Sect 13.1 Voting Methods

Transcription

1 210 Sect 13.1 Voting Methods Objective #1 Understanding and using preferences tables. In deciding what are the top movies for all time or the top players for the Spurs for all time, the outcome can be greatly influenced by the voting method used to determine the results. In this section, we will explore several different methods to determine the top winner. Suppose a group of sports writers are trying to determine the top Spurs player of all time. They are including Tim Duncan (TD), George Gervin (GG), Tony Parker (TP), and David Robinson (DR) in their ranking. Below are the ballots: nd TD 3 rd DR nd TD 3 rd DR nd TD 3 rd DR 48 The ballots above are called preference ballots. Notice that many ballots are the same. For instance, ballots (in plain text) 22, 23, 30, and 36 rank George Gervin 1 st, Tim Duncan 2 nd, David Robinson 3 rd, and Tony Parker 4 th. We can then organize how often a particular ballot occurs in a preference table.

2 Preference Table for the top Spurs players of all time. Find the following: Ex. 1 In the preference table above, a) How many sports writers voted? b) How many writers selected the players in this order: DR, GG, TD, TP? c) How many writers selected George Gervin as their first choice? a) Add the number of votes in the first row: = 48. So, 48 writers voted. b) That selection matches column four, so 10 writers selected that option. c) Columns #3 and #5 have George Gervin as the first choice, then = 18 voted for George Gervin as the first choice. There are four different voting methods that can be used to determine the winner from a preference table. They are: 1) The Plurality Method 2) The Borda Count Method 3) The Plurality-with-Elimination Method, and 4) The Pairwise Comparison Method As we will see, depending which method is chosen can alter the results. Objective #2: Find the winner using the Plurality Method. 211 The plurality method states that the candidate with the most first place votes is the winner even if that person does not have more than 50% of the first place votes. The Plurality Method The candidate with the most first-place votes is the winner. With the plurality method, we only need the 1 st choice row of the preference table to determine the winner.

3 Determine the winner using the plurality method: Ex. 2 Find the top Spurs player of all time using the plurality method. We only need to examine the first choice row of the preference table: 1 st place votes for Tim Duncan: 20 1 st place votes for George Gervin: 18 1 st place votes for Tony Parker: 0 1 st place votes for David Robinson: 10 Since Tim Duncan has the most first place votes, then he is the winner and hence the top Spurs player of all time. Objective #3: Find the winner using the Borda Count Method. 212 The Borda count method assigns points to each candidate based upon how the voters ranked the candidate. The points are then totaled up for each candidate and the candidate that has the most points is the winner. The Borda Count Method Each voter ranks the candidates from the most favorable to the least favorable. For each last-place vote, the candidate is given one point. For each second-to-last-place vote, the candidate is given two points. For each third-to-last-place vote, the candidate is given three points, and so forth. The points are then totaled for each candidate separately. The candidate that has the most points is the winner. With the Borda count method, we will need to use all of the information in the preference table. Determine the winner using the Borda count method: Ex. 3 Find the top Spurs player of all time using the Borda count method.

4 Since there are four choices, each first-place vote is worth four points, each second-place vote is worth 3 points, each third-place vote is worth 2 points, and each last place vote is worth 1 point. 1 st choice TD 20 4 = 80 GG 14 4 = 56 DR 10 4 = 40 GG 4 4 = 16 2 nd choice DR 20 3 = 60 DR 14 3 = 42 GG 10 3 = 30 TD 4 3 = 12 3 rd choice GG 20 2 = 40 TD 14 2 = 28 TD 10 2 = 20 DR 4 2 = 8 4 th choice TP 20 1 = 20 TP 14 1 = 14 TP 10 1 = 10 TP 4 1 = 4 Now total the points for each player separately. Tim Duncan gets: = 140 George Gervin gets: = 142 Tony Parker gets: = 48 David Robinson gets: = 150 Since David Robinson has the highest number of points, he is the top Spurs player of all time. Objective #4: Find the winner using the Plurality-with-Elimination Method With the plurality-with-elimination method, one keeps eliminating the candidates with the fewest first place votes until a candidate has a majority of the first place votes. The Plurality-with-Elimination Method If no candidate has a majority of first-place votes, eliminate the candidate with the fewest first-place votes and move each candidate that is below the eliminated candidate up one place. Repeat this process until there is a candidate with the majority of the first place votes. 213

