Math 1010 ‘Mathematical Thought and Practice’

1. Math 1010‘Mathematical Thought and Practice’ An active learning approach to a liberal arts mathematics course

2. Nell RayburnDavid Cochener Department of Mathematics Austin Peay State University Clarksville, Tennessee

3. Decisions How do we make choices?

4. Types of Decisions • Individual—our opinion is our decision. • Group—Individual opinions are expressed by voting(at least in a democratic society)and some procedure is used to combine these individual preferences for a group decision.

5. Two Basic Questions • What type of election decision procedure should we use to combine individual decisions (preferences) into a group decision? • How can we be sure that what is decided is really what the group wants?

6. Desirable Properties of Majority Rule(Two Alternative Case) • All voters are treated equally. (Swapping marked ballots gives no change) • Both alternatives are treated equally. (If all votes are reversed, so is the winner.) • If a new election were held and a single voter changed from a vote for the previous loser to the previous winner, then the outcome would be the same as before.

7. May’s Theorem • If the number of voters is odd, and if we are interested only in voting procedures that never result in a tie, then majority rule is the only voting system for two alternatives that satisfies the conditions listed on the previous slide.

8. Kenneth O. May (1915-1977) • Mathematician, Political Activist • PhD, Univ. of California, 1946 • Mathematics plays a crucial role in social science!

9. Ordinal Ballots Preference List

10. Ordinal Ballots • List your choices in order with the favorite on top and ‘least favorite’ on bottom Ballots must be • Complete (you must rank all candidates) • Linear (no ties) • Transitive (If you prefer A to B and B to C, then you must prefer A to C.)

11. Preference Schedule

12. Five Election Decision Methods • Plurality • Standard Runoff • Sequential Runoff • Borda Count • Condorcet Winner Criterion

13. Plurality • Whoever has the most votes wins! • Problems? Remember Jesse Ventura??

14. 1998 Minnesota Governor’s Race • Jesse ‘The Body’ Ventura: 38% • Hubert Humphrey III: 33% • Norm Coleman: 29% • The latter two were highly experienced, but somewhat dull compared to Ventura.

15. Standard Runoff • If there is no majority, the two candidates receiving the most votes compete head to head.

16. Sequential Runoff • If no one has a majority, eliminate the candidate(s) with the fewest first place votes, and count again. Continue in this way until someone has a majority. • Also known as Hare elimination.

17. Borda Count • If there are n candidates, assign n – 1 points to a first place choice, n – 2 points to a second place choice,…, 0 points to a last place choice. Sum the points for each candidate – the one with the most points wins. • Problems? Borda Count can violate majority rule!

18. A surprising result! • Borda count may violate majority rule! A has a majority, but B wins Borda count by 21 – 19 over C, with A getting 18 points and D getting 8 points.

19. Condorcet Winner Criterion • Conduct head – to – head contests between each pair of candidates. If any candidate can beat each of the others, he is a Condorcet candidate. Under this method a Condorcet candidate is declared the winner. • May not always be decisive! (produce a winner)

20. Fairness Criteria • 1. Majority Criterion. If there is a choice that has a majority of the first-place votes, then that choice should be the winner of the election. • 2. Condorcet Criterion. If there is a choice that is preferred by the voters over each of the other choices (in a head-to-head matchup), then that choice should be the winner of the election. • 3. Monotonicity Criterion. If choice X is a winner of an election and, in a re-election, all the changes in the ballots are favorable to X, then X should still be a winner. • 4. Independence-of-Irrelevant-Alternatives Criterion. If choice X is a winner of an election, and one(or more) of the other choices is disqualified and the ballots recounted, then X should still be a winner. (Also called Binary Independence.) • IIA can also be stated as: It is impossible for an alternative B to move from non-winner to winner unless at least one voter reverses the order in which he/she had ranked B and the winning alterative. • 5. Pareto Criterion. If every voter prefers alternative X over alternative Y, then the voting method should rank X above Y.

21. Fairness Criteria • **Every method we have studied can violate one or more of these! • Plurality: violates Condorcet, IIA • Standard Runoff: violates Monotonicity, Condorcet • Hare Elimination(Sequential Runoff): violates Monotonicity, Condorcetand IIA • Coombs’ Method: violates Monotonicity and Condorcet • Sequential Pairwise Runoff: violates Pareto • Condorcet’s Method: May not even produce a winner! • Borda Count: violates Majority (and hence Condorcet) and IIA • Is there any election decision procedure we could devise which satisfy these fairness criteria if we have 3 or more candidates and use ordinal ballots to rank the candidates?

22. Arrow’s Theorem The search for the perfect election decision procedure

23. The story so far… • We have studied several election decision procedures designed to produce one or more winners from a slate of 3 or more candidates. • Each procedure has had some desirable features and some undesirable ones (‘quirks’). • We’ve even seen that these methods can give different winners using exactly the same set of ordinal ballots!

24. Enter Kenneth Arrow • Arrow, an economist, wanted to find a completely ‘fair’ election decision procedure. • He began by making a list of a few basic properties that he believed any good election decision procedure should have:

25. Arrow’s Properties • Universality—The decision procedure must be any to process any set of ordinal ballots to produce a winner, and must be able to compare any two alternatives. • Non-dictatorship (no one voter can determine the outcome) • Independence-of-Irrelevant-Alternatives Criterion (Binary Independence) • Pareto Criterion

26. IIA or Binary Independence • It is impossible for an alternative B to move from non-winner to winnerunless at least one voter reverses the order in which he/she had ranked B and the winning alternative. • In other words, whether A or B wins should depend only on how the voters compare A to B, and not on how other alternatives are ranked relative to A or B.

27. Pareto Criterion • If every voter prefers alternative X to alternative Y, then the decision procedure should rank X above Y.

28. Asking the Impossible • In 1951 Arrow published a book Social Values and Individual Choice in which he proved that there does not exist an election procedure which ranks for society 3 or more candidates based on individual preferences and which satisfies the fairness criteria we have listed.