# Possibilities of Voting

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1 Possibilities of Voting MATH 100, Survey of Mathematical Ideas J. Robert Buchanan Department of Mathematics Summer 2018

2 Introduction When choosing between just two alternatives, the results of voting are clear and uncontestable. The choice with the majority (more than half) of votes is selected.

3 Introduction When choosing between just two alternatives, the results of voting are clear and uncontestable. The choice with the majority (more than half) of votes is selected. When choosing between three or more alternatives, it is possible that no choice receives a majority, instead we may select by plurality (more votes than any other choice).

4 Plurality Method Consider an election in which each voter ranks their preferences among the candidates. The plurality method of voting makes use of some (but not all of the ranking information).

5 Plurality Method Consider an election in which each voter ranks their preferences among the candidates. The plurality method of voting makes use of some (but not all of the ranking information). Definition In the plurality method of voting, each voter gives one vote to his or her top-ranked candidate. The candidate with the most votes, a plurality of the votes, wins the election.

6 Example Suppose there are four candidates in an election {A, B, C, D} and thirteen voters. The voters rank the candidates as follows. Ranking Number of Votes A > B > C > D 3 B > C > A > D 1 D > B > C > A 2 C > B > A > D 1 A > D > C > B 2 B > C > D > A 1 C > B > D > A 1 A > B > D > C 2 There are 4! = 24 possible rankings. Which candidate receives the most number 1 rankings?

7 Pairwise Comparison Methods Definition In the pairwise comparison method, each voter gives a complete ranking of the candidates. For each pair of candidates A and B, the number of voters preferring candidate A is compared with the number of voters preferring candidate B. The candidate receiving more votes is awarded one point. If the two candidates receive an equal number of votes, each is awarded a half point. After all pairs have been compared, the candidate with the most points wins the election. If there are n candidates, then the number of pairwise comparisons needed for a pairwise comparison method election is n! n(n 1) nc 2 = =. 2!(n 2)! 2

8 Example (1 of 2) Consider the voter profile compiled earlier. Determine the outcome of the election using the pairwise comparison method. Ranking Number of Votes A > B > C > D 3 B > C > A > D 1 D > B > C > A 2 C > B > A > D 1 A > D > C > B 2 B > C > D > A 1 C > B > D > A 1 A > B > D > C 2

9 Example (2 of 2) Consider the voter profile compiled earlier. Determine the outcome of the election using the pairwise comparison method. Pair Pair A > B: B > A: A > C: C > A: A > D: D > A: B > C: C > B: B > D: D > B: C > D: D > C: Candidate A B C D Points

10 Example (2 of 2) Consider the voter profile compiled earlier. Determine the outcome of the election using the pairwise comparison method. Pair Pair A > B: 7 B > A: 6 A > C: C > A: A > D: D > A: B > C: C > B: B > D: D > B: C > D: D > C: Candidate A B C D Points

11 Example (2 of 2) Consider the voter profile compiled earlier. Determine the outcome of the election using the pairwise comparison method. Pair Pair A > B: 7 B > A: 6 A > C: 7 C > A: 6 A > D: 9 D > A: 4 B > C: 9 C > B: 4 B > D: 9 D > B: 4 C > D: 7 D > C: 6 Candidate A B C D Points

12 Example (2 of 2) Consider the voter profile compiled earlier. Determine the outcome of the election using the pairwise comparison method. Pair Pair A > B: 7 B > A: 6 A > C: 7 C > A: 6 A > D: 9 D > A: 4 B > C: 9 C > B: 4 B > D: 9 D > B: 4 C > D: 7 D > C: 6 Candidate Points A = 3 B = 2 C 1 D 0

13 Borda Method Definition In the Borda method, each voter must give a complete ranking of the candidates. Let n be the number of candidates. Each first-place vote a candidate receives is worth n 1 points. Each second-place vote a candidate receives is worth n 2 points. Each third-place earns n 3 points, fourth place earns n 4 points, and so on until last place earns n n = 0 points. Points are tallied separately for each candidate. The candidate with the highest tally of points in the winner.

