Fair Elections

1. Fair Elections Are they possible?

2. Acknowledgment Many of the examples are taken from Excursions in Modern Mathematics by Peter Tannenbaum and Robert Arnold

3. Presidential Election • Not decided by majority vote • Decided by electoral college • In 2000, Bush won without receiving a majority of votes

4. Senate and House Elections • Majority wins • If no candidate wins a majority, then the outcome depends on the state’s rules

5. Today’s Assumptions • At least two choices • Voters give preference list:A>B means A preferred over B • Vote A B C D Means A>B>C>D

6. Who Wins? • Given everyone’s preference vote, what method should be used to determine winner? • Some bad methods: • My vote wins: Called dictatorship • Ignore the votes and pick one at random: Decision should be deterministic • Always pick A as the winner regardless of vote: Imposition, Method should be non-impositional

7. Vote for Class President

8. Plurality Method • Candidate with most first place votes wins • In case of class president, A wins • We will not worry about ties. (Ways to resolve ties, or just have more than one winner!)

9. AdvantageIf majority of voters place a choice as their first preference, then that choice wins. DisadvantageIgnores the lower preference choices Plurality Method Majority Criterion: If a choice receives a majority of the top preference votes, then that choice should win

10. Plurality Method A wins plurality, but if B goes head to head with either A or C, B wins!

11. Plurality Method • Used for many political elections (common in England, India, US, Canada) • Often used to pick corporate executive officers

12. Condercet Criterion • If any choice wins in a head-to-head comparison over every other choice, then that choice should win. • Plurality does not satisfy Condercet criterion • Means that plurality will not satisfy Condercet criterion for some voting outcome, but for some voting outcomes it might

13. Insincere Voting • Sometimes in order to avoid undesirable winner, voter may put first choice second • Other schemes, may move second choice to last

14. Borda Count • Assign points for each vote base on preference • Example A 3 points D 2 points B 1 point C 0 points

15. Borda Count • Who wins class president? • A 79 • B 106 • C 104 • D 81 • B wins!

16. Borda Count Violates Majority Criterion and Condercet

17. Borda Count • Used in some political elections (Slovenia and Micronesia) • Baseball MPV in AL and NL • Heisman Trophy • Universities often use it for hiring

18. Plurality with Elimination • If a choice has a majority of first place votes, that choice wins • Eliminate choice with fewest first place votes and pretend the election was only among the other choices • Back to 1.

19. Plurality with Elimination • Who wins class president? • D

20. Plurality with Elimination • Three candidates in an election. Word gets out who is voting for who. Consequently, a few people change their votes so they vote for a winner.

21. Advantage Majority Criterion Satisfied Disadvantage Violates Monotonicity Criterion:If votes are changed that only improves the winners position, the winner should not change Plurality with Elimination

22. Plurality with Elimination • Variations used in political elections (France) • Often used in hiring • Olympic Committee uses to determine location of Olympics

23. Pairwise Comparison • For each pair of choices, x and y, count the number of votes that places x before y and the number that places y before x. • One point to the winner, ½ point if tie • Tally all the points and the most points wins

24. Pairwise Comparison • Who wins class president? • A 14, B 23 … B gets 1 point • A 14, C 23 … C gets 1 point • A 14, D 23 … D gets 1 point • B 18, C 19 … C gets 1 point • B 28, D 9 … B gets 1 point • C 25, D 12 … C gets 1 point • C Wins with 3 points!

25. Pairwise Comparison A vrs B B 1 A vrs C A 1 A vrs D A 1 A vrs E A 1 B vrs C C 1 B vrs D B .5 D .5 B vrs E B 1 C vrs D C 1 C vrs E E 1 D vrs E D 1 A: 3 B: 2.5 C: 2 D: 1.5 E: 1 Winner is A But what if C withdraws just before vote?

26. Advantage Satisfies Majority Satisfies Condercet Satisfies Monotonicity Disadvantage Does not satisfy Independence-of-Irrelevant-Alternative Criterion:If x is the winner and a choice other than x is removed, x should still be the winner. Pairwise Comparison

27. Pairwise Comparison • Not currently used in any political election – variation last used in Marquette Michigan in the 1920’s • Variation used by Wikimedia Foundation for election of Trustees • Various other private organizations

28. Who Should be Class President? • Four reasonable methods yielded four different answers! • Critical that before voting takes place, method of determining winner is well established! • But, is there some method that satisfies all the fairness criteria?

29. Is Fairness an Illusion? Theorem: If a voting scheme satisfies the Majority Criterion, it cannot satisfy the Independence-of-Alternative Criterion. Proof: Suppose A wins. If B withdraws, then C wins. Suppose B wins. If C withdraws, then A wins. Suppose C wins. If A withdraws, then B wins.

30. Arrows Theorem Theorem: If the method to determine the winner is deterministic, non-impositional, and it satisfies both Monotonicity and Independence-of-Alternative criteria, then it is a dictatorship. Sometimes people say there is no fair way to determine who wins!

31. Summary

32. Best Method • Let’s vote on which method we think is best. • Before we vote, we have to decide which method to use to determine winner. • Let’s vote on which method to use to determine the best method. • Before we vote on which method to use, we have to …