Chapter 1: The Mathematics of Voting Section 3: The Borda Count Method Thursday, January 19, 2012 The Borda Count Method In an election using the Borda Count Method, the candidate with the most points wins. Points are assigned to each place on a ballot. In an election with N candidates, a first place vote receives N points, a second place vote receives one less, and so forth, so that a last place vote receives 1 point. The points for each candidate are tallied to determine the winner. We call this candidate the Borda candidate. Example 4: Using the preference schedule from Example 3, identify the Borda candidate. Number of voters 49 48 3 1st choice 4pts 2nd choice 3pts B D B 3rd choice 2pts C C D 4th choice 1pt D A A Assign points to each preference as above. Candidate A received 49 1st choice votes, 48 4th choice votes, and 3 4th choice votes. Since a 1st choice vote is worth 4 points and a 4th choice vote is worth 1 point, then A receives points. Repeat this process for each candidate. 49(4)+48(1) +3(1) = 247 Candidate points A 49(4)+48(1)+3(1) = 247 B 49(3)+48(4)+3(3) = 348 C 49(2)+48(2)+3(4) = 206 D 49(1)+48(3)+3(2) = 199 Since B receives the most points, then B is the Borda Candidate.
Example 5: The Mathematics Department is in need of a new professor and have formed a hiring committee. The committee has selected their top four finalists: Dr. Addams, Dr. Buntin, Dr. Casey, and Dr. Davis. After interviewing each candidate, the committee members ranked the four candidates using a preference ballot. They have decided that the Borda candidate will receive the position. The corresponding preference schedule is given below. Who will be hired? Number of voters 6 3 2 1st choice 4pts 2nd choice 3pts 3rd choice 2pts 4th choice 1pts Candidate points B C D C D B D A A A 6(4)+3(1)+2(1) = 29 B 6(3)+3(4)+2(2) = 34 C 6(2)+3(3)+2(4) = 29 D 6(1)+3(2)+2(3) = 18 Since B receives the most points, then B is the Borda Candidate. Question: After the position is filled, Dr. Addams is quite upset. Why? Dr. Addams is upset because he is not only the plurality candidate, but also the majority candidate and Condorcet Candidate. Summary List the pros and cons of the Borda Count Method. PROS takes all preferences into account compromise candidate CONS violates Majority Criterion violates Condorcet Candidate good for elections with multiple candidates violations of fairness criterion are infrequent
Chapter 1: The Mathematics of Voting Section 4: The Plurality with Elimination Method The Plurality with Elimination Method Thursday, January 19, 2012 In an election using the Plurality with Elimination Method (Instant Runoff Voting), candidates with the fewest first place votes are successively eliminated until a candidate has a majority of the first place votes. Example 6: The Math Club is in need of a new president. The four eligible members are Anna, Bob, and Carl, and Diane. After each candidate presented a short presentation on their leadership abilities, the club members ranked the candidates by means of a preference ballot. Below is the corresponding preference schedule. Number of voters 14 10 8 4 1 1st choice 4pts A C D B C 2nd choice 3pts B B C D D 3rd choice 2pts C D B C B 4th choice 1pt D A A A A (a) Who is the Plurality winner? Candidate A has the most votes and is therefore the plurality candidate. (b) Who is the Majority Candidate? There is not a Majority candidate since no candidate has more than half the votes. There are 37 voters so a candidate would need at least 19 votes to have a majority. (c) Who is the Condorcet Candidate? A B A C A D B C B D C D 14 23 14 23 14 23 18 19 28 9 25 12 Candidate C is the Condorcet candidate. (d) Who is the Borda Count winner? Candidate points A 14(4)+10(1)+8(1)+4(1)+1(1) = 79 B 14(3)+10(3)+8(2)+4(4)+1(2) = 106 C 14(2)+10(4)+8(3)+4(2)+1(4) = 104 D 14(1)+10(2)+8(4)+4(3)+1(3) = 81 Since B receives the most points, then B is the Borda Candidate. (e) Who is the Plurality with Elimination winner? Step 1: Tally the 1st choice votes: D 14 4 11 8 Step 2:Eliminate the candidate with the fewest votes - Eliminate B, his votes go to D 14 11 12 Step 3: Repeat - Eliminate C - votes go to D 14 23 Finished:D has a majority and is therefore the Plurality with Elimination candidate
Example 7: Using the preference schedule from Worksheet 1, find the winner using Plurality with Elimination. Number of voters 10 8 10 8 7 8 1st choice A A C D D B 2nd choice B D E C C E 3rd choice C B D B B A 4th choice D C A E A C 5th choice E E B A E D D E 18 8 10 15 0 Eliminate E 18 8 10 15 Eliminate B 26 10 15 A has majority Candidate A is the Plurality with Elimination candidate. Example 8: Using the preference schedule from Example 5, find the winner using Plurality with Elimination. Number of voters 6 3 2 1st choice 2nd choice B C D 3rd choice C C B 4th choice D A A 6 3 2 A has majority Candidate A is the Plurality with Elimination candidate.
Example 9: (Example 1.10 from text) Three cities, Athens, Barcelona, and Calgary, are competing to host the Summer Olympic Games. The final decision is made by a secret vote of the 29 members of the Executive Council of the IOC, and the winner is to be chosen using the plurality with elimination method. Two days before the election, a straw poll is conducted by the Executive Council just to see how things stand. Number of voters 7 8 10 4 1st choice A 2nd choice B C A C 3rd choice C A B B Based on the result of the straw poll, who would win? 11 8 10 Eliminate B, his votes go to C 11 18 C has majority Calgary is the Plurality with Elimination winner. We would expect Calgary to host the Summer games. The results of the straw poll are leaked, causing the 4 delegates in the last column to switch their vote from Athens to Calgary. The results of the election are below. Number of voters 7 8 14 1st choice 2nd choice B C A 3rd choice C A B Where will the Summer Olympics be held? A B C 7 8 14 Eliminate A, his votes go to B 15 14 B has majority Barcelona is the Plurality with Elimination winner and gets to host the games. Does this make sense? If you win an election and, in a reelection, you receive more votes, then you should still win!
The Monotonicity Criterion If candidate X is a winner of an election and, in reelection, the only changes in the ballots are changes that favor candidate X (and only X), then X should remain the winner of the election. Summary List the pros and cons of the Plurality with Elimination Method. PROS satisfies the Majority Criterion CONS violates the Condorcet Criterion feasible for elections where majority is required violates Monotonicity Criterion