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**Chapter 1: Methods of Voting**

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**Arrow’s Impossibility Theorem**

For elections with three or more candidates “ a method for determining election results that is democratic and always fair is a mathematical impossibility

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**1.1 Preference Ballots & Preference Schedules**

Preference Ballot- a ballot where the voters are asked to rank the candidates in order of preference Linear Ballot- a ballot in which ties are not allowed

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1.2 The Plurality Method Plurality Method – all we care about is the first place votes, the candidate with the most votes wins. We don’t need the voters to rank the candidate. Majority candidate – the candidate with a majority (more than half) of the first place votes.

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1.2 The Plurality Method with 3 or more candidates there is no guarantee that there is going to be a majority candidate The Majority criterion – If candidate X has a majority of the first place votes, the candidate X should be the winner of the election.

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1.2 The Plurality Method Condorcet candidate – a candidate preferred by a majority of voters over every other candidate when the candidates are compared in head-to-head comparisons. Not every election has one. The Condorcet criterion – If candidate X is preferred by the voters over each of the other candidates in a head-to-head comparison, then candidate X should be the winner of the election.

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**1.2 Flaws of the Plurality Method**

It sometimes leads us to pick a choice that is loved or hated by the voters instead of one that is preferred by most The ease with which election results can be manipulated by a voter or block of voters through insincere voting.

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**1.2 Flaws of the Plurality Method**

Insincere Voting (Strategic Voting) – If we know the candidate we really want doesn’t have a chance of winning, then rather than “waste our vote” on our favorite candidate we can cast it for a lesser choice who has a better chance of winning.

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The Borda Count Method Each place on the ballot is assigned points. For an election with N candidates, last place = 1 point, second to last place = 2 points, and so on until first place = N points. The points are then added up and the candidate with the most points wins.

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The Borda Count Method This method is widely used in many different settings such as sports, awards, and companies. For example Heisman Trophy winner, NBA Rookie of the Year, NFL MVP, college football polls, music awards, and hiring processes.

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**Flaws of the Borda Count Method**

This method can violate the 2 basic criterion for fairness, the majority criterion and the condorcet criterion. These violations are RARE especially if there are a lot of candidates.

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**1.4 The Plurality-with-Elimination Method (Instant Runoff Voting)**

Instant Runoff Voting – use the preference schedule to eliminate the candidates with the fewest first place votes until 1 candidate has the majority of the votes. The election must use a preference ballot.

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**1.4 The Plurality-with-Elimination Method (Instant Runoff Voting)**

Round 1: Count the first place votes. If there is a majority candidate, then that candidate is the winner. If not eliminate the candidates with the fewest first place votes. Round 2: Cross out the candidate(s) that were eliminated and recount the first place votes. If there is a majority, declare the winner. If not continue the process until you have a majority candidate.

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**1.4 The Plurality-with-Elimination Method (Instant Runoff Voting)**

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

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**1.4 The Plurality-with-Elimination Method (Instant Runoff Voting)**

This method is becoming more popular because you don’t have to hold a run-off election. This method is currently used with the Olympic Committee, and some local elections in California, Vermont, Michigan, and Australia.

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**1.5 The Method of Pairwise Comparisons**

Pairwise comparison – every candidate is matched head-to-head against every other candidate The one with the most votes in the comparison wins. If they are split equally, the comparison ends in a tie.

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**1.5 The Method of Pairwise Comparisons**

The winner of the comparison gets 1 point, the loser gets none; if there was a tie, each candidate gets ½ point. The winner of the election is the one with the most points after all the comparisons are done.

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**1.5 The Method of Pairwise Comparisons**

This method satisfies all 3 of the fairness criteria discussed so far. The Independence-of-Irrelevant criterion (IIA) – If candidate X is a winner of an election and in a recount one of the nonwinning candidates withdraws or is disqualified, then X should still be a winner of the election.

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**1.5 The Method of Pairwise Comparisons**

This method can be quite indecisive, it is not unusual to have multiple ties for first place. As the number of candidates grow, the number of comparisons grows even quicker.

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**1.5 The Method of Pairwise Comparisons**

Sum of consecutive integers formula – Number of Pairwise Comparisons with N Candidates -

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1.6 Rankings Ranking of candidates – who comes first, second, third, etc. With the plurality method - count the number of first place votes, then rank in order from greatest to least. With the borda method – find the total points for each candidate and use that to rank them.

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1.6 Rankings With the plurality-with-elimination method – the first one eliminated is in last place & so on until you get to first place. With the pairwise comparisons – rank the candidates using the total points from the comparisons.

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1.6 Rankings Recursive Ranking – Find the winner, remove the winners name from the preference schedule and make a new one. Continue the process until all places are filled. You can do this using any method.

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Excursions in Modern Mathematics, 7e: 1.5 - 2Copyright © 2010 Pearson Education, Inc. 1 The Mathematics of Voting 1.1Preference Ballots and Preference.

Excursions in Modern Mathematics, 7e: 1.5 - 2Copyright © 2010 Pearson Education, Inc. 1 The Mathematics of Voting 1.1Preference Ballots and Preference.

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