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Voting Methods and Fairness Criteria: A Comprehensive Overview

This chapter explores different voting methods such as preference ballots, linear ballots, preference schedules, and their adherence to fairness criteria like Condorcet Criterion, Majority Criterion, and Independence of Irrelevant Alternatives. It also discusses the limitations of voting methods and introduces Arrow's Impossibility Theorem.

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Voting Methods and Fairness Criteria: A Comprehensive Overview

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  1. Discrete Math CHAPTER ONE 1.1 • Preference Ballot: A ballot in which the voters are asked to rank the candidates in order of preference. • Linear Ballot: A ballot in which ties are not allowed. • Preference Schedule: Organize the ballots by grouping identical ballots. • Transitivity: If A beats B and B beats C, then A will beat C. If we need to know which candidate a voter would vote for if it came down to a choice between two candidates, all we have to do is look at which candidate was placed higher on the voter’s ballot.

  2. Discrete Math CHAPTER ONE 1.2 • Plurality: Candidate with the most first place votes wins. • Majority Rule: In an election the candidate with more than half the votes will win. • The Majority Criterion: If a choice receives a majority of the first place votes in an election, then that choice should be the winner of the election. • Condorcet Method: A candidate who wins every head to head comparison against each of the other candidates wins. To be continued...

  3. Discrete Math CHAPTER ONE 1.2 (Continued...) • Condorcet Criterion: If there is a choice that in a head-to-head comparison is preferred by the voters over each of the other choices, then that choice should be the winner of the election. • Plurality: • Violates the Condorcet criterion. • Weakness: Insincere Voting: • Voter who changes the true order of his or her preferences in the ballot in an effort to influence the outcome.

  4. Discrete Math CHAPTER ONE 1.3 • Borda Count: Each place on a ballot is assigned points, and the candidate with the highest total is the winner. • Violates the Majority Criterion • Violates the Concordet Criterion.

  5. Discrete Math CHAPTER ONE 1.4 • Plurality with Elimination: The candidate with the fewest first place votes is eliminated. Continue this process until a winner is achieved. • Violates the Monotonicity Criterion: If choice X is a winner of an election and, in a reelection, the only changes in the ballots are changes that only favor X, then X should remain a winner of the election. • Violates the Condorcet Criterion.

  6. Discrete Math CHAPTER ONE 1.5 • Method of Pairwise Comparison (Copeland’s Method): Like a round robin tournament in which every candidate is matched one-to-one with every other candidate. A win is worth a point (ties are 1/2) and whoever has the most points is the winner. • Violates the Independence-of-Irrelevant-Alternatives(I.I.A) Criteria: If choice X is a winner of an election and one (or more) of the other choices is disqualified and the ballots recounted, the X should still be a winner of the election. To be continued…

  7. (N-1)N 2 Discrete Math CHAPTER ONE 1.5 (Continued...) • Sometimes can produce outcome where everyone is a winner. • In general, there is no way to break a tie, and in practice, it is important to establish the rules as to how ties are to be broken ahead of time. • Number of Comparison: or nC2

  8. CHAPTER ONE Discrete Math Summary: Plurality Violates: Condorcet Criterion, I.I.A. Borda: Majority, Condorcet, I.I.A. Plurality with Elimination: Condorcet, Monotonocity, I.I.A. Pairwise Comparison: I.I.A. (Independence of Irrelative Alternatives

  9. Discrete Math CHAPTER ONE 1.6: Ranking • Extended Ranking:Ranking each candidate based on methods. • Plurality: Second place goes to the candidate with the second highest first place votes. • Borda: Ranked by highest to lowest point total. • Plurality with Elimination: First candidate eliminated is ranked last. • Pairwise comparison: Ranked By Points. To be continued...

  10. Discrete Math CHAPTER ONE 1.6 (Continued...) • Recursive Ranking: The winner is removed on the preference table and method X is reapplied. This continues until every candidate is ranked. • Plurality with elimination: Candidates are eliminated until the winner is left. The winner is then removed and method continues until all other candidates are ranked. • It is Reasonable to expect that a fair voting method ought to satisfy all of the criteria. • Arrow’s Impossibility Theorem: It is mathematically impossible of a democratic voting method to satisfy all of the fairness criteria.

  11. End of Chapter 1 Discrete Math CHAPTER ONE

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