14.2 Homework Solutions Plurality: Musical play Borda count: Classical play Plurality-with-elimination: Classical play Pairwise Comparison: Classical play Number of Votes st ChoiceMCDCDM 2 nd ChoiceCMCDMD 3 rd ChoiceDDMMCC

May 19, 2010 Math 132: Foundations of Mathematics Amy Lewis Math Specialist IU1 Center for STEM Education

May 19, 2010 Math 132: Foundations of Mathematics 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. May 19, 2010 Math 132: Foundations of Mathematics

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 May 19, 2010 Math 132: Foundations of Mathematics Preference Table for the Smallville Mayoral Election Number of Votes First ChoiceADDC Second ChoiceBBBB Third ChoiceCCAA Fourth ChoiceDACD

Pairwise Comparison Method Bob gets 1 point. May 20, 2010 Math 132: Foundations of Mathematics Preference Table for the Smallville Mayoral Election Number of Votes First ChoiceADDC Second ChoiceBBBB Third ChoiceCCAA Fourth ChoiceDACD Antonio vs. Bob

Pairwise Comparison Method Carmen gets 1 pt. May 19, 2010 Math 132: Foundations of Mathematics Preference Table for the Smallville Mayoral Election Number of Votes First ChoiceADDC Second ChoiceBBBB Third ChoiceCCAA Fourth ChoiceDACD Antonio vs. Carmen

Pairwise Comparison Method Antonio gets 1 pt. May 19, 2010 Math 132: Foundations of Mathematics Preference Table for the Smallville Mayoral Election Number of Votes First ChoiceADDC Second ChoiceBBBB Third ChoiceCCAA Fourth ChoiceDACD Antonio vs. Donna

Pairwise Comparison Method Bob gets 1 point May 19, 2010 Math 132: Foundations of Mathematics Preference Table for the Smallville Mayoral Election Number of Votes First ChoiceADDC Second ChoiceBBBB Third ChoiceCCAA Fourth ChoiceDACD Bob vs. Carmen

Pairwise Comparison Method Bob gets 1 pt. May 19, 2010 Math 132: Foundations of Mathematics Preference Table for the Smallville Mayoral Election Number of Votes First ChoiceADDC Second ChoiceBBBB Third ChoiceCCAA Fourth ChoiceDACD Bob vs. Donna

Pairwise Comparison Method Carmen gets 1 pt. May 19, 2010 Math 132: Foundations of Mathematics Preference Table for the Smallville Mayoral Election Number of Votes First ChoiceADDC Second ChoiceBBBB Third ChoiceCCAA Fourth ChoiceDACD 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 Speaker Preference Table for the Straw Vote # of votes st Choice WSGG 2 nd Choice GWSW 3 rd Choice SGWS Preference Table for the Second Election # of votes st Choice WSG 2 nd Choice GWS 3 rd Choice SGW Using the plurality-with-elimination method, which speaker wins each election? What criterion does this eliminate?

Another Mayoral Election # of votes st ChoiceACDD 2 nd ChoiceBBAA 3 rd ChoiceCDCB 4 th ChoiceDABC 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 st Choice ABBA 2 nd Choice BCDB 3 rd Choice CDCD 4 th Choice DAAC

Voting Methods Voting Method Fairness Criteria Plurality Method Borda Count Method Plurality- with- Elimination Pairwise Comparison Majority Always satisfies May not satisfy Always satisfies Head-to-Head May not satisfy Always satisfies Monotonicity Always satisfies May not satisfy Always satisfies Irrelevant Alternatives 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.

May 19, 2010 Math 132: Foundations of Mathematics Homework NONE! Next Session: Monday, May 24 Last 3 sessions are next week (M, Th, F)!