1 The Mathematics of Voting 1 1 Preference

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 Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics, 7 e: 1. 3 - 2

The Plurality-with-Elimination Method When there are three or more candidates running, it is often 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 is held. 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 a county supervisor. Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics, 7 e: 1. 3 - 3

The Plurality-with-Elimination Method A much more efficient way to implement the same process without needing separate runoff elections is to use preference ballots, since a preference ballot tells us not only which candidate the voter wants to win but also which candidate the voter would choose in a runoff between any pair of candidates. The idea is simple but 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 them gets a majority. Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics, 7 e: 1. 3 - 4

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. Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics, 7 e: 1. 3 - 5

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 firstplace votes, then that candidate is the winner. Otherwise, eliminate the candidate (or candidates if there is a tie) with the fewest firstplace votes. Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics, 7 e: 1. 3 - 6

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. Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics, 7 e: 1. 3 - 7

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. Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics, 7 e: 1. 3 - 8

Example 1. 7 The Math Club Election (Plurality-with-Elimination) Let’s see how the plurality-with-elimination method works when applied to the Math Club election. For the reader’s convenience Table 1 -6 shows the preference schedule again. It is the original preference schedule for the election first shown in Table 1 -1. Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics, 7 e: 1. 3 - 9

Example 1. 7 The Math Club Election (Plurality-with-Elimination) Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics, 7 e: 1. 3 - 10

Example 1. 7 The Math Club Election (Plurality-with-Elimination) Round 1. Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics, 7 e: 1. 3 - 11

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. Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics, 7 e: 1. 3 - 12

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. Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics, 7 e: 1. 3 - 13

Example 1. 7 The Math Club Election (Plurality-with-Elimination) We now have a winner, and lo and behold, it’s neither Alisha nor Boris. The winner of the election, with 23 first-place votes, is Dave! Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics, 7 e: 1. 3 - 14

What’s wrong with the Pluarality-with. Elimination Method? The main problem with the plurality-withelimination method is quite subtle and is illustrated by the next example. Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics, 7 e: 1. 3 - 15

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. Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics, 7 e: 1. 3 - 16

Example 1. 10 There Go the Olympics Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics, 7 e: 1. 3 - 17

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. ) Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics, 7 e: 1. 3 - 18

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. Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics, 7 e: 1. 3 - 19

Example 1. 10 There Go the Olympics The results of the official vote are shown in Table 1 -10. The only changes between the straw poll in Table 1 -9 and the official vote are the 4 votes that were switched in favor of Calgary. (To get Table 1 -10, switch A and C in the last column of Table 1 -9 and then combine columns 3 and 4 - they are now the same - into a single column. ) Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics, 7 e: 1. 3 - 20

Example 1. 10 There Go the Olympics Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics, 7 e: 1. 3 - 21

Example 1. 10 There Go the Olympics When we apply the plurality-with-elimination method to Table 1 -10, Athens gets eliminated in the first round, and the 7 votes originally going to Athens go to Barcelona in the second round. Barcelona, with 15 votes in the second round gets to host the next Summer Olympics! Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics, 7 e: 1. 3 - 22

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! Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics, 7 e: 1. 3 - 23

The Monotonicity Criterion Example 1. 10 illustrates what in voting theory is known as a violation of the monotonicity criterion. THE MONOTONICITY CRITERION If candidate X is a winner of an election and, in a reelection, the only changes in the ballots are changes that favor X (and only X), then X should remain a winner of the election. Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics, 7 e: 1. 3 - 24

What’s Wrong with Plurality-with. Elimination • It violates the monotonicity criterion • It violates the Condorcet criterion Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics, 7 e: 1. 3 - 25

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 Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics, 7 e: 1. 3 - 26