Lesson 6 Monotonicity and IIA Criteria Solutions PDF

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Notes for exams and practice problems....

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Voting Methods

Lesson 6

Today in class we reviewed the majority and Condorcet criteria and began our discussion of the final two fairness criteria. The Unit 1: Practice Problems is available and will be due tomorrow, September 10 at 6:00 pm. The PDF of the homework problems and the dropbox can be found under “Course Activities” on Canvas. Lesson Objectives • Describe the last two fairness criteria: the Monotonicity and Independence-of-Irrelevant-Alternatives Criteria. • Demonstrate the ability to test the five voting methods against the Monotonicity Criterion and the Independence-of-Irrelevant-Alternatives Criterion. • Identify which voting methods will always satisfy the four fairness criteria and which voting methods will sometimes satisfy the four fairness criteria.

Review of the First Two Fairness Criteria Example 1: A group of tenth graders were asked what their favorite kind of date would be and to rank the following choices: Picnic, Movie, Kayaking, and Concert. The ballots were analyzed and the preference table below was created. Number of Ballots 1st 2nd 3rd 4th

5 Concert Movie Kayaking Picnic

4 Picnic Concert Movie Kayaking

5 Picnic Concert Kayaking Movie

10 Kayaking Concert Picnic Movie

7 Movie Picnic Kayaking Concert

1. Does the Pairwise Comparison method satisfy the Majority Criterion? Explain your answer thoroughly.

2. Does the Borda Count method satisfy the Majority Criterion? Explain your answer thoroughly. Number of Ballots

5

4

5

10

7

1st

Concert

Picnic

Picnic

Kayaking

Movie

2nd

Movie

Concert

Concert

Concert

Picnic

3rd

Kayaking

Movie

Kayaking

Picnic

Kayaking

4th

Picnic

Kayaking

Movie

Movie

Concert

3. Does the Plurality method satisfy the Condorcet Criterion? Explain your answer thoroughly.

4. Does the Instant Runoff method satisfy the Condorcet Criterion? Explain your answer thoroughly.

Monotonicity Criterion and Independence-of-Irrelevant-Alternatives (IIA) Criterion Monotonicity Criterion: This fairness criterion states that if Candidate X is the winner, then Candidate X would still be the winner if a voter ranked Candidate X higher in their preference ballot regardless of the voting method used. (In other words, a winning candidate should not be changed by a voter moving them up in their ballot.) Independence-of-Irrelevant-Alternatives (IIA) Criterion: This fairness criterion states that if Candidate X is the winner, then Candidate X would still be the winner had one or more of the irrelevant alternatives (i.e., losing candidates) not been in the race regardless of the voting method used. (In other words, the winner should not be changed by an irrelevant alternative candidate dropping from the election.) Example 2: After ordering at various fast food restaurants, selected customers were sent a survey. Among the questions in the survey, people were asked to rank four fast food restaurants by which sold the best french fries. Their responses are listed below: Number of Votes 1st 2nd 3rd

7 Chick-fil-A McDonald’s Checkers

8 McDonald’s Checkers Chick-fil-A

10 Checkers Chick-fil-A McDonald’s

2 Chick-fil-A Checkers McDonald’s

About a month after the results came in, the same customers were asked to submit their preferences again which caused a slight change in the preference table (shown below): Number of Votes 1st 2nd 3rd

7 Chick-fil-A McDonald’s Checkers

8 McDonald’s Checkers Chick-fil-A

10 Checkers Chick-fil-A McDonald’s

2 Checkers Chick-fil-A McDonald’s

1. Does the Plurality method satisfy the Monotonicity Criterion? Explain your answer thoroughly.

2. Does the Instant Runoff method satisfy the Monotonicity Criterion? Explain your answer thoroughly.

3. Does the Pairwise Comparison method satisfy the Monotonicity Criterion? Explain your answer thoroughly.

4. Does the Borda Count method satisfy the Monotonicity Criterion? Explain your answer thoroughly. Number of Votes 1st

7 Chick-fil-A

8 McDonald’s

10 Checkers

2 Chick-fil-A

2nd

McDonald’s

Checkers

Chick-fil-A

Checkers

3rd

Checkers

Chick-fil-A

McDonald’s

McDonald’s

Number of Votes 1st

7 Chick-fil-A

8 McDonald’s

10 Checkers

2 Checkers

2nd

McDonald’s

Checkers

Chick-fil-A

Chick-fil-A

3rd

Checkers

Chick-fil-A

McDonald’s

McDonald’s

Example 3: The Los Angeles LAXers are the newest expansion team in the NFL and are awarded the first pick in the upcoming draft. The draft committee (made up of coaches, scouts, and team executives) has narrowed down the list to five candidates: Allen, Byers, Castillo, Dixon, and Evans. After many meetings, the draft committee is ready to make their first vote for the team’s top pick in the draft. Number of Votes 1st 2nd 3rd 4th 5th

2 Allen Dixon Castillo Byers Evans

6 Byers Allen Castillo Dixon Evans

4 Byers Allen Dixon Evans Castillo

1 Castillo Byers Allen Dixon Evans

1 Castillo Dixon Allen Byers Evans

4 Dixon Allen Evans Castillo Byers

4 Evans Castillo Dixon Byers Allen

After 24 hours of research, the draft committee decided to drop Castillo from the list of players. The table below reflects this change. Number of Votes 1st 2nd 3rd 4th

2 Allen Dixon Byers Evans

6 Byers Allen Dixon Evans

4 Byers Allen Dixon Evans

1 Byers Allen Dixon Evans

1 Dixon Allen Byers Evans

4 Dixon Allen Evans Byers

4 Evans Dixon Byers Allen

1. Does the Plurality method satisfy the IIA Criterion? Explain your answer thoroughly.

2. Does the Instant Runoff method satisfy the IIA Criterion? Explain your answer thoroughly.

3. Does the Pairwise Comparison method satisfy the IIA Criterion? Explain your answer thoroughly.

4. Does the Borda Count method satisfy the IIA Criterion? Explain your answer thoroughly.

No. of Votes 1st

2 Allen

6 Byers

4 Byers

1 Castillo

1 Castillo

4 Dixon

4 Evans

2nd

Dixon

Allen

Allen

Byers

Dixon

Allen

Castillo

3rd

Castillo

Castillo

Dixon

Allen

Allen

Evans

Dixon

4th

Byers

Dixon

Evans

Dixon

Byers

Castillo

Byers

5th

Evans

Evans

Castillo

Evans

Evans

Byers

Allen

No. of Votes 1st

2 Allen

6 Byers

4 Byers

1 Byers

1 Dixon

4 Dixon

4 Evans

2nd

Dixon

Allen

Allen

Allen

Allen

Allen

Dixon

3rd

Byers

Dixon

Dixon

Dixon

Byers

Evans

Byers

4th

Evans

Evans

Evans

Evans

Evans

Byers

Allen

Now, we have shown with several examples which voting methods will always satisfy the four fairness criteria and which voting methods will sometimes satisfy the four fairness criteria.

Fairness Criteria Majority Condorcet Monotonicity IIA

Plurality Method Always Satisfies May Violate Always Satisfies May Violate

Instant Runoff Always Satisfies May Violate May Violate May Violate

Pairwise Comparison Always Satisfies Always Satisfies Always Satisfies May Violate

Borda Count May Violate May Violate Always Satisfies May Violate...