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MAT 105 Spring 2008

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There are many more methods for determining the winner of an election with more than two candidates We will only discuss a few more: sequential pairwise voting Hare system plurality runoff

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Idea: We like pairwise voting (where we can use majority rule), but if we look at all pairwise elections, we sometimes don’t get a winner In sequential pairwise voting, we put the candidates in order on a list, called the agenda

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We pit the first two candidates on the agenda against each other. The winner moves on to face the next candidate on the list, and so on. The candidate remaining at the end is the winner. This process resembles a tournament bracket, and has the advantage that, unlike Condorcet’s method, we always get a winner

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Let’s use sequential pairwise voting with this profile and the agenda A, B, C, D VotersPreference Order 4A > B > D > C 3C > A > B > D 3B > D > C > A A B C D A C D A beats B, 7-3 C beats A, 6-4 D beats C, 7-3

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If we look closely at this agenda, we notice that every single voter prefers B over D, and yet D was our winner! In fact, by cleverly choosing the right agenda, we could make any of the four candidates win this election Sequential pairwise voting does not satisfy the Pareto condition VotersPreference Order 4A > B > D > C 3C > A > B > D 3B > D > C > A

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If every voter prefers one candidate over another, then the latter candidate should not be among the winners of the election Named for Vilfredo Pareto (1848-1923), Italian economist Does plurality satisfy the Pareto condition?

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Also known as Instant Runoff Voting, this system is used for various elections in the US, Canada, the UK, Ireland, and Australia Repeatedly delete candidates that are “least preferred” in the sense of being at the top of the fewest ballots. If there is a tie, eliminate all of the tied candidates, until there is no one left to eliminate

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In this example, A has 5 first-place votes, B has 5 first-place votes, and C has 4 first-place votes, so C is eliminated Now A has 5 first-place votes, and B has 9, so A is eliminated B is the only candidate left, so B is the winner VotersPreference Order 5A > B > C 4C > B > A 3B > C > A 2B > A > C VotersPreference Order 5A > B 4B > A 3 2

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This time, A has 5 first-place votes, and B and C are tied with 4, so B and C are both eliminated at the same time This leaves only A to win the election VotersPreference Order 5A > B > C 4C > B > A 3B > C > A 1B > A > C

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Now let’s modify the profile from the previous example, so that the 1 voter with preference B > A > C now has preference A > B > C Notice that this change moves the winner higher on that voter’s ballot VotersPreference 6A > B > C 4C > B > A 3B > C > A VotersPreference 6A > C 4C > A 3 C wins!

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A was the winner of the original election, and one of the voters changed his ballot to move A higher, causing A to lose This shows that the Hare system is not monotone

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A voting system is monotone if whenever a candidate is a winner, and a new election is held where the only change is for some voter to move that winner higher on his/her ballot, then the original winner should remain the winner The Hare system is not monotone, but despite this drawback it is one of the more common alternative voting systems in use today

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Hold a plurality election, but if no candidate receives a majority, we hold a runoff election The runoff election is between the two candidates who received the most first-place votes in the original election In case of ties, there might be more than two candidates with the most first-place votes, so we use plurality to decide a winner between those candidates only

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In this profile, A gets 5 first-place votes, B gets 5 first-place votes, and C only gets 4 The runoff is between A and B B wins the runoff 9 votes to 5 VotersPreference Order 5A > B > C 4C > B > A 3B > C > A 2B > A > C

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In this profile, A has 4 first-place votes, B has 3, C has 3, and D has 2 The runoff is between A, B, and C We use plurality to decide the winner; keep in mind that the 2 voters who like D best get to vote in the runoff! B wins the runoff with 5 votes VotersPreference Order 4A > B > C > D 3C > D > B > A 3B > C > D > A 2D > B > A > C

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