Understanding Fair Elections: Mathematical Principles and Voting Methods
Explore the fascinating intersection of mathematics and democratic elections. This guide introduces key concepts like preference ballots and Arrow’s Impossibility Theorem, explaining the criteria for fair elections, such as majority rule and the Condorcet criterion. You'll learn about different voting methods, including the plurality method, Borda count, and pairwise comparison, and their implications for determining election winners. Understand how these mathematical principles ensure fairness in electoral systems and the challenges posed by the quest for a perfect voting method.
Understanding Fair Elections: Mathematical Principles and Voting Methods
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Presentation Transcript
Mathematics • The study of symbols, shapes, algorithms, sets, and patterns, using logical reasoning and quantitative calculation. • Quantitative Reasoning: Interpreting, understanding, making judgments, and applying mathematical concepts to analyze and solve problems from various backgrounds.
Preference Ballot: Ballots in which a voter is asked to rank all candidates in order of preference
preference schedule (page 5) - When we organize preference ballots by grouping together like ballots we have a preference schedule.
Arrow’s Impossibility Theorem (page 3) - A method for determining election results that is democratic and always fair is a mathematical impossibility.
CRITERIA FOR A FAIR ELECTION majority rule - in a democratic election between two candidates, the one with the majority (more than half) of the votes wins. 1st criteria for a fair election: The Majority Criterion (page 6) - If a choice receives a majority of the first place votes in an election, then that choice should be the winner of the election.
2nd criteria for a fair election The Condorcet Criterion (page 8) - If there is a choice that in a head-to-head comparison is preferred by the voters over every other choice, then that choice should be the winner of the election. A candidate that wins every head-to-head comparison with the other candidates is called a Condorcet candidate.
3rd criteria for a fair election: The Monotonicity Criterion (page 15). 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.
4th criteria for a fair election: The Independence-of-Irrelevant-Alternatives Criterion (page 18). If choice X is a winner of an election and one (or more) of the other choices is removed and the ballots recounted, then X should still be a winner of the election.
Arrow’s Impossibility Theorem (page 3) - A method for determining election results that is democratic and always fair is a mathematical impossibility. • Methods used to find the winner of an election: • Plurality Method • Borda Count Method • Plurality-with-Elimination Method • Method of Pairwise Comparison
I. THE PLURALITY METHOD plurality method (page 6) - the candidate (or candidates) with the most first place votes wins. A plurality does not imply a majority but a majority does imply a plurality.
Example 1.3. The Band Election (page 7) What’s wrong with the plurality method? If we compare the Hula Bowl to any other bowl on a head-to-head basis, the Hula Bowl is always the preferred choice.
What’s wrong with the plurality method? 2nd criteria for a fair election The Condorcet Criterion (page 8) - If there is a choice that in a head-to-head comparison is preferred by the voters over every other choice, then that choice should be the winner of the election.
Homework • Read pages 1 – 11 • Page 30: 1, 2, 3, 6, 11, 12, 13, 14, 18a
