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## A Mathematical View of Our World

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**A Mathematical View of Our World**1st ed. Parks, Musser, Trimpe, Maurer, and Maurer**Chapter 3**Voting and Elections**Section 3.1Voting Systems**• Goals • Study voting systems • Plurality method • Borda count method • Plurality with elimination method • Pairwise comparison method • Discuss tie-breaking methods**3.1 Initial Problem**• The city council must select among 3 locations for a new sewage treatment plant. • A majority of city councilors say they prefer site A to site B. • A majority of city councilors say they prefer site A to site C. • In the vote site B is selected. • Did the councilors necessarily lie about their preferences before the election? • The solution will be given at the end of the section.**Voting Systems**• The following voting methods will be discussed: • Plurality method • Borda count method • Plurality with elimination method • Pairwise comparison method**Plurality Method**• When a candidate receives more than half of the votes in an election, we say the candidate has received a majority of the votes. • When a candidate receives the greatest number of votes in an election, but not more than half, we say the candidate has received a plurality of the votes.**Question:**Suppose in an election, the vote totals are as follows. Andy gets 4526 first-place votes. Lacy gets 1901 first-place votes. Peter gets 2265 first-place votes. Choose the correct statement. a. Andy has a majority. b. Andy has a plurality only.**Plurality Method, cont’d**• In the plurality method: • Voters vote for one candidate. • The candidate receiving the most votes wins. • This method has a couple advantages: • The voter chooses only one candidate. • The winner is easily determined.**Plurality Method, cont’d**• The plurality method is used: • In the United States to elect senators, representatives, governors, judges, and mayors. • In the United Kingdom and Canada to elect members of parliament.**Example 1**• Four persons are running for student body president. The vote totals are as follows: • Aaron: 2359 votes • Bonnie: 2457 votes • Charles: 2554 votes • Dion: 2288 votes • Under the plurality method, who won the election?**Example 1, cont’d**• Solution: With 2554 votes, Charles has a plurality and wins the election. • Note that there were a total of 9658 votes cast. • A majority of votes would be at least 4830 votes. Charles did not receive a majority of votes.**Example 2**• Three candidates ran for Attorney General in Delaware in 2002. The vote totals were as follows: • Carl Schnee: 103,913 votes • Jane Brady: 110,784 • Vivian Houghton: 13,860 • What percent of the votes did each candidate receive and who won the election?**Example 2, cont’d**• Solution: A total of 228,557 votes were cast. • Schnee received • Brady received • Houghton received • Brady received a plurality and is the winner.**Borda Count Method**• In the Borda count method: • Voters rank all of the m candidates. • Votes are counted as follows: • A voter’s last choice gets 1 point. • A voter’s next-to-last choice gets 2 points. • … • A voter’s first choice gets m points. • The candidate with the most points wins.**Borda Count Method, cont’d**• The main advantage of the Borda count method is that it uses more information from the voters. • A variation of the Borda count method is used to select the winner of the Heisman trophy.**Example 3**• Four persons are running for student body president. Voters rank the candidates as shown in the table below. • Under the Borda count method, who is elected?**Example 3, cont’d**• Solution: Convert the votes to points.**Example 3, cont’d**• Solution: Total the points for each person: • Aaron: 9436 + 4104 + 5572 + 3145 = 22,257 • Bonnie: 9828 + 10,497 + 4948 + 1228 = 26,501 • Charles: 10,216 + 7101 + 3468 + 3003 = 23,788 • Dion: 9152 + 7272 + 5328 + 2282 = 24,034 • Bonnie has the most points and is the winner.**Example 3, cont’d**• Note that in this same election: • Charles won using the plurality method because he had more first place votes than any other candidate. • Bonnie won using the Borda count method because her point total was highest, due to having many second-place votes.**Plurality with Elimination Method**• In the plurality with elimination method: • Voters choose one candidate. • The votes are counted. • If one candidate receives a majority of the votes, that candidate is selected. • If no candidate receives a majority, eliminate the candidate who received the fewest votes and do another round of voting.**Plurality with Elimination, cont’d**• Cont’d: • This process is repeated until someone receives a majority of the votes and is declared the winner. • The plurality with elimination method is used: • To select the location of the Olympic games. • In France to elect the president.**Plurality with Elimination, cont’d**• Rather than needing to potentially conduct multiple votes, the voters can be asked to rank all candidates during the first election. • A preference table is used to display these rankings.**Example 4**• Four persons are running for department chairperson. The 17 voters ranked the candidates 1st through 4th. • Under plurality with elimination, who is the winner?**Example 4, cont’d**• Solution: Some voters had the same preference ranking. Identical rating have been grouped to form the preference table below. • The number at the top of each column indicates the number of voters who shared that ranking.**Example 4, cont’d**• Solution, cont’d: The first-place votes for each candidate are totaled: • Alice: 6; Bob: 4; Carlos: 4; Donna: 3 • No candidate received a majority, 9 votes. • Donna, who has the fewest first-place votes, is eliminated.