14 Example (1 of 3) Consider the voter profile compiled earlier. Determine the outcome of the election using the Borda method. Ranking Number of Votes A > B > C > D 3 B > C > A > D 1 D > B > C > A 2 C > B > A > D 1 A > D > C > B 2 B > C > D > A 1 C > B > D > A 1 A > B > D > C 2

15 Example (2 of 3) Number 1st Place 2nd Place 3rd Place 4th Place Ranking of Votes Earns 3 Pts Earns 2 Pts Earns 1 Pt Earns 0 Pts A > B > C > D 3 A B C D B > C > A > D 1 B C A D D > B > C > A 2 D B C A C > B > A > D 1 C B A D A > D > C > B 2 A D C B B > C > D > A 1 B C D A C > B > D > A 1 C B D A A > B > D > C 2 A B D C

16 Example (3 of 3) Candidate Points A (7)(3) + (0)(2) + (2)(1) = 23 B C D

17 Example (3 of 3) Candidate Points A (7)(3) + (0)(2) + (2)(1) = 23 B (2)(3) + (9)(2) + (0)(1) = 24 C (2)(3) + (2)(2) + (7)(1) = 17 D (2)(3) + (2)(2) + (4)(1) = 14 Candidate B rather than A wins the election conducted by the Borda method.

18 Hare Method Definition In a Hare method election, each voter gives one vote to his or her favorite candidate in the first round. If a candidate receives a majority of the votes, he or she is declared the winner. If no candidate receives a majority of the votes, the candidate (or candidates) with the fewest votes is (are) eliminated, and a second election is conducted, in the same way, on the remaining candidates. The rounds of voting continue to eliminate candidates until one candidate receives a majority of the votes. That candidate wins the election.

19 Example (1 of 4) Consider the voter profile compiled below. Determine the outcome of the election using the Hare method. Ranking Number of Votes T > K > H > B > C 18 C > H > K > B > T 12 B > C > H > K > T 10 K > B > H > C > T 9 H > C > K > B > T 4 H > B > K > C > T 2 Total 55

20 Example (1 of 4) Consider the voter profile compiled below. Determine the outcome of the election using the Hare method. Ranking Number of Votes T > K > H > B > C 18 C > H > K > B > T 12 B > C > H > K > T 10 K > B > H > C > T 9 H > C > K > B > T 4 H > B > K > C > T 2 Total 55 In round 1, no candidate receives a majority of the votes. H receives 6, the fewest and is eliminated.

21 Example (2 of 4) Ranking Number of Votes T > K > B > C 18 C > K > B > T 12 B > C > K > T 10 K > B > C > T 9 C > K > B > T 4 B > K > C > T 2 Total 55

22 Example (2 of 4) Ranking Number of Votes T > K > B > C 18 C > K > B > T 12 B > C > K > T 10 K > B > C > T 9 C > K > B > T 4 B > K > C > T 2 Total 55 In round 2, no candidate receives a majority of the votes. K receives 9, the fewest and is eliminated.

23 Example (3 of 4) Ranking Number of Votes T > B > C 18 C > B > T 12 B > C > T 10 B > C > T 9 C > B > T 4 B > C > T 2 Total 55

24 Example (3 of 4) Ranking Number of Votes T > B > C 18 C > B > T 12 B > C > T 10 B > C > T 9 C > B > T 4 B > C > T 2 Total 55 In round 3, no candidate has a majority of votes. Candidate C has 16, the fewest, and is eliminated.

25 Example (4 of 4) Ranking Number of Votes T > B 18 B > T 12 B > T 10 B > T 9 B > T 4 B > T 2 Total 55

26 Example (4 of 4) Ranking Number of Votes T > B 18 B > T 12 B > T 10 B > T 9 B > T 4 B > T 2 Total 55 In round 4, candidate B has 37 votes, a majority and is declared the winner.

27 Example A committee consists of 13 people and they are selecting a chairperson. Three people have been nominated for chairperson (A, B, and C). The rankings of the nominees by the 13 committee members are tabulated below. Ranking Number of Votes A > B > C 4 B > C > A 2 B > A > C 4 C > A > B 3 Enter the winner of the chairperson position using your i>clicker using the 1. plurality method, 2. pairwise comparison method, 3. Borda method, 4. Hare method.

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