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Borda Count, Pairwise Comparison, Approval Voting, Fairness Criteria

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Borda Count, Pairwise Comparison, Approval Voting, Fairness Criteria Day 2: Borda count, Pairwise Comparison, Approval Voting, Fairness Criteria Recall: Consider the voting system below. Who is the winner under the IRV method? 10 of the voters who had originally voted in the order Brown, Adams, Carter change their vote to favor the presumed winner changing those votes to Adams, Brown, Carter. Who is the winner under the IRV method? Monotonicity Criterion If voters change their votes to increase the preference for a candidate, it should not harm that candidate’s chances of winning. Boda count: (Another voting method, named for Jean-Charles de Borda) In this method, points are assigned to candidates based on their ranking; 1 point for last choice, 2 points for second-to-last choice, and so on. The point values for all ballots are totaled, and the candidate with the largest point total is the winner. Ex1) A camping club is deciding where to go on their next trip. The preference table is shown below, Find the winner under Borda count method. A = Arches, G = Grand Canyon, Y = Yosemite, Z = Zion Number of voters 12 5 4 9 6 10 1st Choice Y G Z A A Z 2nd Choice Z Y A Y Z G 3rd Choice G A G G Y A 4th Choice A Z Y Z G Y Since there are 4 choices 1th Place gets 4 points 2th Place gets 3 points 3th Place gets 2 points 4th Place gets 1point Number of voters 12 5 4 9 6 10 1st Choice Y G Z A A Z 4Point 12.4=48 5.4=20 4.4=16 9.4=36 6.4=24 10.4=40 2nd Choice Z Y A Y Z G 3Point 12.3=36 5.3=15 4.3=12 3rd Choice G A G G Y A 2Point 5.2=1o 10.2=20 4th Choice A Z Y Z G Y 1Point 12.1=12 Total points: Arches = 36 + 24 + 12 +10 +10 +20 +12 = 114 Grand Canyon = 106 Yosemite = 116 Zion = 124 Under the Borda count method Zion is the winner What’s Wrong with Borda Count? One potential flaw of Borda count is a candidate could receive a majority of the first- choice votes and still lose the election. Ex2) Total voters: 20 Number of voters 9 2 8 1 !"# st A: = 55% (majority) 1 Choice A A B C #$ 2nd Choice B C C B B: 40% C: 5% 3rd Choice C B A A A would win by majority by still lose the election! Number of 9 2 8 1 voters Total points: st 1 Choice A A B C A: 27 + 6 + 8 + 1 = 42 , (3Points) 27 6 24 3 B: 46 , nd 2 Choice B C C B C: 32 (2Points) 18 4 16 2 rd 3 Choice C B A A This brings up another fairness criterion. (1Points) 9 2 8 1 2 Majority Criterion: If a choice has a majority (over 50%) of votes, that choice should be the winner. Borda count is sometimes described as a consensus-based voting system, since it can sometimes choose a more broadly acceptable option over the one with majority support. In the example above, Tacoma is probably the best compromise location. This is a different approach than plurality and instant runoff voting that focus on first-choice votes; Borda Count considers every voter’s entire ranking to determine the outcome. Because of this consensus behavior, Borda Count, or some variation of it, is commonly used in awarding sports awards. Variations are used to determine the Most Valuable Player in baseball, to rank teams in NCAA sports, and to award the Heisman trophy. Note: Normally, we think of Democratic voting as one person, one vote. However, since no voting system is perfect, various voting methods use different techniques to try to fairly select a winner. Most voting systems require ranking of candidates, but not all voting methods. Copeland’s method (Pairwise Comparison): In this method, each pair of candidates is compared, using all preferences to determine which of the two is more preferred. The more preferred candidate is awarded 1 point. If there is a tie, each candidate is awarded ½ point. After all pairwise comparisons are made, all candidates points are totaled, and the one with the most points is declared the winner. Ex2) Find the winner of this election under Pairwise Comparison (Copeland’s method). Number of voters 10 6 11 15 1st Choice D C A B 2nd Choice A D C D 3rd Choice C B B A 4th Choice B A D C Step1: With 4 candidates, there are 6 comparisons: A vs B: 21 votes to 21 votes……. There is a tie so A gets 1/2, B gets 1/2 A vs C: 36 votes to 6 votes………A gets 1 A vs D: 11 votes to 31 votes……..D gets 1 B vs C: 15 votes to 27 votes…….C gets 1 B vs D: 26 votes to 16 votes……. B gets 1 C vs D: 17 votes to 25 votes……. D gets 1 Step2: Point Totals: 1 A: 1/2 + 1 =1 2 points 1 B: 1/2 + 1 =1 2 points C: 1 point D: 1 + 1 = 2 points Step3: D wins with 2 points. 