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Chapter 9:Social Choice: the Impossible Dream Chapter 9:Social Choice: The Impossible Dream September 18, 2013 Chapter 9:Social Choice: The Impossible Dream Last Time Last time we talked about Voting systems Majority Rule Condorcet's Method Plurality Borda Count Sequential Pairwise Voting Chapter 9:Social Choice: The Impossible Dream Condorcet's Method Definition A ballot consisting of such a rank ordering of candidates is called a preference list ballot because it is a statement of the preferences of the individual who is voting. Description of Condorcet's Method With the voting system known as Condorcet's method, a candidate is a winner precisely when he or she would, on the basis of the ballots cast, defeat every other candidate in a one-on-one contest using majority rule. Chapter 9:Social Choice: The Impossible Dream Example Consider the following set of preference lists: Number of Voters(9) Rank 3 1 1 1 1 1 1 First A A B B C C D Second D C C C B D B Third C B D A D B C Fourth B D A D A A A Winner: C Chapter 9:Social Choice: The Impossible Dream Pros and Cons of Condorcet's Method Pro: Takes all preferences into account. Con: Condorcet's Voting Paradox With three or more candidates, there are elections in which Condorcet's method yields no winners. In particular, the following ballots constitute an election in which Condorcet's method yields no winner. Chapter 9:Social Choice: The Impossible Dream Plurality Voting Plurality Voting Each voter pick their top choice. Chapter 9:Social Choice: The Impossible Dream Example Consider the following set of preference lists: Number of Voters(9) Rank 3 1 1 1 1 1 1 First A A B B C C D Second D B C C B D C Third B C D A D B B Fourth C D A D A A A Winner: A Chapter 9:Social Choice: The Impossible Dream Pros and Cons of Plurality Voting May's Theorem Among all two-candidate voting systems that never result in a tie, majority rule is the only one that treats all voters equally, treats both candidates equally, and is monotone. Condorcet Winner Criterion A voting system is said to satisfy the Condorcet winner criterion(CWC) provided that, for every possible sequence of preference list ballots, either (1) there is no Condorcet winner or (2) the voting system produces exactly the same winner for this election as does Condorcet's Method. Manipulability A voting system is subject to manipulabity if there are elections in which it is to a voter's advantage to submit a ballot that misrepresents his or her true preferences. Chapter 9:Social Choice: The Impossible Dream Borda Count Rank Method A rank method of voting assigns points in a nonincreasing manner to the ordered candidates on each voter's preference list ballot and then sums these points to arrive at the group's final ranking. Borda Count The rank method in which there are n candidates with each first-place vote worth n − 1 points, each second-place vote worth n − 2 points, and so on down to each last-place vote worth 0 points is known as the Borda Count. The point totals are referred to as a candidate's Borda score. Chapter 9:Social Choice: The Impossible Dream Example Consider the following set of preference lists: Number of Voters(9) Rank 3 1 1 1 1 1 1 First A A B B C C D Second D B C C B D C Third B C D A D B B Fourth C D A D A A A Winner: B Chapter 9:Social Choice: The Impossible Dream Pros and Cons of Borda Count Pro: Takes all preferences into account. Con: Independence of Irrelevant Alternatives A voting system is said to satisfy independence of prevalent alternatives (IIA) if it is impossible for a candidate X to move from nonwinner status to winner status unless at least one voter reverses the order in which he or she had X and the winning candidate ranked. Chapter 9:Social Choice: The Impossible Dream Sequential Pairwise Voting Description of Sequential Pairwise Voting Sequential pairwise voting starts with an agenda and pits the first candidate against the second in a one-on-one contest. The winner then moves on to confront the third candidate in the list, one on one. Losers are deleted. This process continues throughout the entire agenda, and the one remaining at the end wins. Chapter 9:Social Choice: The Impossible Dream Example Consider the following set of preference lists: Number of Voters(9) Rank 3 1 1 1 1 1 1 First A A B B C