Download Comparison of Voting Systems: IRV, Borda, Copeland, Approval Voting & Fairness Criteria and more Exams Discrete Mathematics in PDF only on Docsity!
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 1 st Choice Y G Z A A Z 2 nd Choice Z Y A Y Z G 3 rd Choice G A G G Y A 4 th Choice A Z Y Z G Y
Since there are 4 choices 1 th Place gets 4 points 2 th Place gets 3 points 3 th Place gets 2 points 4 th Place gets 1point Number of voters 12 5 4 9 6 10 1 st Choice 4Point
Y
G
Z
A
A
Z
nd Choice 3Point
Z
Y
A
Y Z G
rd Choice 2Point
G A
5.2=1o
G G Y A
th Choice 1Point
A
Z Y Z G Y
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 A: !"# #$ = 55% (majority) B: 40% C: 5%
A would win by majority by still lose the election!
Number of voters
st Choice (3Points)
A
A
B
C
nd Choice (2Points)
B
C
C
B
rd Choice (1Points)
C
B
A
A
Total points: A: 27 + 6 + 8 + 1 = 42 , B: 46 , C: 32 This brings up another fairness criterion. Number of voters 9 2 8 1 1 st Choice A A B C 2 nd Choice B C C B 3 rd Choice C B A A
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.
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 1 st Choice A B C 2 nd Choice B C B 3 rd Choice C A A Using this preference schedule if we assume the voters first two choices are considered approved, find the winner using approval voting. 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.
MATH 1030 In-Class Activities - Voting Theory Consider the following preference schedule.
voters 10 10 12 8
1st D A C B 2nd A C B D 3rd B B D A 4th C D A C i. Find each candidate's score using Copeland's Method (a.k.a. Pair-wise Comparison ). ii. Which candidate wins by Copeland's Method? iii. Is there a Condorcet Candidate? iv. Which candidate gets the lowest score by Copeland's Method? v. Remove the candidate with the lowest score and rewrite the preference schedule.
voters 10 10 12 8
1st 2nd 3rd vi. Find each candidate's score using Copeland's Method. vii. Now who wins?
Consider the IIA 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. i. In complete sentences, explain how the IIA Fairness Criterion relates to the above election. Consider the following quote from your text. "After finishing dinner, Sidney Morganbesser decides to order dessert. The waitress tells him he has two choices: apple pie and blueberry pie. Sidney orders the apple pie. After a few minutes the waitress comes back and says that they also have cherry pie. Morganbesser replies, "In that case I'll have the blueberry pie." ii. In complete sentences, explain how the IIA Fairness Criterion relates to the dessert order.