3 What’s Wrong with Copeland’s method? Copeland’s Method satisfies: – Condorcet Criterion – Majority Criterion – Monotonicity Criterion However, Copeland’s Method is not perfect. Sometimes this method can violate the following Fairness Criterion. The Independence of Irrelevant Alternatives (IIA) Criterion: • If a non-winning choice is removed from the ballot, it should not change the winner of the election. • Equivalently, if choice A is preferred over choice B, introducing or removing a choice C should not cause B to be preferred over A. Approval Voting : Approval voting does NOT ask voters to rank the candidates. Voters approve or disapprove of each candidate. The results are tallied, and the option with the most approval is the winner. Ex3) The table below summarizes the results of an approval vote among A, B, and D. Each column shows the number of people with a certain approval vote. Approval are marked with X. Find the winner under the approval voting method. Number of voters 24 21 24 23 22 25 A X X X B X X X C X X X D X X X Approvals: A: 24 + 21 + 25 = 70, B: 69, C: 67, D: 71 then D is winner under approval method Ex4) Consider 3 candidates running for Mayor. Let’s assume Candidate A and Candidate B have similar political views. 35% approve only C. 32% approve A first and B second. 32% approve B first and A second. 1% approve only A. Use approval voting to select the winner. A: 32% + 32% + 1% = 65%, B: 64%, C:35% So Candidate A wins. 4 A group of friends is trying to decide upon a movie to watch. Three choices are provided, and each person is asked to mark with an “X” which movies they are willing to watch. The results are: Totaling the results, we find Titanic received 5 approvals Scream received 6 approvals The Matrix received 7 approvals. In this vote, The Matrix would be the winner. What’s Wrong with approval voting? Sometimes approval voting tends to elect the least disliked candidate. Ex5) Number of voters 39 8 3 Using this preference schedule if we 1st Choice A B C assume the voters first two choices are 2nd Choice B C B considered approved, find the winner using approval voting. 3rd Choice C A A We just ignore third row. A is approved by 39 voters. B is approved by 39 + 8 + 3 = 50 voters. C is approved by 8 + 3 = 11 voters. So Candidate B wins. Which mean not seem right because notice this how only 8 voters selected the B as the first choice and 39 voters selected A as the first choice and A has a majority of the first choice votes and therefore would have a majority win but under approval voting method Candidate B wins. So you can see why some times the approval voting method selected the least disliked Candidate as the winner. In this example nobody really disliked B, but only 8 voters want B to win. What’s Wrong with Approval Voting: • Approval Voting can easily violate the Majority Criterion. • Approval Voting is susceptible to strategic insincere voting. 5 No Perfect Voting Method: A mathematical economist, Kenneth Arrow, was able to prove in 1949 that there is no voting method that will satisfy all the fairness criteria we have discussed. Arrow’s Impossibility Theorem states, roughly, that it is not possible for a voting method to satisfy every fairness criterion that we’ve discussed. Theorem Explanation The choice with the most first-preference votes is declared the Plurality Method winner. The choice with the least first-place votes is eliminated. Any Instant Runoff votes for that candidate are redistributed. This continues until a Voting (IRV) choice has a majority. Points are assigned to candidates based on their ranking; 1 point for last choice, 2 points for second to last choice and so on. The Borda Count point values for each candidate are totaled and the candidate with the largest point total is the winner. Each pair of candidates is compared. The more preferred candidate is awarded 1 point. If there is a tie, both candidates Copeland’s Method are awarded ½ a point. After all pairwise comparisons are made the candidate with the most points is declared the winner.
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