1 The Mathematics of Voting

2. 1 The Mathematics of Voting 1.1 Preference Ballots and Preference Schedules 1.2 The Plurality Method 1.3 The Borda Count Method 1.4 The Plurality-with-Elimination Method (Instant Runoff Voting) 1.5 The Method of Pairwise Comparisons 1.6 Rankings

3. The Borda Count Method • 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

4. Example 1.5 The Math Club Election (Borda Method) 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 pointvalues under each column based on first place worth 4 points, second place worth3 points, third place worth 2 points, and fourth place worth 1 point.

5. Example 1.5 The Math Club Election (Borda Method)

6. Example 1.5 The Math Club Election (Borda Method) Tally the points: A gets: 56 + 10 + 8 + 4 + 1 = 79 points B gets: 42 + 30 + 16 + 16 + 2 = 106 points C gets: 28 + 40 + 24 + 8 + 4 = 104 points D gets: 14 + 20 + 32 + 12 + 3 = 81 points The Borda winner of this election is Boris! (Wasn’t Alisha the winner of thiselection under the plurality method?)

7. What’s wrong with the Borda Method? 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.

8. Example 1.6 The School Principal Election 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.

9. Example 1.6 The School Principal Election

10. Example 1.6 The School Principal Election A simple count tells us that Mr. B is the Borda winner with 32 points. 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

11. What’s wrong with the Borda Method? The Borda Method violates two basic criteria of fairness: • Majority criterion • Condorcet criterion Despite its flaws, experts in voting theory consider the Borda count method oneof the best, if not the very best, method for deciding elections with many candidates.

12. In Defense of the Borda count method? (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.

13. Borda count method in Real Life • 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