C D Second D B C C B D C Third B C D A D B B Fourth C D A D A A A With Agenda A; B; C; D Winner: D With Agenda D; C; B; A Winner: B Chapter 9:Social Choice: The Impossible Dream Pros and Cons of Borda Count Pro: Efficient way to determine a winner. Always a unique winner with an odd number of voters. Con: Agenda dependent Pareto Condition A voting system is said to satisfy the Pareto condition provided that in every election in which every voter prefers one candidate X to another candidate Y , the latter candidate Y is not among the winners. Chapter 9:Social Choice: The Impossible Dream Hare System Description of the Hare System The Hare system proceeds to arrive at a winner by repeatedly deleting candidates that are \least preferred\ in the sense of being at the top of the fewest ballots. If a single candidate remains after all others have been eliminated, he or she alone is the winner. If two or more candidates remain and all of these remaining candidates would be eliminated in the next round (because they all have the same number of first-place votes), then these candidates are declared to be tied for the win. Chapter 9:Social Choice: The Impossible Dream Example Consider the following set of preference lists: Number of Voters(9) Rank 3 1 1 1 1 1 1 First A A B B C C D Second D B C C B D C Third B C D A D B B Fourth C D A D A A A Winner: C Chapter 9:Social Choice: The Impossible Dream Possible Problems with the Hare System Monotonicity A voting system for three or more candidates is said to satisfy monotoncity provided that, for every election, if some candidate X is a winner and a new election is held in which the only ballot change made is for some voter to move this winning candidate X higher on his or her ballot (and to make no other changes), then X will remain a winner. Another reason: Chicago lost being the host city to the 2016 Summer Olympics because of this system. Chapter 9:Social Choice: The Impossible Dream Example Consider the following set of preference lists: Number of Voters(12) Rank 5 4 3 1 First A C B B Second B B C A Third C A A C Winner: A Number of Voters(12) Rank 5 4 3 1 First A C B A Second B B C B Third C A A C Winner: C Chapter 9:Social Choice: The Impossible Dream Plurality Runoff Method Description of Plurality Runoff Method Plurality runoff is the voting system in which there is a runoff (that is, a new election using the same ballots) between the two candidates receiving the most first place votes. If there are ties, then the runoff is among either those tied for the most first-place votes, or the lone candidate with the most first-place votes along with those tied for the second-most first-place votes (and plurality voting is used). Chapter 9:Social Choice: The Impossible Dream Example Consider the following set of preference lists: Number of Voters(9) Rank 3 1 1 1 1 1 1 First A A B B B C D Second D B C C C D C Third B C D A D B B Fourth C D A D A A A Winner: B Chapter 9:Social Choice: The Impossible Dream Hare System vs Plurality Runoff Method Consider the following set of preference lists: Number of Voters(13) Rank 4 4 3 2 First A B C D Second B A D C Third C C A A Fourth D D B B Plurality Runoff Method Winner: A Hare System Winner: C Chapter 9:Social Choice: The Impossible Dream Practice Problems 1 Does the Condorcet's rule satisfy the Pareto Condition? Monotonicity? Why? 2 Does plurality voting satisfy the Pareto Condition? Monotonicity? Why? 3 Does Borda count voting satisfy the Pareto Condition? Monotonicity? Why? 4 Does sequential pairwise voting satisfy the CWC? Monotonicity? Why? 5 Does Hare system voting satisfy the Pareto Condition? 6 Does plurality voting satisfy IIA? Chapter 9:Social Choice: The Impossible Dream Practice Problems Consider the following set of preference lists: Number of Voters(7) Rank 2 2 1 1 1 First A B A C D Second D D B B D Third C A D D A Fourth B C C A C Calculate the winner using 1 plurality voting. 2 the Borda count. 3 the Hare system. 4 sequential pairwise voting with the agenda B; D; C; A. Chapter 9:Social Choice: The Impossible Dream Practice Problems Consider the following set of preference lists: Number of Voters(7) Rank 2 2 1 1 1 First C E C D A Second E B A E E Third D D D A C Fourth A C E C D Fifth B A B B B Calculate the winner using 1 plurality voting.
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