12.2 Defects in Voting Methods
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1 12.2 Defects in Voting Methods
2 Recall the different Voting Methods: 1. Plurality - one vote to one candidate, the others get nothing The remaining three use a preference ballot, where all candidates are ranked.
3 2. Borda Count points assigned to candidates for being 1 st, 2 nd, etc and then the candidate with the highest total points wins. 3. Plurality-with-Elimination remove the candidate with the least 1 st place votes. Repeat until only one remains, the winner. 4. Pairwise Comparison For every pair of candidates, give 1 point to the one who would win in a two candidate election, ½ point if they tie. The one with the highest total points is the winner.
4 We will analyze how these well intended methods can go wrong.
5 Majority = more than half This works for the plurality method.
6 For Elimination method, if B gets more than half of the 1 st place votes, B will never be eliminated and thus will win. The Elimination method satisfies this criterion.
7 Consider the pairwise comparison method. If B is the 1 st choice more for more than half the ballots, then B will always win 1 point for each pair tested (with B in it). So B will get the highest points and win. Pairwise comparison satisfies the majority criterion.
8 Testing Borda Count method. Voter 1: A B C D Voter 2: B D C A Voter 3: A B D C
9 Testing Borda Count method. Voter 1: A B C D Voter 2: B D C A Voter 3: A B D C A gets = 9 points B gets = 10 points C gets = 5 points D gets = 6 points B wins, but A had the most 1 st place votes. Borda Count violates the Majority Condition!
10 This is an idea behind the pairwise comparison method. So it works.
11 Testing Pluriality method. Voter 1: C A B Voter 2: B A C Voter 3: A B C Voter 4: B A C Voter 5: B A C Voter 6: C A B Voter 7: A B C
12 Testing Pluriality method. Voter 1: C A B Voter 2: B A C Voter 3: A B C Voter 4: B A C Voter 5: B A C Voter 6: C A B Voter 7: A B C B wins plurality (most 1 st place votes) A beats B as a pair, and A beats C as a pair. Plurality fails Condorcet's Criterion.
13 This is similar to the Elimination method.
14 Testing Pluriality method. Voter 1: A B C Voter 2: C B A Voter 3: A C B Voter 4: B A C Voter 5: B C A Voter 6: C B A Voter 7: A B C
15 Testing Pluriality method. Voter 1: A B C Voter 2: C B A Voter 3: A C B Voter 4: B A C Voter 5: B C A Voter 6: C B A Voter 7: A B C Remove C and now B has more 1 st place votes than A. Plurality fails this condition.
16 Example: The following table summarizes the preference ballots cast for candidates A, B, C, and D: Determine the winner of this election using pairwise comparison. Do the results of the election change if any of the losing candidates are removed? (continued on next slide) 2010 Pearson Education, Inc. All rights reserved. Section 12.2, Slide 16
17 Solution: We note that A is the winner in a head-to-head vote. (continued on next slide) 2010 Pearson Education, Inc. All rights reserved. Section 12.2, Slide 17
18 Removing B and C, we see that D defeats A by 10 votes to 8. This method does not satisfy the independence-of-irrelevant-alternatives criterion Pearson Education, Inc. All rights reserved. Section 12.2, Slide 18
19 Plurality satisfies this criterion since if a candidate who wins gets more votes, that candidate still wins.
20 The Monotonicity Criterion Example: An election for president of a club has (C)hang, (K)wami, and (W)oytek as candidates. Plurality-with-elimination is being used to determine the winner. Three supporters of W, who had preferred C, decide to support her in the election. W tells the new supporters to vote for C instead. If the three voters indicated in the highlighted column in the table (next slide) change their votes to W first, C second, and K third, why should this cause W concern? (continued on next slide) 2010 Pearson Education, Inc. All rights reserved. Section 12.2, Slide 20
21 The Monotonicity Criterion Solution: K has the fewest 1 st place votes and is eliminated. W wins the election. (continued on next slide) 2010 Pearson Education, Inc. All rights reserved. Section 12.2, Slide 21
22 The Monotonicity Criterion Now consider the situation if the three voters had changed their votes. In this case, C has the least votes and is eliminated. (continued on next slide) 2010 Pearson Education, Inc. All rights reserved. Section 12.2, Slide 22
23 The Monotonicity Criterion With C eliminated, K now wins the election Pearson Education, Inc. All rights reserved. Section 12.2, Slide 23
24 The Monotonicity Criterion 2010 Pearson Education, Inc. All rights reserved. Section 12.2, Slide 24
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