InfoBallots and SchedulesPluralityBordaPlurality with EliminationPairwise ComparisonsInfo Ballots and Schedules Plurality Borda Plurality with Elimination Pairwise ComparisonsThe Mathematics of VotingBeth Kirby and Carl LeeUniversity of KentuckyMA 111Fall 2009Voting UKInfo Ballots and Schedules Plurality Borda Plurality with Elimination Pairwise ComparisonsInfoBallots and SchedulesPluralityBordaPlurality with EliminationPairwise ComparisonsVoting UKInfo Ballots and Schedules Plurality Borda Plurality with Elimination Pairwise ComparisonsCourse InformationText: Peter Tannenbaum, Excursions in Modern Mathematics,second custom edition for the University of Kentucky, Pearson.Course Website:http://www.ms.uky.edu/∼lee/ma111fa09/ma111fa09.htmlVoting UKInfo Ballots and Schedules Plurality Borda Plurality with Elimination Pairwise Comparisons1.1 Preference Ballots and Preference SchedulesVoting UKInfo Ballots and Schedules Plurality Borda Plurality with Elimination Pairwise ComparisonsWho Wins the Election?Below is a preference schedule giving the voter preferences inan election.Number of Voters 8 4 3 21st choice A B B D2nd choice C D C C3rd choice B C D B4th choice D A A AWho should win the election? Why?Voting UKInfo Ballots and Schedules Plurality Borda Plurality with Elimination Pairwise ComparisonsWho Wins the Election?Number of Voters 8 4 3 21st choice A B B D2nd choice C D C C3rd choice B C D B4th choice D A A AIIf everyone votes for their first choice, who gets the mostvotes? (This is the Plurality Method.)A.IIf you do repeated voting for top choices, eliminating lastplace candidates until someone receives the majority ofthe votes, who wins? (This is the Plurality withElimination Method.) B.Voting UKInfo Ballots and Schedules Plurality Borda Plurality with Elimination Pairwise ComparisonsWho Wins the Election?Number of Voters 8 4 3 21st choice A B B D2nd choice C D C C3rd choice B C D B4th choice D A A AIIf everyone votes for their first choice, who gets the mostvotes? (This is the Plurality Method.) A.IIf you do repeated voting for top choices, eliminating lastplace candidates until someone receives the majority ofthe votes, who wins? (This is the Plurality withElimination Method.) B.Voting UKInfo Ballots and Schedules Plurality Borda Plurality with Elimination Pairwise ComparisonsWho Wins the Election?Number of Voters 8 4 3 21st choice A B B D2nd choice C D C C3rd choice B C D B4th choice D A A AIIf everyone votes for their first choice, who gets the mostvotes? (This is the Plurality Method.) A.IIf you do repeated voting for top choices, eliminating lastplace candidates until someone receives the majority ofthe votes, who wins? (This is the Plurality withElimination Method.)B.Voting UKInfo Ballots and Schedules Plurality Borda Plurality with Elimination Pairwise ComparisonsWho Wins the Election?Number of Voters 8 4 3 21st choice A B B D2nd choice C D C C3rd choice B C D B4th choice D A A AIIf everyone votes for their first choice, who gets the mostvotes? (This is the Plurality Method.) A.IIf you do repeated voting for top choices, eliminating lastplace candidates until someone receives the majority ofthe votes, who wins? (This is the Plurality withElimination Method.) B.Voting UKInfo Ballots and Schedules Plurality Borda Plurality with Elimination Pairwise ComparisonsWho Wins the Election?Number of Voters 8 4 3 21st choice A B B D2nd choice C D C C3rd choice B C D B4th choice D A A AIIf you compare candidates head to head, which candidatedoes the majority of voters prefer to each of the others?(Such a candidate is called a Condorcet candidate.)C.Voting UKInfo Ballots and Schedules Plurality Borda Plurality with Elimination Pairwise ComparisonsWho Wins the Election?Number of Voters 8 4 3 21st choice A B B D2nd choice C D C C3rd choice B C D B4th choice D A A AIIf you compare candidates head to head, which candidatedoes the majority of voters prefer to each of the others?(Such a candidate is called a Condorcet candidate.) C.Voting UKInfo Ballots and Schedules Plurality Borda Plurality with Elimination Pairwise ComparisonsPreference BallotsEach voter prepares a preference ballot, in which thecandidates are ranked in order of preference.Example of a filled-out preference ballot:Ballot1st A2nd C3rd B4th DWe will assume that every voter can prepare a preferenceballot with no ties. Such a ballot is called a linear ballot.Voting UKInfo Ballots and Schedules Plurality Borda Plurality with Elimination Pairwise ComparisonsPreference BallotsEach voter prepares a preference ballot, in which thecandidates are ranked in order of preference.Example of a filled-out preference ballot:Ballot1st A2nd C3rd B4th DWe will assume that every voter can prepare a preferenceballot with no ties. Such a ballot is called a linear ballot.Voting UKInfo Ballots and Schedules Plurality Borda Plurality with Elimination Pairwise ComparisonsPreference SchedulesThe preference ballots from an election are collected andtallied in a preference schedule.Example of a preference schedule:Number of Voters 8 4 3 21st choice A B B D2nd choice C D C C3rd choice B C D B4th choice D A A AVoting UKInfo Ballots and Schedules Plurality Borda Plurality with Elimination Pairwise ComparisonsTransitivityIt is natural to assume that a voter’s preferences are transitive.That is to say, if a voter prefers candidate A over candidate B,and prefers candidate B over candidate C, then the voterprefers candidate A over candidate C.Voting UKInfo Ballots and Schedules Plurality Borda Plurality with Elimination Pairwise ComparisonsElimination of CandidateIt is also natural to assume that the relative preferences of avoter are not affected by the elimination of one or more of thecandidates—the voter would just move the lower-rankedremaining candidates higher on the preference ballot.Voting UKInfo Ballots and Schedules Plurality Borda Plurality with Elimination Pairwise ComparisonsSome QuestionsIIn an election with three candidates, what is themaximum number of columns possible in the preferenceschedule?IIn an election with four candidates, what is the maximumnumber of columns possible in the preference schedule?IIn an election with N candidates, what is the maximumnumber of columns possible in the preference schedule?Voting UKInfo Ballots and Schedules Plurality Borda Plurality with Elimination Pairwise ComparisonsSome QuestionsIIn an election with three candidates, what is themaximum number of columns possible in the preferenceschedule?IIn