5 Determine the winner using the plurality-with-elimination method: Ex. 4 Find the top Spurs player of all time using the plurality-withelimination method. Since there were a total of 48 ballots, a player will need 25 first place votes in order to have a majority. The number of first place votes are: 1 st place votes for Tim Duncan: 20 1 st place votes for George Gervin: 18 1 st place votes for Tony Parker: 0 1 st place votes for David Robinson: 10 Since none of the players have a majority, eliminate the candidate with the fewest first place votes which would be Tony Parker. Now, the number of first place votes are: 1 st place votes for Tim Duncan: 20 1 st place votes for George Gervin: 18 1 st place votes for David Robinson: 10 Since none of the players have a majority, eliminate the candidate with the fewest first place votes which would be David Robinson. 1 st choice TD GG GG GG 2 nd choice GG TD TD TD 1 st place votes for Tim Duncan: 20 1 st place votes for George Gervin: 28 Since George Gervin has the majority of the first place votes, then he is the winner and hence the top Spurs player of all time. 214

6 Notice that we had three different winners depending on which method we used. That is why it is important to know the rules prior to the voting process. Objective #5: Find the winner using the Pairwise Comparison Method. 215 In the pairwise comparison method, each candidate is compared one-onone with every other candidate. The winner in the comparison is given 1 point. In case of a tie, each candidate is given ½ of point. Once all of the comparisons are completed, the candidate with the most points is the winner. To find the number of comparisons needed for n candidates is combination of n candidates taken two at a time: nc 2 = n! (n 2)!2! = n(n 1)(n 2)! (n 2)!2! = n(n 1) 2 1 = n(n 1) 2 If we have four candidates, then the number of comparisons needed is: 4C 2 = 4! (4 2)!2! = 4(3) 2 = 6 comparisons The Pairwise Comparison Method After the voters rank all of the candidates, a series of comparisons is made where each candidate is compared to each of the other candidates. The preferred candidate in each comparison is given 1 point; in case of a tie, each candidate gets ½ of a point. The candidate with the most points is the winner. Determine the winner using the pairwise comparison method: Ex. 5 Find the top Spurs player of all time using the pairwise comparison method. Since we have four candidates, we will need to do 4 C 2 = 6 comparisons. The comparisons are TD vs. GG, TD vs. TP, TD vs. DR, GG vs. TP, GG vs. DR, and TP vs. DR. TD vs. GG

7 1 st choice TD GG GG 2 nd choice GG TD 3 rd choice GG TD TD 4 th choice 20 people prefer Tim Duncan and = 28 prefer George Gervin, so George Gervin gets 1 point. TD vs. TP 1 st choice TD 2 nd choice TD 3 rd choice TD TD = 48 people prefer Tim Duncan and 0 prefer Tony Parker, so Tim Duncan gets 1 point. TD vs. DR 1 st choice TD DR 2 nd choice DR DR TD 3 rd choice TD TD DR 4 th choice = 24 people prefer Tim Duncan and = 24 prefer David Robinson, so Tim Duncan and David Robinson each get ½ point. 216 GG vs. TP 1 st choice GG GG 2 nd choice GG 3 rd choice GG = 48 people prefer George Gervin and 0 prefer Tony Parker, so George Gervin gets 1 point. GG vs. DR 1 st choice GG DR GG 2 nd choice DR DR GG 3 rd choice GG DR 4 th choice

8 = 18 people prefer George Gervin and = 30 prefer David Robinson, so David Robinson gets 1 point. TP vs. DR 1 st choice DR 2 nd choice DR DR 3 rd choice DR = 48 people prefer David Robinson and 0 prefer Tony Parker, so David Robinson gets 1 point. Total points: Points for Tim Duncan: 1 + ½ = 1½ Points for George Gervin: = 2 Points for Tony Parker: 0 Points for David Robinson: ½ = 2½ Since David Robinson has the most points, then he is the winner and hence the top Spurs player of all time. The Plurality Method The candidate with the most first-place votes is the winner. The Borda Count Method Each voter ranks the candidates from the most favorable to the least favorable. For each last-place vote, the candidate is given one point. For each second-to-last-place vote, the candidate is given two points, and so forth. The points are then totaled for each candidate separately. The candidate that has the most points is the winner. The Plurality-with-Elimination Method If no candidate has a majority of first-place votes, eliminate the candidate with the fewest first-place votes and move each candidate that is below the eliminated candidate up one place. Repeat this process until there is a candidate with the majority of the first place votes. The Pairwise Comparison Method After the voters rank all of the candidates, a series of comparisons is made where each candidate is compared to each of the other candidates. The preferred candidate in each comparison is given 1 point; in case of a tie, each candidate gets ½ of a point. The candidate with the most points is the winner. 217