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. Each place on a ballot is assigned points With N candidates, 1 point for last place, 2 points for second from last, and so on First-place vote is worth N points Tally points for each candidate separately Candidate with highest total is winner Candidate is called the Borda winner The Borda Count Method

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. Let’s use the Borda count method to choose the winner of the Math Appreciation Society election first introduced in Example 1.1. Table 1-4 shows the point values under each column based on first place worth 4 points, second place worth 3 points, third place worth 2 points, and fourth place worth 1 point. Example 1.5The Math Club Election (Borda Method)

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. Example 1.5The Math Club Election (Borda Method)

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. Tally the points: Example 1.5The Math Club Election (Borda Method) A gets: = 79 points B gets: = 106 points C gets: = 104 points D gets: = 81 points The Borda winner of this election is Boris! (Wasn’t Alisha the winner of this election under the plurality method?)

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. In contrast to the plurality method, the Borda count method takes into account all the information provided in the voters’ preference ballots, and the Borda winner is the candidate with the best average ranking - the best compromise candidate if you will. On its face, the Borda count method seems like an excellent way to take full consideration of the voter’s preferences, so indeed, what’s wrong with it? The next example illustrates some of the problems with the Borda count method. What’s wrong with the Borda Method?

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. The last principal at Washington Elementary School has just retired and the School Board must hire a new principal. The four finalists for the job are Mrs. Amaro, Mr. Burr, Mr. Castro, and Mrs. Dunbar (A, B, C, and D, respectively). After interviewing the four finalists, each of the 11 school board members gets to rank the candidates by means of a preference ballot, and the Borda winner gets the job. Table 1-5 shows the preference schedule for this election. Example 1.6The School Principal Election

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. Example 1.6The School Principal Election

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. A simple count tells us that Mr. B is the Borda winner with 32 points. Example 1.6The School Principal Election What about Mrs. A? Majority candidate, 6 of 11 first-place votes, and Condorcet candidate, with 6 first-place votes beats each candidate in head-to-head comparison

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. The Borda Method violates two basic criteria of fairness: What’s wrong with the Borda Method? Majority criterion Condorcet criterion Despite its flaws, experts in voting theory consider the Borda count method one of the best, if not the very best, method for deciding elections with many candidates.

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. (1)Although violations of the majority criterion can happen, they do not happen very often, and when there are many candidates such violations are rare; and (2)violations of the Condorcet criterion automatically follow violations of the majority criterion, since a majority candidate is automatically a Condorcet candidate. In Defense of the Borda count method?

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. individual sports awards (Heisman Trophy winner, NBA Rookie of the Year, NFL MVP, etc.) college football polls music industry awards hiring of school principals, university presidents, and corporate executives Borda count method in Real Life