(c) 2013 Janice L. Epstein Voting Methods 1
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1 (c) 2013 Janice L. Epstein Voting Methods 1 Majority Rule: Each voter votes for one candidate. The candidate with the majority of the votes wins. n n + 1 When there are n votes, the majority is + 1 [n even] or [n odd] 2 2 Plurality Method: Each voter votes for one candidate. The candidate with the most votes is the winner. Borda Count: Each voter ranks the n candidates with n points assigned to the first choice, n 1 to the second choice and so on. The candidate with the most points wins. Runoff Election: If there is no majority, another vote is taken after eliminating one or more of the candidates. The top two candidates could be in the runoff, or the Hare Method could be used. Hare Method: If there is no majority winner, then the candidate with the fewest number of first place votes is eliminated. The election is held again and if no majority winner is found, the candidate with the fewest first place votes is eliminated and the election held again. Repeat until a majority winner is found. Pairwise Comparison Method: Each voter ranks the candidates. Each candidate is compared to each of the other candidates and the candidate who is preferred gets one point. The candidate with the most points wins. Tournament Method: Compare the entire slate of candidates two at a time, in a pre-determined order. The candidate with the fewest votes is eliminated and the winner goes on to compare with the third candidate. These pairwise comparisons continue until a winner is found. Approval Method: Each voter votes for all the candidates they approve of. The candidate with the most votes wins.
2 (c) 2013 Janice L. Epstein Voting Methods 2 FAIRNESS CRITERIA: Majority: If a candidate receives a majority of the first place votes, then that candidate should be declared the winner. Condorcet: If a candidate is favored when compared one-on-one with every other candidate, then that candidate should be declared the winner. Monotonicity: A candidate who wins a first election and then gains additional support without losing any of the original support should also win a second election. Irrelevant Alternatives: If a candidate is declared the winner of an election and in a second election one or more of the candidates is removed, then the previous winner should still be declared the winner. Note that there may be a tie. With two candidates and an even number of votes, it is possible that each received n/2 votes. The method to break the tie should be in place before the election! Ways to break a tie: Flip a coin Use the number of first place votes. Introduce a new voter [the Senate uses the VP].
3 (c) 2013 Janice L. Epstein Voting Methods 3 Example: Consider an election for Chief with 3 candidates, X, Y and Z. There were 12 voters and the voters were allowed to rank their choices for Chief. The results were Choices XYZ XZY YXZ YZX ZYX ZXY # votes Who won? A majority requires Borda Count: 1 st place votes = 3 points, 2 nd place votes = 2 points, 3 rd place votes = 1 point. X has (3 pts) + (2 pts) + (1 pt) = points Y has (3 pts) + (2 pts) + (1 pt) = points Z has (3 pts) + (2 points) + (1 pt) = points Heisman trophy uses Borda count. Baseball MVP uses modified Borda count with a first place vote worth 14, 2 nd place is 9, 3 rd place is th place is 1.
4 (c) 2013 Janice L. Epstein Voting Methods 4 Hare Method: Eliminate the candidate with the fewest first place votes. Move their 2 nd place vote to first place. Here we would eliminate Y. Choices XYZ XZY YXZ YZX ZYX ZXY # votes X has 1 st place votes and Z has 1 st place votes so wins with the Hare Method. Pairwise Comparison Method: X over Y: Y over X: X over Z: Z over X: Y over Z: Z over Y: gets a point gets a point gets a point Tournament Method: There are 3 ways to set up the tournament. (X vs. Y) winner vs. Z gives ( vs. ) so advances. ( vs. Z) is ( vs. ) so wins. (X vs. Z) winner vs. Y gives ( vs. ) so advances. ( vs. Y) is ( vs. ) so wins (Y vs. Z) winner vs. X gives ( vs. ) so advances. ( vs. X) is ( vs. ) so wins
5 (c) 2013 Janice L. Epstein Voting Methods 5 Approval Method: Voters mark all the options they find acceptable. The option chosen most often wins. Example: A family is deciding what to serve for dinner on Saturday and Sunday. Mom draws up a list and the votes are Liver and Onions Lamb Stew Fish Sticks Fried Chicken Hamburgers Spaghetti Mom Dad Boy Girl 1 Girl 2 Total VOTING DILEMMAS: 1. Majority Criterion: An election had the following results XYZ YZX ZYX There were 3 rankings with no votes and a total of 31 voters. X: (3 points) + (2 points) + (1 point) = Y: (3 points) + (2points) + (1 point) = Z: (3 points) + (2 points) + (1 point) =
6 (c) 2013 Janice L. Epstein Voting Methods 6 2. Condorcet Criterion: An election had the following results DABC ACBD BCAD CBDA CBAD Pairwise Winner? Make a table, A B C D A B C D A vs. B is vs. so wins. A vs. C is vs. so wins. A vs. D is vs. so wins. B vs. C is vs. so wins. B vs. C is vs. so wins. C vs. D is vs. so wins. is the pairwise [Condorcet] winner. Who is the majority winner? Who is the plurality winner?
7 (c) 2013 Janice L. Epstein Voting Methods 7 DABC ACBD BCAD CBDA CBAD Borda count: 1 st place is 4 points, 2 nd place is 3 points, 3 rd place is 2 points and 4 th place is 1 point. A =, B =, C =, and D = is the winner by Borda count. Hare method: Eliminate candidate with the fewest 1 st place votes. That is in this election. DABC ACBD BCAD CBDA CBAD has 1 st place votes, has 1 st place votes and has 1 st place votes. is the winner Pairwise method: Use the table and A has points, C has points, and B has points, D has points. wins. This method will never violate the Condorcet criteria.
8 (c) 2013 Janice L. Epstein Voting Methods 8 3. Monotonicity Criteria: Straw vote and then a binding vote. IOC voting for the Winter Olympics to be held in Quebec (Q), Salt Lake City (L), Ostersund (T) or Sion (S). A total of 87 votes [fictional]. Day 1 TLSQ LQTS QSTL TQSL TSLQ Is there a majority winner? Need votes. Use the Hare method. T has L has Q has S has Eliminate and still no majority. Eliminate. Now has and has. wins on the first day. That night the Salt Lake City reps convince the people who voted for to move to the top of their list. Day 2 TLSQ LQTS QSTL QTSL QTSL T has 1 st place votes L has 1 st place votes Q has 1 st place votes eliminated Next is eliminated and we have with 1 st place votes and has 1 st place votes and wins.
9 (c) 2013 Janice L. Epstein Voting Methods 9 4. Irrelevant Alternatives Criteria: Have 5 candidates and votes (BDCEA) (BDEAC) (EDABC) (ACEBD) (DECBA) (CBDEA) (CEDBA)
10 (c) 2013 Janice L. Epstein Voting Methods 10 SINGLE TRANSFERABLE VOTE A group of 20 people want to choose 3 movies to watch from a set of 5 movies. Each person could list their top 2 picks. The movies are 1 st choice 2 nd choice 4 votes were X-men Shrek 2 votes were Shrek X-Men 8 votes were Iron Man Up 4 votes were Iron Man The Dark Knight 1 vote was Up 1 vote was The Dark Knight What movies do they watch?
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