Voting Fairness Idea: Condorcet Criterion (CO)
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1 Voting Fairness Idea: ondorcet riterion (O) Definition (Voting Fairness Idea: ondorcet riterion (O)) voting system satisfies the ondorcet riterion if the ondorcet andidate always wins. In the ballots below,
2 Voting Fairness Idea: ondorcet riterion (O) Definition (Voting Fairness Idea: ondorcet riterion (O)) voting system satisfies the ondorcet riterion if the ondorcet andidate always wins. In the ballots below, we get the following comparisons: Matchup Points Matchup Points Matchup Points vs. 1for vs. 1for vs. 1for andidate is the ondorcet andidate with 2 points, the most possible in a 3-candidate election. To satisfy the ondorcet riterion, a voting system must have andidate as the winner.
3 ondorcet riterion Example 15 cademy Voters get together to compare their preferences for the 2012 est Picture. The films under consideration are rgo, Life of Pi, and Zero Dark Thirty. Preference for the voters are summarized below Votes place Z L place L L Z place Z Which film wins if the Plurality method is used? Is there a condorcet candidate?
4 2012 est Picture 4 Example 15 cademy Voters get together to compare their preferences for the 2012 est Picture. The films under consideration are rgo, Life of Pi, and Zero Dark Thirty. Preference for the voters are summarized below Votes place Z L place L L Z place Z Which film is eliminated in the first round of the Plurality with Elimination method? Is there a ondorcet candidate?
5 2013 N Football: Version 8 Example (2013 N Football 8) 37 Sports Writers made predictions for the top 4 rankings in 2013 N football. Their predictions for labama (L), Louisville (LV), Ohio State University (OSU), and Oregon (OR) are given below: Number of Votes Place OSU OR LV L OR Place L L OR LV LV Place OR LV L OR L 4th Place LV OSU OSU OSU OSU Do you remember which team wins if the orda method is used? Is there a ondorcet candidate?
6 SIDE
7 Voting Fairness Idea: O- MJ onnection If a candidate gets more than half of all -place votes, then that candidate has more -place votes than any other candidate.
8 Voting Fairness Idea: O- MJ onnection If a candidate gets more than half of all -place votes, then that candidate has more -place votes than any other candidate. Therefore, this majority candidate will will win all head-to-head matchups against every other candidate.
9 Voting Fairness Idea: O- MJ onnection If a candidate gets more than half of all -place votes, then that candidate has more -place votes than any other candidate. Therefore, this majority candidate will will win all head-to-head matchups against every other candidate. In other words, the Majority andidate is LWYS the ondorcet andidate!
10 Voting Fairness Idea: O- MJ onnection If a candidate gets more than half of all -place votes, then that candidate has more -place votes than any other candidate. Therefore, this majority candidate will will win all head-to-head matchups against every other candidate. In other words, the Majority andidate is LWYS the ondorcet andidate! Theorem (O-MJ onnection) ny voting system where the ondorcet riterion (O) holds is a voting system where the Majority riterion (MJ) holds as well.
11 Voting Fairness Idea: O- MJ onnection If a candidate gets more than half of all -place votes, then that candidate has more -place votes than any other candidate. Therefore, this majority candidate will will win all head-to-head matchups against every other candidate. In other words, the Majority andidate is LWYS the ondorcet andidate! Theorem (O-MJ onnection) ny voting system where the ondorcet riterion (O) holds is a voting system where the Majority riterion (MJ) holds as well. Turning the logic around, if the Majority riterion (MJ) DOES NOT hold in a voting system, then the ondorcet riterion (O) DOES NOT hold as well.
12 Voting Fairness Idea: Indep. of Irr. lternatives (II) Definition (Voting Fairness Idea: Ind. of Irrel. lternatives riterion (II)) If any candidate wins a first election, and one of the irrelevant losing candidates drops out before the second election, then the previous winner should also win the second election.
13 Voting Fairness Idea: Indep. of Irr. lternatives (II) Definition (Voting Fairness Idea: Ind. of Irrel. lternatives riterion (II)) If any candidate wins a first election, and one of the irrelevant losing candidates drops out before the second election, then the previous winner should also win the second election. In the ballots below, suppose there is a voting system in which andidates and are in contention to win the election, but andidate does not have a chance at winning. voting system satisfies the Independence of Irrelevant lternatives riterion (II) only if the outcome of the election does not change from absence of andidate.
14 Voting Fairness Idea: Indep. of Irr. lternatives (II) Definition (Voting Fairness Idea: Ind. of Irrel. lternatives riterion (II)) If any candidate wins a first election, and one of the irrelevant losing candidates drops out before the second election, then the previous winner should also win the second election. In the ballots below, suppose there is a voting system in which andidates and are in contention to win the election, but andidate does not have a chance at winning. voting system satisfies the Independence of Irrelevant lternatives riterion (II) only if the outcome of the election does not change from absence of andidate. Ralph Nader and Ross Perot are real-world examples of irrelevant alternatives or spoiler candidates in
15 nother New One Direction Fan lub 1 Example (nother New One Direction Fan lub 1) The Lexington One Direction fan club has 42 members. No one likes Louis so the the four choices for best band member are: Harry, Liam, Niall, and Zayn. Here s how the 42 person club votes: Votes place H N Z N L place L H H L H place N L L Z Z 4th place Z Z N H N Which candidate wins if the Plurality method is used? Which candidate(s) is/are most irrelevant if the Plurality method is used?
16 nother New One Direction Fan lub 1 Example (nother New One Direction Fan lub 1) The Lexington One Direction fan club has 42 members. No one likes Louis so the the four choices for best band member are: Harry, Liam, Niall, and Zayn. Here s how the 42 person club votes: Votes place H N Z N L place L H H L H place N L L Z Z 4th place Z Z N H N Which candidate wins if the Plurality method is used? Which candidate(s) is/are most irrelevant if the Plurality method is used? Does the outcome of the election change if one (or more) of the irrelevant candidates drop out?
