The Mathematics of Voting Chapter 1

Preference Ballot A Ballot in which the voters are asked to rank the candidates in order of preference. 1. Brownies 2. Cupcakes 3. Cookies 4. Donuts

Preference Schedule A way to organize preference ballots by grouping together identical ballots. Transfer your ballot to a sticky note.

Fairness Criterion Majority Criterion – If candidate X has a majority (more than 50%) of the first-place votes, then candidate X should be the winner of the election.

Fairness Criterion Condorcet Criterion - If candidate X is preferred by the voters over each of the other candidates in a head-to-head comparison, the candidate X should be the winner of the election.

Fairness Criterion Monotonicity Criterion –If candidate X is a winner of an election and, in a reelection, the only changes in the ballots are changes that favor X (and only X), then X should remain a winner of the election.

Fairness Criterion Independence-of-Irrelevant Alternatives Criterion (IIA) – If candidate X is a winner of an election and in a recount one of the non-winning candidates is removed from the ballots, then X should still be a winner of the election.

Voting Methods Plurality Method – determines the winner of an election by counting only the 1 st place votes. ◦Majority Rule – the candidate with more than half the votes is the winner of the election.  Always the case in a two candidate election.  Not always the case when more than 2 candidates.

Voting Methods Number of voters st ChoiceACDBC 2 nd ChoiceBBCDD 3 rd ChoiceCDBCB 4 th ChoiceDAAAA

Plurality Method ProsCons Majority Criterion is always satisfied when only two candidates Gives most wanted candidate. May not be clear majority when >2 candidates. Fails to take into account voter’s preference beyond first choice. Susceptible to manipulation via insincere voting.

Voting Methods Borda Count Method – Each place on the ballot is assigned points; 1 pt. for last place, 2 pts. for 2 nd place & so on to N pts. For 1 st place candidate. The points are tallied and the candidate with the highest amount of points is the winner.

Voting Methods Number of voters st ChoiceACDBC 2 nd ChoiceBBCDD 3 rd ChoiceCDBCB 4 th ChoiceDAAAA

Borda Count Method ProsCons Takes into account all preferences. Gives best compromise candidate. May violate both Majority and Condorcet Criterion.

Voting Methods Number of voters st ChoiceAACDD B 2 nd ChoiceBDECC E 3 rd ChoiceCBDBB A 4 th ChoiceDCAEA C 5 th choiceEEBAE D

Plurality with Elimination (Instant Run-off Voting – IRV) A method that creates a majority candidate by eliminating candidates with the fewest 1 st place votes.

Plurality with Elimination 1. Count the 1 st place votes, if no majority cross off the candidate with the fewest 1 st place votes; all other candidates move up a spot. 2. Recalculate the 1 st place votes. 3. If there is still no majority candidate, repeat steps 1 & 2 until you have a majority.

Straw Poll Results # of Voters st ChoiceABCA 2 nd ChoiceBCAC 3 rd ChoiceCABB

Final Election Results # of Voters st ChoiceABC 2 nd ChoiceBCA 3 rd ChoiceCAB

Plurality w/ Elimination Method ProsCons Satisfies Majority Criterion. Saves time and $ b/c no run-off election is needed. May violate both Monotonicity and Condorcet Criterion.

Pairwise Comparision (Copeland’s Method) Between candidate X & Y, the vote goes to whichever of the two candidates is listed higher on the ballot.

Pairwise Comparison Method 1. Compare 2 candidates (X & Y) at a time, whoever is highest in the column wins all votes in that column. 2. Tally all votes for X & Y; whoever has the most votes is assigned 1 pt, loser gets 0 pt., and each get ½ pt. if there is a tie. 3. Repeat process with all other candidates (head-to-head). 4. The Candidate with the most points is the winner.

Pairwise Comparison

Voting Methods Number of voters st ChoiceACDBC 2 nd ChoiceBBCDD 3 rd ChoiceCDBCB 4 th ChoiceDAAAA

Pairwise Comparison Method ProsCons Satisfies Condorcet Criterion Gives candidate preferred by majority of voters over the other candidates. Can be impractical to calculate by hand. Can violate IIA Criterion.

Ranking Candidates Shows candidates in order of voter’s preference. Sometimes used to fill multiple positions. ◦Top three get municipality seats. ◦1 st gets Pres, 2 nd VP, 3 rd Secretary, 4 th Treasurer

Extended Plurality Only looks at first place votes to rank. Number of voters st ChoiceACDBC 2 nd ChoiceBBCDD 3 rd ChoiceCDBCB 4 th ChoiceDAAAA

Extended Borda Count Ranks candidates from Highest to lowest Borda Count numbers.

Extended Plurality w/Elimination First candidate eliminated is ranked last, the 2 nd candidate eliminated is ranked 2 nd to last, and so on until you have the majority winner who is ranked first.

Extended Pairwise Comparison Ranks candidates from highest to lowest number from pairwise comparisions.

Rankings Number of voters st ChoiceACDBC 2 nd ChoiceBBCDD 3 rd ChoiceCDBCB 4 th ChoiceDAAAA