Excursions in Modern Mathematics, 7e: 1.3 - 2Copyright © 2010 Pearson Education, Inc. 1 The Mathematics of Voting 1.1Preference Ballots and Preference.

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Excursions in Modern Mathematics, 7e: 1.3 - 2Copyright © 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: 1.3 - 3Copyright © 2010 Pearson Education, Inc. 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. The Plurality-with-Elimination Method

Excursions in Modern Mathematics, 7e: 1.3 - 4Copyright © 2010 Pearson Education, Inc. 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. The Plurality-with-Elimination Method

Excursions in Modern Mathematics, 7e: 1.3 - 5Copyright © 2010 Pearson Education, Inc. 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. The Plurality-with-Elimination Method

Excursions in Modern Mathematics, 7e: 1.3 - 6Copyright © 2010 Pearson Education, Inc. 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. The Plurality-with-Elimination Method

Excursions in Modern Mathematics, 7e: 1.3 - 7Copyright © 2010 Pearson Education, Inc. 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. The Plurality-with-Elimination Method

Excursions in Modern Mathematics, 7e: 1.3 - 8Copyright © 2010 Pearson Education, Inc. 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. The Plurality-with-Elimination Method

Excursions in Modern Mathematics, 7e: 1.3 - 9Copyright © 2010 Pearson Education, Inc. 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. Example 1.7The Math Club Election (Plurality-with-Elimination)

Excursions in Modern Mathematics, 7e: 1.3 - 10Copyright © 2010 Pearson Education, Inc. Example 1.7The Math Club Election (Plurality-with-Elimination)

Excursions in Modern Mathematics, 7e: 1.3 - 11Copyright © 2010 Pearson Education, Inc. Round 1. Example 1.7The Math Club Election (Plurality-with-Elimination)

Excursions in Modern Mathematics, 7e: 1.3 - 12Copyright © 2010 Pearson Education, Inc. Round 2. B’s 4 votes go to D, the next best candidate according to these 4 voters. Example 1.7The Math Club Election (Plurality-with-Elimination)

Excursions in Modern Mathematics, 7e: 1.3 - 13Copyright © 2010 Pearson Education, Inc. Round 3. C’s 11 votes go to D, the next best candidate according to these 11 voters. Example 1.7The Math Club Election (Plurality-with-Elimination)

Excursions in Modern Mathematics, 7e: 1.3 - 14Copyright © 2010 Pearson Education, Inc. 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! Example 1.7The Math Club Election (Plurality-with-Elimination)

Excursions in Modern Mathematics, 7e: 1.3 - 15Copyright © 2010 Pearson Education, Inc. The main problem with the plurality-with- elimination method is quite subtle and is illustrated by the next example. What’s wrong with the Pluarality-with- Elimination Method?

Excursions in Modern Mathematics, 7e: 1.3 - 16Copyright © 2010 Pearson Education, Inc. 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. Example 1.10There Go the Olympics

Excursions in Modern Mathematics, 7e: 1.3 - 17Copyright © 2010 Pearson Education, Inc. Example 1.10There Go the Olympics

Excursions in Modern Mathematics, 7e: 1.3 - 18Copyright © 2010 Pearson Education, Inc. 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.) Example 1.10There Go the Olympics

Excursions in Modern Mathematics, 7e: 1.3 - 19Copyright © 2010 Pearson Education, Inc. 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. Example 1.10There Go the Olympics

Excursions in Modern Mathematics, 7e: 1.3 - 20Copyright © 2010 Pearson Education, Inc. 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.) Example 1.10There Go the Olympics

Excursions in Modern Mathematics, 7e: 1.3 - 21Copyright © 2010 Pearson Education, Inc. Example 1.10There Go the Olympics

Excursions in Modern Mathematics, 7e: 1.3 - 22Copyright © 2010 Pearson Education, Inc. 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! Example 1.10There Go the Olympics

Excursions in Modern Mathematics, 7e: 1.3 - 23Copyright © 2010 Pearson Education, Inc. 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! Example 1.10There Go the Olympics

Excursions in Modern Mathematics, 7e: 1.3 - 24Copyright © 2010 Pearson Education, Inc. 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. THE MONOTONICITY CRITERION

Excursions in Modern Mathematics, 7e: 1.3 - 25Copyright © 2010 Pearson Education, Inc. It violates the monotonicity criterion It violates the Condorcet criterion What’s Wrong with Plurality-with- Elimination

Excursions in Modern Mathematics, 7e: 1.3 - 26Copyright © 2010 Pearson Education, Inc. 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 Plurality-with-Elimination method in Real Life a.k.a. Instant Runoff Voting

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