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A Mathematical View of Our World

A Mathematical View of Our World

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A Mathematical View of Our World

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  1. A Mathematical View of Our World 1st ed. Parks, Musser, Trimpe, Maurer, and Maurer

  2. Chapter 3 Voting and Elections

  3. Section 3.1Voting Systems • Goals • Study voting systems • Plurality method • Borda count method • Plurality with elimination method • Pairwise comparison method • Discuss tie-breaking methods

  4. 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.

  5. Voting Systems • The following voting methods will be discussed: • Plurality method • Borda count method • Plurality with elimination method • Pairwise comparison method

  6. 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.

  7. 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.

  8. 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.

  9. 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.

  10. 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?

  11. 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.

  12. 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?

  13. 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.

  14. 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.

  15. 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.

  16. 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?

  17. Example 3, cont’d • Solution: Convert the votes to points.

  18. 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.

  19. 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.

  20. 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.

  21. 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.

  22. 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.

  23. 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?

  24. 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.

  25. 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.

  26. 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.

  27. 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.

  28. 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.

  29. 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.

  30. 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.

  31. Pairwise Comparison, cont’d • The pairwise comparison method is also called the Condorcet method.

  32. 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?

  33. 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.

  34. 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.

  35. Example 5, cont’d • Solution, cont’d: • Alice is preferred to Carlos 9 to 8, so Alice receives another point.

  36. Example 5, cont’d • Solution, cont’d: • Carlos is preferred to Bob 10 to 7, so Carlos receives one point.

  37. 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.

  38. 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

  39. 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.

  40. 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.

  41. 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.

  42. 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.

  43. 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.

  44. 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

  45. 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.

  46. 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”.

  47. 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

  48. The Majority Criterion • If a candidate is the first choice of a majority of voters, then that candidate should be selected.

  49. Question: Candidate A won an election with 3000 of the 8500 votes. Was the majority criterion necessarily violated? a. yes b. no

  50. 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.