Section 14

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MGF1107: Section 14.2 Flaws of Voting In the previous section, we learned about four voting methods. We also discovered that the four methods can produce the same winner or different winners. So, although the methods appear to provide reasonable ways to determine winners, there are flaws with each method. Mathematicians and political scientists have agreed that a voting method should meet the following four fairness criteria in order for the voting method to be considered fair. The Head-to-Head criterion is also known as the Condorcet criterion . This criterion is named after the Marquis de Condorcet. The Marquis de Condorcet (1743-1794) was one of the most influential mathematicians during the American and French revolutions. He discovered, much to his disappointment that occasionally there can be no clear and fair way to declare the winner of an election. ( https://en.wikipedia.org/wiki/Marquis_de_Condorcet ) All from etext pages 900-905 Example: Applying the majority criterion The members of the Fayetteville Town Board are holding an election to select a company to maintain the property at the town park. The choices are Green World Landscaping (G), Lawn and Garden Haven (L), and Jubilant Gardens (J). The 13 board members rank the three choices use the Borda count method to make their selection. The preference table is shown below.
from etext page 900 The Borda count method resulted in Lawn and Garden Haven being selected. Does the winner have a majority of first-place votes? Is the majority criterion satisfied? Solution: Lawn and Garden has 6 out of 13 first place votes, which is less than 50%. So the winner of the election, Lawn and Garden Have does not have a majority of the first-place votes. In fact, a majority of the town board members, 7 out of 13, selected Green World Landscaping. Therefore, the Majority criterion is not satisfied. This example demonstrates that the Borda count method has the potential to violate the Majority criterion. Example: The preference table below shows the outcomes of 33 votes. Decide which candidate would be chosen with each of the following voting methods. Also discuss whether or not the method violates the majority criterion. from etext page 900 a. Plurality method The plurality method awards the election to A since it received the most votes for first place: 17 out of 33 total votes cast. This is also a majority (51.5%). The majority criterion states the candidate with the majority of first place votes must win. In general, a candidate who holds a majority of the first-place votes also holds a plurality of first-place votes. Therefore , the plurality method never violates the majority criterion. However, this does not mean that the plurality method always produces a winner with a majority of the votes.
b. Borda count method Each first place vote is worth 3 points, 2 nd place vote 2 points and 3 rd place 1 point. Candidate A: (17 x 3) + (16 x 1) = 67 points Candidate B: (9 x 3) + (24 x 2) = 75 points Candidate C: (7 x 3) + (9 x 2) + (17 x 1) = 56 points Candidate B wins using the Borda count method. Since candidate A holds a majority of the first place votes, this method violates the majority criterion. The Borda count method has the potential for violating the majority criterion. It does not violate in every case but the potential is there. c. Plurality with elimination This method awards the election to Candidate A since this candidate received the majority of the first place votes. No elimination of candidates is needed. In general, a candidate who holds a majority of first-place votes is awarded the election without having to hold a second election, or without having to consider eliminating a candidate and re- aligning voters’ choices. Therefore, the plurality with elimination method never violates the majority criterion. d. Pairwise comparison method There are 3 candidates so there are 3(2)/2 = 3 comparisons Candidate A vs. Candidate B Candidate A: 17 votes Candidate B: 9 + 7 = 16 votes Candidate A gets one point Candidate A vs. Candidate C Candidate A: 17 points Candidate B: 9+7 = 16 points Candidate A gets one point Candidate B vs. Candidate C Candidate B: 17 + 9 = 26 points Candidate C: 7 points Candidate B gets one point Totals: Candidate A: 2 points Candidate B: 1 points Candidate C: 0 points
This method awards the election to candidate A. In general, if a candidate holds a majority of the first place votes, this candidate always wins every pairwise comparison. Thus, the pairwise comparison method never violates the majority criterion. Of the four methods, only the Borda count method can violate the majority criterion. Example: Applying the head to head criterion Suppose that four candidates [Alvarez (A), Buchannan (B), Czechanski (C), and Davis (D)] are running for mayor of Springwater. The election involves ranking the candidates with the results shown below: from etext page 902 a. Is there one candidate who is favored over all others using a head-to-head comparison? 4 candidates so 4(3)/2 = 6 comparisons Alvarez vs. Buchannan: Alvarez: 129 + 90 = 219 votes Buchanan: 87 + 78 + 42 = 207 votes Alvarez wins Alvarez vs. Czechanski: Alvarez: 129 + 90 = 219 points Czechanski: 87 + 78 + 42 = 207 votes Alvarez wins Alvarez vs. Davis: Alvarez: 90 + 87 + 42 = 219 votes Davis: 129 + 78 = 207 votes Alvarez wins Buchannan vs. Czechanski: Buchannan: 129 + 87 = 216 votes Czechanski: 90+ 78 + 42 = 210 votes Buchannan wins Buchannan vs. Davis: Buchannan: 90 + 87 + 78 + 42 = 297 votes Davis: 129 votes Buchannan wins Czechanski vs. Davis: Czechanski: 90+ 87 + 78 + 42 = 297 votes Davis: 129 votes Czechanski wins
Summary: Alvarez is favored over all others. b. Who wins the election if the plurality method is used? Does this result violate the head-to-head criterion? Using the plurality method, Davis wins because he received 129 first-place votes. Therefore, this result violates the head-to-head comparison. This example demonstrates that the plurality method has the potential to violate the head-to-head criterion. c. Who wins if the Borda count method is used? Does this result violate the head-to- head criterion? There are four candidates so first-place is 4 points, second place 3 points, etc. Alvarez: (3 x 129) + (4 x 90) + (2 x 87) + (1 x 78) + (2 x 42) = 1083 Buchannan: (2 x 129) + (2 x 90) + (4 x 87) + (3 x 78) + (3 x 42) = 1146 Czechanski: (1 x 129) + (3 x 90) + (3 x 87) + (4 x 78) + (4 x 42) = 1140 Davis: (4 x 129) + (1 x 90) + (1 x 87) + (2 x 78) + (1 x 42) = 891 Buchannan is elected. Therefore, this result violates the head-to-head criterion. This example demonstrates that the Borda count method has the potential to violate the head-to-head criterion. d. Who wins if the plurality with elimination method is used? Does this result violate the head-to-head criterion? Using the plurality with elimination method, Buchannan is eliminated in the first round. The preference table is updated below: # of votes 129 90 87 78 42 1 st D A C C C 2 ND A C A D A 3 RD C D D A D Now Alvarez has 90 first place votes, Czechanski has 207 and Davis has 129; Alvarez is eliminated. The updated preference table is shown below: # of votes 129 90 87 78 42 1 st D C C C C 2 ND C D D D D
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