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Discrete Math

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.

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Discrete Math

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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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