**Example 4, cont’d**• Solution, cont’d: A new preference table, without Donna, must be created. • Donna is eliminated from each column. • Any candidates ranked below Donna move up.**Example 4, cont’d**• Solution, cont’d: The first-place votes for each candidate are totaled: • Alice: 7; Bob: 4; Carlos: 6 • No candidate received a majority. • Bob, who has the fewest first-place votes, is eliminated.**Example 4, cont’d**• Solution, cont’d: A new preference table, without Bob, must be created. • Bob is eliminated from each column. • Any candidates ranked below Bob move up.**Example 4, cont’d**• Solution, cont’d: The first-place votes for each candidate are totaled: • Alice: 9; Carlos: 8 • Alice received a majority and is the winner.**Pairwise Comparison Method**• In the pairwise comparison method: • Voters rank all of the candidates. • For each pair of candidates X and Y, determine how many voters prefer X to Y and vice versa. • If X is preferred to Y more often, X gets 1 point. • If Y is preferred to X more often, Y gets 1 point. • If the candidates tie, each gets ½ a point. • The candidate with the most points wins.**Pairwise Comparison, cont’d**• The pairwise comparison method is also called the Condorcet method.**Example 5**• Three persons are running for department chair. The 17 voters rank all the candidates, as shown in the preference table below. • Under the pairwise comparison method, who wins the election?**Example 5, cont’d**• Solution: There are 3 pairs of candidates to compare: • Alice vs. Bob • Alice vs. Carlos • Bob vs. Carlos • For each pair of candidates, delete the third candidate from the preference table and consider only the two candidates in question.**Example 5, cont’d**• Solution, cont’d: • Alice receives 10 first-place votes, while Bob only receives 7. • We say Alice is preferred to Bob 10 to 7. • Alice receives one point.**Example 5, cont’d**• Solution, cont’d: • Alice is preferred to Carlos 9 to 8, so Alice receives another point.**Example 5, cont’d**• Solution, cont’d: • Carlos is preferred to Bob 10 to 7, so Carlos receives one point.**Example 5, cont’d**• Solution, cont’d: The final point totals are: • Alice: 2 points • Bob: 0 points • Carlos: 1 point • Alice wins the election.**Question:**Candidate B is the winner of an election with the following preference table . What voting method could have been used to determine the winner? a. Plurality method b. Borda count method c. Plurality with elimination method d. Pairwise comparison method**Voting Methods, cont’d**• The four voting systems studied here can produce different winners even when the same voter preference table is used. • Any of the four methods can also produce a tie between two or more candidates, which must be broken somehow.**Tie Breaking**• A tie-breaking method should be chosen before the election. • To break a tie caused by perfectly balanced voter support, election officials may: • Make an arbitrary choice. • Flipping a coin • Drawing straws • Bring in another voter. • The Vice President votes when the U. S. Senate is tied.**3.1 Initial Problem Solution**• A majority of city councilors said they preferred site A to site B and also site A to site C. If B won the election, did they necessarily lie? • Solution: • The councilors would not have to lie in order for this to happen. This situation can occur with some voting methods.**Initial Problem Solution, cont’d**• For example, this situation could occur if the voting method used was plurality with elimination. • Suppose 11 councilors ranked the sites as shown in the table below.**Initial Problem Solution, cont’d**• Notice that in this scenario: • Site A is preferred to site B 7 to 4. • Site A is preferred to site C 7 to 4. • However, in the vote count: • Site A, with the fewest first-place votes, is eliminated. • In the second round of voting, site B wins.**Section 3.2Flaws of the Voting Systems**• Goals • Study fairness criteria • The majority criterion • Head-to-head criterion • Montonicity criterion • Irrelevant alternatives criterion • Study fairness of voting methods • Arrow impossibility theorem • Approval voting**3.2 Initial Problem**• The Compromise of 1850 averted civil war in the U.S. for 10 years. • Henry Clay proposed the bill, but it was defeated in July 1850. • A short time later, Stephen Douglas was able to get essentially the same proposals passed. • How is this possible? • The solution will be given at the end of the section.**Flaws of Voting Systems**• We have seen that the choice of voting method can affect the outcome of an election. • Each voting method studied can fail to satisfy certain criteria that make a voting method “fair”.**Fairness Criteria**• The fairness criteria are properties that we expect a good voting system to satisfy. • Four fairness criteria will be studied: • The majority criterion • The head-to-head criterion • The monotonicity criterion • The irrelevant alternatives criterion**The Majority Criterion**• If a candidate is the first choice of a majority of voters, then that candidate should be selected.**Question:**Candidate A won an election with 3000 of the 8500 votes. Was the majority criterion necessarily violated? a. yes b. no**The Majority Criterion, cont’d**• For the majority criterion to be violated: • A candidate must have more than half of the votes. • This same candidate must not win the election. • Note: • This criterion does not say what should happen if no candidate receives a majority. • This criterion does not say that the winner of an election must win by a majority.