1 1 n n Discrete Math Preference Ballot

1. 1 n n Discrete Math 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.

Discrete Math 1. 2 n n 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. . .

Discrete Math 1. 2 (Continued. . . ) n n 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. n Weakness: Insincere Voting: n n Voter who changes the true order of his or her preferences in the ballot in an effort to influence the outcome.

Discrete Math 1. 3 n Borda Count: Each place on a ballot is assigned points, and the candidate with the highest total is the winner. n n Violates the Majority Criterion Violates the Concordet Criterion.

Discrete Math 1. 4 n n n 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.

Discrete Math 1. 5 n 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. n 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…

Discrete Math 1. 5 (Continued. . . ) n n n 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: (N-1)N 2 or n C 2

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

Discrete Math 1. 6: Ranking n Extended Ranking: Ranking each candidate based on methods. n n 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. . .

Discrete Math 1. 6 (Continued. . . ) n Recursive Ranking: The winner is removed on the preference table and method X is reapplied. This continues until every candidate is ranked. n n n 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.

Discrete Math