Voting systems Chi-Kwong Li

Outline... How do we define a “good” voting system? Voting between two candidates Voting among three or more candidates Arrow’s Impossibility Theorem Discussion

Social choice theory: How can we measure individual interests and preferences and combine them into one collective decision? Goal: Finding an outcome that reflects “the will of the people”

Preference List Ballot A rank order of candidates: often pictured as a vertical list with the most preferred candidate on top and the least preferred on the bottom 1 Green 2 Orange 3 Yellow 4 Red 5 Purple

Choosing Between Two Candidates Majority rule: Each voter indicates a preference for one of the two candidates and the candidate with the most votes wins. Assumption: The number of voters is odd.

Advantages of Majority Rule All voters are treated equally. 2. Both candidates are treated equally. 3. It is monotone: If a new election were held and a single voter changed her ballot from the loser of the previous election to the winner, but everyone else voted exactly as before, the outcome of the new election would be the same.

Can we find a better voting system? May’s Theorem Among all possible 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. Proven by Kenneth May in 1952 Mark Fey extended the theorem to an infinite number of voters in 2004

What about elections with three or more candidates? Several different possibilities for voting systems exist: Condorcet’s Method Plurality Voting The Borda Count Sequential pair-wise voting The Hare System

1. Condorcet’s Method First A B C Second Third A candidate is a winner if he or she would defeat every other candidate in a one-on-one contest using majority rule. First A B C Second Third A defeats B (2 to 1) A defeats C (2 to 1) Therefore, A wins! B defeats C (2 to 1)

Condorcet’s Voting Paradox With three or more candidates, there are elections in which Condorcet’s method yields no winners! First A B C Second Third A defeats B (2 to 1) C defeats A (2 to 1) No winner! B defeats C (2 to 1)

2. Plurality Voting Only first place winners are considered The candidate with the most votes wins Fails to satisfy the Condorcet Winner Criterion, e.g. 2000 US presidential election Manipulability Condorcet’s Winner Criterion For every possible sequence of preference ballots, either there is no Condorcet winner, or the voting system produces exactly the same winner for this election as does Condorcet’s Method.

3. The Borda Count Assigns points to each voter’s rankings and then sums these points to arrive at a group’s final ranking. Each first place vote is worth n-1 points, each second place vote is worth n-2 points, and so on down. Method: count the number of occurrences of other candidate names that are below this candidates name. Applications: senior class rank, sports hall of fame, track meets, etc. First A B Second C Third

Problem with the Borda Count... Does not satisfy the property known as “independent of irrelevant alternatives”. Independence of Irrelevant Alternatives It is impossible for a candidate B to move from non-winner status to winner status unless at least one voter reverses the order in which he or she had B and the winning candidate ranked.

Failure of the IIA First A B Second C Third Borda scores: A = 6, B = 7, C = 2  B is the winner! First A C Second B Third Borda scores: A = 6, B = 5, C = 4  A is the winner But no one has changed his or her mind about whether B is preferred to A!

4. Sequential Pairwise voting Start with a (non-ordered) list of the candidates. Pit the first candidate against the second in a one-on-one contest The winner then moves on to the third candidate in the list, one-on-one. Continue this process through the entire list until only one remains at the end. Example: choosing a favorite color:

Problem with Sequential Pairwise Voting... It fails to satisfy the “Pareto Condition.” Pareto Condition If everyone prefers one candidate A to another candidate B, then B should not be the winner!

5. The Hare System “[The Hare System] is among the greatest improvements yet made in the theory and practice of government.” ~ John Stuart Mill 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, it alone is the winner (otherwise, it is a tie).

Applying the Hare System Rank 5 4 3 1 First A C B Second Third Eliminate B, C. So, A is the winner. However, if the last column changes to [A,B,C]^T, then….. Rank 5 4 3 1 First A C Second Then, C wins!

Problem with the Hare System It fails to satisfy the property of monotonicity. Monotonicity If a candidate is a winner, and a new election is held in which the only ballot change made is for some voter to move the former winning candidate higher on his or her ballot, then the original winner should remain the winner!

A summary of voting systems for three or more candidates Problem Condorcet’s Method Not always a winner Plurality Voting Fails to satisfy the Condorcet Winner Criterion, manipulability Borda Count Fails to satisfy the Independence of Irrelevant Alternatives (IIA) property Sequential Pairwise Voting Fails to satisfy the Pareto Condition Hare System Fails to satisfy monoticity

Can we do better? Is it possible to find a voting system for three or more candidates as “ideal” as majority rule for two candidates? Arrow’s Impossibility Theorem With three or more candidates and any number of voters, there does not exist - and there will never exist - a voting system that always produces a winner, satisfies the Pareto condition and independence of irrelevant alternatives, and is not a dictatorship.

Another possibility? Approval Voting We have seen that any search for an idealvoting system of the kind we have discussed is doomed to failure. One alternative possibility is approval voting: Instead of sing a preference list ballot, each voter is allowed to give one vote to as many of the candidates as he or she finds acceptable. No limit is set on the number of candidates for whom an individual can vote. The winner under approval voting is the candidate who receives the larges number of approval votes. Approval voting is used to elect new members to the National Academy of Science and the Baseball Hall of Fame.

Discussion Other applications? Which method do you think is most easily manipulated? Which might be a good method for electing the US president, as an alternative to the Electoral College, if any?