Info Ballots and Schedules Plurality Borda Plurality with Elimination Pairwise Comparisons The Mathematics of Voting Beth Kirby and Carl Lee University of Kentucky MA 111 Fall 2009 Voting UK Info Ballots and Schedules Plurality Borda Plurality with Elimination Pairwise Comparisons Info Ballots and Schedules Plurality Borda Plurality with Elimination Pairwise Comparisons Voting UK Info Ballots and Schedules Plurality Borda Plurality with Elimination Pairwise Comparisons Course Information Text 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 html Voting UK Info Ballots and Schedules Plurality Borda Plurality with Elimination Pairwise Comparisons 1 1 Preference Ballots and Preference Schedules Voting UK Info Ballots and Schedules Plurality Borda Plurality with Elimination Pairwise Comparisons Who Wins the Election Below is a preference schedule giving the voter preferences in an election Number of Voters 8 1st choice A 2nd choice C 3rd choice B 4th choice D 4 B D C A 3 B C D A 2 D C B A Who should win the election Why Voting UK Info Ballots and Schedules Plurality Borda Plurality with Elimination Pairwise Comparisons Who Wins the Election Number of Voters 8 1st choice A 2nd choice C 3rd choice B 4th choice D I Voting 4 B D C A 3 B C D A 2 D C B A If everyone votes for their first choice who gets the most votes This is the Plurality Method UK Info Ballots and Schedules Plurality Borda Plurality with Elimination Pairwise Comparisons Who Wins the Election Number of Voters 8 1st choice A 2nd choice C 3rd choice B 4th choice D I Voting 4 B D C A 3 B C D A 2 D C B A If everyone votes for their first choice who gets the most votes This is the Plurality Method A UK Info Ballots and Schedules Plurality Borda Plurality with Elimination Pairwise Comparisons Who Wins the Election Number of Voters 8 1st choice A 2nd choice C 3rd choice B 4th choice D I I Voting 4 B D C A 3 B C D A 2 D C B A If everyone votes for their first choice who gets the most votes This is the Plurality Method A If you do repeated voting for top choices eliminating last place candidates until someone receives the majority of the votes who wins This is the Plurality with Elimination Method UK Info Ballots and Schedules Plurality Borda Plurality with Elimination Pairwise Comparisons Who Wins the Election Number of Voters 8 1st choice A 2nd choice C 3rd choice B 4th choice D I I Voting 4 B D C A 3 B C D A 2 D C B A If everyone votes for their first choice who gets the most votes This is the Plurality Method A If you do repeated voting for top choices eliminating last place candidates until someone receives the majority of the votes who wins This is the Plurality with Elimination Method B UK Info Ballots and Schedules Plurality Borda Plurality with Elimination Pairwise Comparisons Who Wins the Election Number of Voters 8 1st choice A 2nd choice C 3rd choice B 4th choice D I Voting 4 B D C A 3 B C D A 2 D C B A If you compare candidates head to head which candidate does the majority of voters prefer to each of the others Such a candidate is called a Condorcet candidate UK Info Ballots and Schedules Plurality Borda Plurality with Elimination Pairwise Comparisons Who Wins the Election Number of Voters 8 1st choice A 2nd choice C 3rd choice B 4th choice D I Voting 4 B D C A 3 B C D A 2 D C B A If you compare candidates head to head which candidate does the majority of voters prefer to each of the others Such a candidate is called a Condorcet candidate C UK Info Ballots and Schedules Plurality Borda Plurality with Elimination Pairwise Comparisons Preference Ballots Each voter prepares a preference ballot in which the candidates are ranked in order of preference Example of a filled out preference ballot Ballot 1st A 2nd C 3rd B 4th D Voting UK Info Ballots and Schedules Plurality Borda Plurality with Elimination Pairwise Comparisons Preference Ballots Each voter prepares a preference ballot in which the candidates are ranked in order of preference Example of a filled out preference ballot Ballot 1st A 2nd C 3rd B 4th D We will assume that every voter can prepare a preference ballot with no ties Such a ballot is called a linear ballot Voting UK Info Ballots and Schedules Plurality Borda Plurality with Elimination Pairwise Comparisons Preference Schedules The preference ballots from an election are collected and tallied in a preference schedule Example of a preference schedule Number of Voters 8 1st choice A 2nd choice C 3rd choice B 4th choice D Voting 4 B D C A 3 2 B D C C D B A A UK Info Ballots and Schedules Plurality Borda Plurality with Elimination Pairwise Comparisons Transitivity It 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 voter prefers candidate A over candidate C Voting UK Info Ballots and Schedules Plurality Borda Plurality with Elimination Pairwise Comparisons Elimination of Candidate It is also natural to assume that the relative preferences of a voter are not affected by the elimination of one or more of the candidates the voter would just move the lower ranked remaining candidates higher on the preference ballot Voting UK Info Ballots and Schedules Plurality Borda Plurality with Elimination Pairwise Comparisons Some Questions I Voting In an election with three candidates what is the maximum number of columns possible in the preference schedule UK Info Ballots and Schedules Plurality Borda Plurality with Elimination Pairwise Comparisons Some Questions I I Voting In an election with three candidates what is the maximum number of columns possible in the preference schedule In an election with four candidates what is the maximum number of columns possible in the preference schedule UK Info Ballots and Schedules Plurality Borda Plurality with Elimination Pairwise Comparisons Some Questions I I I Voting In an election with three candidates what is the maximum number of columns possible in the preference schedule In an election with four candidates what is the maximum number of columns possible in the preference schedule In an election with N candidates what is the maximum number of columns possible in the preference schedule UK Info Ballots and Schedules Plurality Borda Plurality with Elimination Pairwise Comparisons Some Questions Answers I If there are three candidates
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