅	A ✅	D
3rd	C	A	B	A	D	B
4th	D	D	D	B	C	A

In this scenario,

A= 18 + 1 = 19

C= 7 + 6 + 11 + 7 = 31

Let's look at another scenario. In C vs. D, C is the winner

C vs. D
Number of Students	18	7	6	11	1	7
1st	A	B	C ✅	D ✅	B	C ✅
2nd	B	C ✅	A	C	A	D
3rd	C ✅	A	B	A	D ✅	B
4th	D	D	D	B	C	A

C = 18 + 7 + 6 + 7 =38

D = 11 + 1 = 12

Pepsi is the sequential pairwise winner using the agenda A, B, C, D.

The flaw in the sequential pairwise method lies in the agenda. The agenda could change the winner of an election.

Assume we have the following agenda, B, C, D, A

B vs C B is the winner (26pts vs 24pts)

B vs D B is the winner (32pts vs 18pts)

B vs A A is the winner (35pts vs 15pts)

Coke is the sequential pairwise winner using the agenda B, C, D, A.

Agenda B, C, D, A
	A	B	C	D
A	-	1	0	1
B	0	-	1	1
C	1	0	-	1
D	0	0	0	-

Runner is the vertical labels and the competitor is the horizontal labels.

For example, C vs A is a win for C so a 1 is placed in the box.

Which drink is the winner using the Condorcet method?

None, no drink beats all other candidates.

This is known as Condorcet's voting paradox, a case with 3 or more candidates in which the method yields no winner.

Borda Count Method

In some cases, we may want to know who won second and third place too. The Borda Count method is a ranking method that assigns points in a non-increasing manner to the ordered candidate on each voter's preference list ballot and then sums these points to arrive at a group's final ranking.

Borda Count Method
Number of Students	18	7	6	11	1	7
1st	A	B	C	D	B	C
2nd	B	C	A	C	A	D
3rd	C	A	B	A	D	B
4th	D	D	D	B	C	A

Find the winner of the favorite drink using the Borda Count Method. Assign 4pts to first place votes, 3pts for second place, 2pt for third place, and 1pt for fourth place votes.

A: 4(18) + 3(7) + 2(18) + 1(7) = 136

B: 4(8) + 3(18) + 2(13) + 1(11) = 123

C: 4(13) + 3(18) + 2(18) + 1(1) = 143

D:C 4(11) + 3(7) + 2(1) + 1(31) = 98

The winner using the Borda Count Method is Pepsi.

An easy way to calculate the Borda Count Winner is to use a matrix operation. Let the Borda points for the preference schedules form matrix A and the frequency of each schedule form matrix B. Matrix A * Matrix B = the points for each group.

Matrix multiplication of [Matrix A] and [Matrix B] resulting in a column matrix representing the points for each candidate. The resulting points are 136, 123, 143, and 98, with candidate C winning with 143 points.

Approval Voting Method

Approval voting is a method in which voters can vote for as many candidates as they wish. Each approval vote is worth one point and the candidate with the most approval votes wins. This method works well in elections in which more than one candidate can win. In an election using Approval Voting, adding or removing candidates or alternatives does not change the point totals of the other candidates or alternatives. Using the same preference schedules, let's assume the voters only approve their top two choices.

The winner using the Approval method would be :

A: 18+6+1=25

B: 18+7+1=26

C:7+6+11+7=31

D:11+7=18

Approval Method
Number of Students	18	7	6	11	1	7
1st	A	B	C	D	B	C
2nd	B	C	A	C	A	D

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