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 Plurality-with-Elimination Method When there are three or more candidates running, it isoften the case that no candidate gets a majority. Typically, the candidate or candidates with the fewest first-place votes are eliminated, and a runoff election isheld. Since runoff elections are expensive to both the candidates and the municipality, this is an inefficient and a cumbersome method for choosing a mayor or acounty supervisor.

4. The Plurality-with-Elimination Method A much more efficient way to implement the same process without needingseparate runoff elections is to use preference ballots, since a preference ballot tellsus not only which candidate the voter wants to win but also which candidate thevoter would choose in a runoff between any pair of candidates. The idea is simplebut powerful: From the original preference schedule for the election we can eliminate the candidates with the fewest first-place votes one at a time until one of themgets a majority.

5. The Plurality-with-Elimination Method This method has become increasingly popular and is nowadays fashionably known as instant runoff voting (IRV). Other names had been used in the past and in other countries for the same method, including plurality-with-elimination and the Hare method. We will call it the plurality-with-elimination method - it is the most descriptive of the three names.

6. The Plurality-with-Elimination Method Round 1: Count the first-place votes for each candidate, just as you would in the plurality method. If a candidate has a majority of first-place votes, then that candidate is the winner. Otherwise, eliminate the candidate (or candidates if there is a tie) with the fewest first-place votes.

7. The Plurality-with-Elimination Method Round 2: Cross out the name(s) of the candidates eliminated from the preference schedule and recount the first-place votes. (Remember that when a candidate is eliminated from the preference schedule, in each column the candidates below it move up a spot.) If a candidate has a majority of first-place votes, then declare that candidate the winner. Otherwise, eliminate the candidate with the fewest first-place votes.

8. The Plurality-with-Elimination Method Round 3, 4 . . . . Repeat the process, each time eliminating one or more candidates until there is a candidate with a majority of first-place votes. That candidate is the winner of the election.

9. Example 1.7 The Math Club Election (Plurality-with-Elimination) Let’s see how the plurality-with-eliminationmethod works when applied to the Math Clubelection. For the reader’s convenience Table 1-6shows the preference schedule again. It is theoriginal preference schedule for the election firstshown in Table 1-1.

10. Example 1.7 The Math Club Election (Plurality-with-Elimination)

11. Example 1.7 The Math Club Election (Plurality-with-Elimination) Round 1.

12. Example 1.7 The Math Club Election (Plurality-with-Elimination) Round 2. B’s 4 votes go to D, the next best candidate according to these 4 voters.

13. Example 1.7 The Math Club Election (Plurality-with-Elimination) Round 3. C’s 11 votes go to D, the next best candidate according to these 11 voters.

14. Example 1.7 The Math Club Election (Plurality-with-Elimination) We now have a winner, and lo and behold, it’s neitherAlisha nor Boris. The winner of the election, with 23first-place votes, is Dave!

15. What’s wrong with the Pluarality-with-Elimination Method? The main problem with the plurality-with-elimination method is quite subtle andis illustrated by the next example.

16. Example 1.10 There Go the Olympics Three cities, Athens (A), Barcelona (B), and Calgary (C), are competing to host the Summer Olympic Games. The final decision is made by a secret vote of the 29 members of the Executive Council of the International Olympic Committee, and the winner is to be chosen using the plurality-with-elimination method. Two days before the actual election is to be held, a straw poll is conducted by the Executive Council just to see how things stand. The results of the straw poll are shown in Table 1-9.

17. Example 1.10 There Go the Olympics

18. Example 1.10 There Go the Olympics Based on the results of the straw poll, Calgary is going to win the election. (In the first round Athens has 11 votes, Barcelona has 8, and Calgary has 10. Barcelona is eliminated, and in the second round Barcelona’s 8 votes go to Calgary. With 18 votes in the second round, Calgary wins the election.)

19. Example 1.10 There Go the Olympics Although the results of the straw poll are supposed to be secret, the word gets out that it looks like Calgary is going to host the next Summer Olympics. Since everybody loves a winner, the four delegates represented by the last column of Table 1-9 decide as a block to switch their votes and vote for Calgary first and Athens second. Calgary is going to win, so there is no harm in that, is there? Well, let’s see.

20. Example 1.10 There Go the Olympics The results of the official vote are shown inTable 1-10. The only changes between the straw pollin Table 1-9 and the official vote are the 4 votes thatwere switched in favor of Calgary. (To get Table 1-10,switch A and C in the last column of Table 1-9 andthen combine columns 3 and 4 - they are now thesame - into a single column.)

21. Example 1.10 There Go the Olympics

22. Example 1.10 There Go the Olympics When we apply the plurality-with-eliminationmethod to Table 1-10, Athens gets eliminated in thefirst round, and the 7 votes originally going toAthens go to Barcelona in the second round.Barcelona, with 15 votes in the second round gets tohost the next Summer Olympics!

23. Example 1.10 There Go the Olympics How could this happen? How could Calgary lose an election it was winning in the straw poll just because it got additional first-place votes in the official election? While you will never convince the conspiracy theorists in Calgary that the election was not rigged, double-checking the figures makes it clear that everything is on the up and up - Calgary is simply the victim of a quirk in the plurality-with-elimination method: the possibility that you can actually do worse by doing better!

24. The Monotonicity Criterion Example1.10 illustrates what in voting theory is known as a violation of the monotonicitycriterion. THE MONOTONICITY CRITERION If candidate X is a winner of an election and, in a reelection, the onlychanges in the ballots are changes that favor X (and only X), then Xshould remain a winner of the election.

25. What’s Wrong with Plurality-with-Elimination • It violates the monotonicity criterion • It violates the Condorcet criterion

26. Plurality-with-Elimination method in Real Life a.k.a. Instant Runoff Voting • International Olympic Committee to choose host cities • Since 2002, San Francisco, CA • Since 2005, Burlington, VT • In process, Berkeley, CA • In process, Ferndale, MI • Australia to elect members of the House of Representatives