WHO do we elect?

1. WHO do we elect? • a look at voting methods Robert Cutler ★ Purdue University ★ Jan 29, 2013

2. Definitions • Voting system or Electoral system • A method by which voters choose a winner from among one or more candidates • Algorithmic in nature • Evaluated using voting system criteria

3. Definitions • Election • The votes of all of the voters to determine a winner based on the rules of a voting system • Candidate • A choice option for the voter • Voter • A person choosing from among the candidates

4. Definitions • Winner • The overall choice of candidate determined by the voting system based on the preferences of the voters • Ballot • The choices of a voter in an election

5. Simple Majority • Each voter gets one vote • The winner is the candidate with more than 50% of the votes • Example: • 20 voters • Candidates: Coke, Pepsi • Coke: 11 votes; Pepsi: 9 votes • Coke is the winner with 55% of the vote

6. Plurality • Each voter gets one vote • The winner is the candidate with most votes (even if not >50%) • Example: • 20 voters • Candidates: Coke, Pepsi, Sprite • Coke: 5 votes; Pepsi: 8 votes; Sprite: 7 votes • Pepsi is the winner with only 40% of the vote

7. What if…? • Voters have preferences • What if they order their preferences? • Example: • 20 voters; candidates: Coke, Pepsi, Sprite • 7 voters like Pepsi better than Sprite and both better than Coke [P > S > C: 7] • And so on…

8. What if…? • P > S > C: 7P > C > S: 1C > S > P: 4C > P > S: 1S > C > P: 6S > P > C: 1 • By plurality, Pepsi still wins by the same margin over Sprite and Coke (40% to 35% to 25%) as before • But…if Sprite were not in the mix, more people still prefer Coke to Pepsi (55% to 45%) as before • Yet…more people prefer Sprite to Pepsi (55% to 45%) • And way more people prefer Sprite to Coke (70% to 30%)

9. Who SHOULD WIN?

10. One option: Runoff • P > S > C: 7P > C > S: 1C > S > P: 4C > P > S: 1S > C > P: 6S > P > C: 1 • Eliminate all but top two vote getters. “Re-vote” based on preferences. • In the first round, Pepsi and Sprite win and Coke is eliminated • Of the 5 votes Coke received, 4 prefer Sprite; 1 prefers Pepsi • So in the second round, Sprite beats Pepsi by a 11 to 9 margin (55% to 45%)

11. But wait, it gets better! • Now let’s add Dr. Pepper to the mix… • We’ll simplify somewhat, but keep the same basic preferences:P > D > S > C: 7C > D > S > P: 4S > D > C > P: 6D > C > S > P: 3

12. Plurality • P > D > S > C: 7C > D > S > P: 4S > D > C > P: 6D > C > S > P: 3 • Pepsi still wins in plurality voting with 35%

13. Runoff • P > D > S > C: 7C > D > S > P: 4S > D > C > P: 6D > C > S > P: 3 • Coke and Dr. Pepper are eliminated in the first round. • Sprite gets all 7 of their votes to win in the second round 65% to 35%.

14. Another option: Instant Runoff • P > S > C: 7P > C > S: 1C > S > P: 4C > P > S: 1S > C > P: 6S > P > C: 1 • Eliminate the lowest vote getter. “Re-vote” losing voters choices based on preferences. Continue until there is a winner. • With 3 candidates, exactly the same as Runoff

15. Instant Runoff • P > D > S > C: 7C > D > S > P: 4S > D > C > P: 6D > C > S > P: 3 • Dr. Pepper is eliminated in the first round. All 3 votes go to Coke. Pepsi: 7 votes; Sprite: 6 votes; Coke: 7 votes. • Sprite is eliminated in the second round. All 6 votes go to Coke (Dr. Pepper already eliminated). • Coke wins with 13 votes (65% to 35%)!

16. One more option: Borda Count • P > S > C: 7P > C > S: 1C > S > P: 4C > P > S: 1S > C > P: 6S > P > C: 1 • For each voter assign 1 point to the first place candidate; 2 points to the second place candidate; and so on… • Candidate with lowest score wins • Pepsi (8×1 + 2×2 + 10×3 = 42)Coke (5×1 + 7×2 + 8×3 = 43)Sprite (7×1 + 11×2 + 2×3 = 35) ➙ winner!

