14.2 Homework Solutions Plurality: Musical play Borda count: Classical play Plurality-with-elimination: Classical play Pairwise Comparison: Classical play

14.2 Homework Solutions

•Plurality: Musical play•Borda count: Classical play•Plurality-with-elimination: Classical play•Pairwise Comparison: Classical play

Number of Votes 10 6 6 4 2 2

1st Choice M C D C D M

2nd Choice C M C D M D

3rd Choice D D M M C C

May 19, 2010 Math 132: Foundations of Mathematics

Math 132:Math 132:Foundations of MathematicsFoundations of Mathematics

Amy LewisMath Specialist

IU1 Center for STEM Education


14.2 Flaws of Voting Methods

• Use 4 different criterion methods to determine a voting system’s fairness.

• Understand Arrow’s Impossibility Theorem.

First of all…

• I was twice a liar yesterday!

Pairwise Comparison Method• The preference table is used to make a series of

comparisons in which each candidate is compared to each of the other candidates.

• For each pair of candidates, X and Y, use the table to determine how many voters prefer X to Y and vice versa.

• If a majority prefer X to Y, then X receives 1 point. If a majority prefer Y to X, then Y receives 1 point. If the candidates tie, then each receives ½ point.

• After all comparisons have been made, the candidate receiving the most points is the winner.


Pairwise Comparison Method

• How many comparisons do we need to make?– Antonio vs. Bob– Antonio vs. Carmen– Antonio vs. Donna– Bob vs. Carmen– Bob vs. Donna– Carmen vs. Donna


Preference Table for the Smallville Mayoral Election

Number of Votes 130 120 100 150

First Choice A D D C

Second Choice B B B B

Third Choice C C A A

Fourth Choice D A C D


• Bob gets 1 point.








Antonio vs. Bob


• Carmen gets 1 pt.








Antonio vs. Carmen


• Antonio gets 1 pt.








Antonio vs. Donna


• Bob gets 1 point








Bob vs. Carmen


• Bob gets 1 pt.








Bob vs. Donna


• Carmen gets 1 pt.








Carmen vs. Donna

Pairwise Comparison Method• Who wins?

– Antonio: 1 point– Bob: 3 points– Carmen: 2 points– Donna: 0 points…so sad.

• Bob wins! Again!• My answer didn’t change, but the correct

reason did…May 19, 2010 Math 132: Foundations of Mathematics

Criterion to Determine a Voting System’s Fairness

• The Majority Criterion– If a candidate receives a majority of first-place

votes in an election, then that candidate should win the election.

• The Head-to-Head Criterion– If a candidate is favored when compared

separately—that is, head-to-head—with every other candidate, then that candidate should win the election.

Criterion to Determine a Voting System’s Fairness

• The Monotonicity Criterion– If a candidate wins an election and, in a reelection,

the only changes are changes that favor the candidate, then that candidate should win the reelection.

• The Irrelevant Alternatives Criterion– If a candidate wins an election and, in a recount, the

only changes are that one or more of the other candidates are removed from the ballot, then that candidate should still win the election.

Finding a Speaker

• The 58 members of the Student Activity Council are meeting to elect a keynote speaker.– Bill Gates– Howard Stern– Oprah Winfrey

• They take a straw vote before an actual election

Finding a SpeakerPreference Table for the Straw Vote

# of votes 20 16 14 81st Choice W S G G2nd Choice G W S W3rd Choice S G W S

Preference Table for the Second Election

# of votes 28 16 141st Choice W S G2nd Choice G W S3rd Choice S G W

Using the plurality-with-elimination method, which speaker wins each election?

What criterion does this eliminate?

Another Mayoral Election# of votes 150 90 90 301st Choice A C D D2nd Choice B B A A3rd Choice C D C B4th Choice D A B C

• Use the pairwise comparison method to determine the winner of the election.

• What if B & C drop out of the race. Now who wins?

• What fairness criterion does this violate?

Electing a Principal

• Use the Borda count method to find the new principal.• Who has a majority of the first-place votes?• Which criterion does this violate?

Preference Table for Selecting a New Principal# of Votes 6 4 2 21st Choice A B B A2nd Choice B C D B3rd Choice C D C D4th Choice D A A C

Voting MethodsVoting Method

Fairness Criteria

Plurality Method

Borda Count

Method

Plurality-with-

Elimination

Pairwise Comparison

Majority Always satisfies

May not satisfy

Always satisfies

Always satisfies

Head-to-Head May not satisfy

May not satisfy

May not satisfy

Always satisfies

Monotonicity Always satisfies

Always satisfies

May not satisfy

Always satisfies

Irrelevant Alternatives

May not satisfy

May not satisfy

May not satisfy

May not satisfy

Arrow’s Impossibility Theorem

• It is mathematically impossible for any democratic voting system to satisfy each of the four fairness criteria.

• There does not exist, and will never exist, any democratic voting system that satisfies all of the fairness criteria.


Homework

NONE!

Next Session: Monday, May 24Last 3 sessions are next week (M, Th, F)!

Documents

14.2 Homework Solutions Plurality: Musical play Borda count: Classical play Plurality-with-elimination: Classical play Pairwise Comparison: Classical play