Mathematical Thinking Chapter 9 Voting Systems
Voting Systems A voting system is a rule for transforming a set of individual preferences into a single group decision. What are the desirable properties of a fair voting system? Which voting systems have these properties? Do perfectly fair voting systems exist? Voting theory lies at the intersection of mathematics, computer science, economics, and political science.
Voting Systems Terminology: voters choose among different options or candidates. Simplifying assumptions: Each voter is able to rank the candidates without any ties. Sometimes we assume an odd number of voters. There are two basic cases to consider: Two candidates. Everything is cut-and-dried. Three or more candidates. Subtle and complex.
Majority Rule Fairness properties All voters are treated equally: if any two voters were to exchange marked ballots before submitting them, the outcome of the election would not change.
Majority Rule Fairness properties All voters are treated equally: if any two voters were to exchange marked ballots before submitting them, the outcome of the election would not change. Both candidates are treated equally: if a new election were held and each voter reversed his or her vote, the outcome would be the opposite.
Majority Rule Fairness properties All voters are treated equally: if any two voters were to exchange marked ballots before submitting them, the outcome of the election would not change. Both candidates are treated equally: if a new election were held and each voter reversed his or her vote, the outcome would be the opposite. Monotone: Suppose X beats Y by majority rule in some election. If a voter s preference for Y were modified to be a preference for X, the outcome would not change.
May s Theorem (1952) 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. In other words, majority rule is the only fair voting system for two candidates. Food for thought: Prof. X Prof. Y Prof. Z Janet A A F Jasper B B B
Preference Ballots A preference ballot is a ranking of the candidates. By convention, it is written vertically. An election is a set of preference ballots. A B B C C C C A B A A B Who wins this election?
Plurality Voting The candidate with the greatest number of first place votes is the winner. A B B C C C C A B A A B
Plurality Voting Who wins? 6 5 3 1 GB AG RN PB AG RN AG GB PB GB GB AG RN PB PB RN But wait...
Condorcet Method The winner is the candidate who is preferred by a majority of voters over each other candidate. 6 5 3 1 GB AG RN PB AG RN AG GB PB GB GB AG RN PB PB RN Equivalently: the winner would defeat each other candidate in a one-on-one contest determined by majority rule.
1980 Election for US Senate in NY D Amato, Holtzman, Javits 22 23 15 29 9 4 D D H H J J H J D J H D J H J D D H
Condorcet Voting Paradox A B C B C A C A B
Condorcet Winner Criterion A voting system satisfies the CWC if it is consistent with the Condorcet method: it selects the Condorcet winner whenever there is one. A voting system that satisfies the CWC is essentially an extended version of the Condorcet method that deals with ties in some way. Does Plurality Voting satisfy the CWC?
Rank Methods Each position on a preference balance is worth a certain number of points. Used for sports polls, Hall of Fame elections, and political elections in some countries. Track meet with four schools: Event 1 Event 2 Event 3 Event 4 Event 5 5 CL MI BU BU SH 3 BU SH SH MI CL 2 MI CL MI CL MI 1 SH BU CL SH BU
Borda Count Used for sports polls, Hall of Fame elections, and political elections in some countries. For n candidates, first place is worth n 1 points, second place is worth n 2 points, and so on. 2 A A A B B 1 B B B C C 0 C C C A A
Borda Count Who is the racquetball player of the day? 2 A A A B B 1 B B B C C 0 C C C A A According to Borda count...? According to Plurality voting...?
Borda Count Who wins? 2 A A A C C 1 B B B B B 0 C C C A A 2 A A A B B 1 B B B C C 0 C C C A A And now?
Independence of Irrelevant Alternatives A voting system satisfies the IIA if it is impossible for a candidate X to move from losing to winning status unless the relative order of X and the winner is reversed on at least one ballot.
Sequential Pairwise Voting Agenda: A, B, C, D A C B B A D D B C C D A D wins, but...
Pareto Condition A voting system satisfies the Pareto Condition if it is impossible for a candidate X to win if there is another candidate whom every voter prefers over X.
Pareto Condition A voting system satisfies the Pareto Condition if it is impossible for a candidate X to win if there is another candidate whom every voter prefers over X. If the voting system used C C A to decide this election A A B satisfies the Pareto B B C Condition, then we know B cannot be the winner. But that does not mean A wins. For example, if plurality voting is used, C wins.
Hare System A runoff method 2016 Summer Olympics in Rio de Janeiro Election of public officials in Ireland Least preferred = fewest first-place votes A A B B C C C C C A B B A A B Round 1: C eliminated Round 2: B eliminated 2 2 1 1 1 C D C B A A A D D B B C A A D D B B C C Round 1: A & B eliminated Round 2: C eliminated
Hare System Failure of monotonicity 5 4 3 1 A C B B B B C A C A A C 5 4 3 1 A C B A B B C B C A A C
Plurality Runoff 4 3 3 2 1 A B C D E B A A B D C C B C C D D D A B E E E E A 4 4 3 2 A B C D B A D C C C A A D D B B Both of these elections show that Plurality Runoff is not the same as the Hare System. The first election shows that the Hare System is not monotone.
Does a completely fair voting system exist?
Arrow s Impossibility Theorem Kenneth Arrow, Nobel Laureate Proved in his 1951 Ph.D. thesis. First appeared in 1950: A Difficulty in the Concept of Social Welfare, Journal of Political Economy Popularized in 1951 book: Social Choice and Individual Values (Wiley & Sons)
Arrow s Impossibility Theorem The only possible voting system for three or more candidates that always produces a winner satisfies the Pareto Condition satisfies the IIA is a dictatorship.
Weak Version There is no voting system for three or more candidates and an odd number of voters that always produces a winner. satisfies the CWC. satisfies the IIA. Proof sketch in book. Implications for social welfare functions?
Organ Transplant Patients are ranked using multiple criteria, including: waiting time probability of success probability of future donors being suitable Each criterion is like a voter. Patient X ranked higher on each individual criterion than Y, but Y is selected. Patient X has higher priority that Y. One of the criteria is changed, but the order of X and Y according to that criterion remains the same, and yet Y now has higher priority.
Approval Voting Current uses: Election to the Baseball Hall of Fame Election to the National Academy of Sciences Choosing the Secretary General of the United Nations It is practical and has much to commend it, but it is not perfect (the Impossibility Theorem applies). Limits the expression of voter preferences.