17. Borda Count • P > D > S > C: 7C > D > S > P: 4S > D > C > P: 6D > C > S > P: 3 • Pepsi (7×1 + 0×2 + 0×3 + 13×4 = 59)Coke (4×1 + 3×2 + 6×3 + 7×4= 56)Sprite (6×1 + 0×2 + 14×3 + 0×4 = 48)Dr. Pepper (3×1 + 17×2 + 0×3 + 0×4 = 37) ➙ winner!

18. WTF? • Plurality: Pepsi wins • Runoff: Sprite wins • Instant Runoff: Coke wins • Borda Count: Dr. Pepper wins!

19. WTF? • All methods seem “fair” • The plurality winner is preferred by more people than anyone else • The runoff winner is one of the top two preferred candidates and more preferred than the other when voters are asked to choose between them • The instant runoff winner is based on eliminating candidates one-by-one based on their preferences • The Borda count winner is based on the ranking of all candidates by all voters

20. Measures of “fairness” • We look at goals of an election and set criteria according to certain categories: • Absolute result criteria • Relative result criteria as voters change • Relative result criteria as nominees change • Administration criteria • Voter criteria

21. Absolute Result Criteria • Majority criterion: If one candidate is preferred by a majority of the voters, then that candidate must win. • Fails: Borda Count • Mutual majority criterion: If there is some subset of candidates such that the majority of voters prefer every candidate of the subset to every candidate outside the subset, then the winner must be in the subset. • Fails: Borda Count, Plurality

22. Absolute Result Criteria • Condorcet winner: The candidate who, when compared with every other candidate, is preferred by more voters. • Condorcet criterion: Chooses the Condorcet winner if one exists. • Satisfies: Majority vote • Fails: Borda Count, Plurality, Instant Runoff

23. Relative Result Criteria • Monotonicity criterion: A candidate should not be harmed if it is given higher preference by some voters. • In other words, if I change my ballot to rank winner x higher, x should not then lose the election • Satisfies: Plurality, Borda Count • Fails: Runoff, Instant Runoff

24. Relative Result Criteria • Participation criterion: The addition of a ballot where candidate A is strictly preferred to candidate B should not change the winner from candidate A to candidate B. • Satisfies: Plurality, Borda Count • Fails: Instant Runoff, Any Condorcet method!

25. Relative Result Criteria • Independence of irrelevant alternatives criterion: The addition of a candidate Y to an election where candidate X wins should not cause some third candidate Z to win. • Fails: Plurality

26. Administration Criteria • Can we find a winner in polynomial time? • Can we detect cheating (sum of tallies at polling stations) in polynomial time?

27. Voter Criteria • Ease of voting • Too many candidates • Alphabetical ordering of candidates • Understanding of tabulation method • Can we rank candidates equally?

28. Other Methods/Criteria • Criteria • Condorcet loser • Reversal symmetry • Cloneproof • Equal rankings allowed • Methods • Approval • Copeland • Kemeny-Young • Minimax • Range voting • Ranked pairs • Schulze

29. Arrow’s Impossibility Theorem • Can’t have a voting system that uses voters’ ranked preferences of candidates into a group winner while meeting three “fairness” criteria:1) If every voter prefers X over Y, then the group prefers X over Y2) If every voter’s preference between X and Y remains unchanged, then the group’s preference between X and Y also remains unchanged (even if other pairwise preferences change)3) No single voter can always determine the group’s preference.

30. What does this mean? • Some say “No voting system is fair” • Trivially seen in “rock-paper-scissors” preferences • Fairness depends on: • Culture • History • Goals of election

31. What does this mean? • Bigger issues: • Voters who do not vote honestly (i.e., for candidates other than their preference • “I voted for Romney even though I prefer Gingrich because Romney has a better chance against Obama.” • “I voted for Santorum even though I prefer Gingrich because if Santorum drops out, then Romney will win in the two-person race.”

32. Next Thursday • Prepare a 15 min presentation on one or two methods of tabulating votes. • Make sure to share your method choices by emailing everyone so we don’t end up with the same methods!