Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. 1 The Mathematics of Voting The Paradoxes of Democracy Vote! In a democracy, the rights and duties of citizenship are captured in that simple one-word mantra. We vote in presidential elections, gubernatorial elections, local elections, school bonds, stadium bonds, American Idol selections, and initiatives large and small.
Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. 1 The Mathematics of Voting The Paradoxes of Democracy The paradox is that the more opportunities we have to vote, the less we seem to appreciate and understand the meaning of voting. Why should we vote? Does our vote really count? How does it count?
Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. 1 The Mathematics of Voting Voting Theory First half is voting; Second half is counting. Arrow’s impossibility theorem: A method for determining election results that is democratic and always fair is a mathematical impossibility.
Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. 1 The Mathematics of Voting 1.1Preference Ballots and Preference Schedules 1.2The Plurality Method 1.3 The Borda Count Method 1.4The Plurality-with-Elimination Method (Instant Runoff Voting) 1.5The Method of Pairwise Comparisons 1.6Rankings
Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. The Math Appreciation Society (MAS) is a student organization dedicated to an unsung but worthy cause, that of fostering the enjoyment and appreciation of mathematics among college students. The Tasmania State University chapter of MAS is holding its annual election for president. There are four candidates running for president: Alisha, Boris, Carmen, and Dave (A, B, C, and D for short). Each of the 37 members of the club votes by means of a ballot indicating his or her first, second, third, and fourth choice. The 37 ballots submitted are shown on the next slide. Once the ballots are in, it’s decision time. Who should be the winner of the election? Why? Example 1.1The Math Club Election
Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. Example 1.1The Math Club Election
Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. Essential ingredients of every election: Example 1.1The Math Club Election Voters Candidates (electing people); Choice (nonhuman alternatives-cities, colleges, pizza toppings, etc.) Ballots: Preference Ballot: rank in order of preference Linear Ballot: ties are not allowed
Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. Preference Schedule Example 1.1The Math Club Election Only a few different ways to rank results: organize in a preference schedule
Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. Transitivity and Elimination of Candidates Example 1.1The Math Club Election Transitive: Voter prefers A over B and B over C then automatically prefers A over C Elimination: Relative preferences are not affected by the elimination of one or more candidates
Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. In the next section we return to the business of deciding the outcome of elections in general and the Math Club election in particular. Example 1.1The Math Club Election