17 2013 N Football: Version 9 Example (2013 N Football 9) The predictions of 37 Sports Writers (given below) for labama (L), Louisville (LV), Ohio State University (OSU), and Oregon (OR) makelabama(l) theordawinner. Number of Votes Place OSU OR LV L OR Place L L OR LV LV Place OR LV L OR L 4th Place LV OSU OSU OSU OSU Ohio State (OSU) is not even close to winning if the orda method is used. Which team wins using the orda method if OSU is completely dropped from consideration?
18 2013 N Football: Version 9 Example (2013 N Football 9) The predictions of 37 Sports Writers (given below) for labama (L), Louisville (LV), Ohio State University (OSU), and Oregon (OR) makelabama(l) theordawinner. Number of Votes Place OSU OR LV L OR Place L L OR LV LV Place OR LV L OR L 4th Place LV OSU OSU OSU OSU Ohio State (OSU) is not even close to winning if the orda method is used. Which team wins using the orda method if OSU is completely dropped from consideration? Louisville (LV) is not even close to winning if the orda method is used. Which team wins using the orda method if LV is completely dropped from consideration?
19 School oard Election 1 Example (School oard Election 1) Four candidates,,, andd are running for a seat for the School oard Executive ommittee. The School oard voters give the following preferences: Number of votes place D place place 4th place D D D Is andidate irrelevant under the Plurality with Elimination method?
20 School oard Election 1 Example (School oard Election 1) Four candidates,,, andd are running for a seat for the School oard Executive ommittee. The School oard voters give the following preferences: Number of votes place D place place 4th place D D D Is andidate irrelevant under the Plurality with Elimination method? What about andidate?
21 School oard Election 1 Example (School oard Election 1) Four candidates,,, andd are running for a seat for the School oard Executive ommittee. The School oard voters give the following preferences: Number of votes place D place place 4th place D D D Is andidate irrelevant under the Plurality with Elimination method? What about andidate? Who wins if the Plurality with Elimination is used?
22 School oard Election 1 Example (School oard Election 1) Four candidates,,, andd are running for a seat for the School oard Executive ommittee. The School oard voters give the following preferences: Number of votes place D place place 4th place D D D Is andidate irrelevant under the Plurality with Elimination method? What about andidate? Who wins if the Plurality with Elimination is used? Who will win if andidate drops out of the election?
23 School oard Election 2 Example (School oard Election 2) di erent School oard has four candidates,,, andd running for a seat for the School oard Executive ommittee. The School oard voters give the following preferences: Number of votes place D place D place D 4th place D D Who wins the election if the Pairwise omparison method is used?
24 School oard Election 2 Example (School oard Election 2) di erent School oard has four candidates,,, andd running for a seat for the School oard Executive ommittee. The School oard voters give the following preferences: Number of votes place D place D place D 4th place D D Who wins the election if the Pairwise omparison method is used? Which candidate(s) can be considered irrelevant? If andidate D drops out, will the winner of the election change in any way?
25 SIDE
26 Recap Definition (Voting Fairness Idea: Majority riterion (MJ)) voting system satisfies the Majority riterion if a candidate with over 50% of first-place votes automatically wins the election.
27 Recap Definition (Voting Fairness Idea: Majority riterion (MJ)) voting system satisfies the Majority riterion if a candidate with over 50% of first-place votes automatically wins the election. Definition (Voting Fairness Idea: ondorcet riterion (O)) voting system satisfies the ondorcet riterion if the ondorcet andidate always wins.
28 Recap Definition (Voting Fairness Idea: Majority riterion (MJ)) voting system satisfies the Majority riterion if a candidate with over 50% of first-place votes automatically wins the election. Definition (Voting Fairness Idea: ondorcet riterion (O)) voting system satisfies the ondorcet riterion if the ondorcet andidate always wins. Definition (Voting Fairness Idea: Monotonicity riterion (MO)) voting system satisfies the Monotonicity riterion if an improvement in a given candidate s vote, without changing the relative quality of the other candidate s votes, does not hurt the given candidate s chance of winning the election.
29 Recap Definition (Voting Fairness Idea: Majority riterion (MJ)) voting system satisfies the Majority riterion if a candidate with over 50% of first-place votes automatically wins the election. Definition (Voting Fairness Idea: ondorcet riterion (O)) voting system satisfies the ondorcet riterion if the ondorcet andidate always wins. Definition (Voting Fairness Idea: Monotonicity riterion (MO)) voting system satisfies the Monotonicity riterion if an improvement in a given candidate s vote, without changing the relative quality of the other candidate s votes, does not hurt the given candidate s chance of winning the election.
30 Fairness Idea: rrow s Theorem Theorem (rrow s Impossibility Theorem) In an election with more than two candidates, there is NO FIR method of voting that will simultaneously satisfy the Majority criterion, the ondorcet criterion, the monotonicity criterion, and the Independence of Irrelevant lternatives criterion.
31 Fairness Idea: rrow s Theorem Theorem (rrow s Impossibility Theorem) In an election with more than two candidates, there is NO FIR method of voting that will simultaneously satisfy the Majority criterion, the ondorcet criterion, the monotonicity criterion, and the Independence of Irrelevant lternatives criterion. Here is a summary of how each of the Voting Methods fails the Fairness riteria: MO MJ O II Plurality X X X X orda X X X X Plurality w/ Elimination X X X X Pairwise omparison X X X X (Xmeans always satisfied while X can fail to satisfy)
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