1. We vote, but do we elect whom we really want? Don Saari Institute for Mathematical Behavioral Sciences University of California, Irvine, CA dsaari@uci.edu What math can offer: Beyond ad hoc approaches, goal should be to find systematic approaches where ideas transfer to other areas. So much goes wrong in this area! So many mysteries!! So, what goes wrong with voting indicates what goes wrong elsewhere in the social sciences in particular economics, business, engineering, etc. Aggregation rule

2. Party time! 6 5 4 Milk,Wine,Beer Beer, Wine, Milk Wine, Beer, Milk Plurality Milk-6, Beer-5, Wine-4 MilkWine 6 9 MilkBeer 6 9 BeerWine 5 10 Pairwise Wine, Beer, Milk Beer? Runoff election Rather than voter preferences, an election outcome can reflect the choice of an election method! Why? That is the basic issue addressed today Milw. Wash, Boston Boston, Wash, Milw Wash, Boston, Milw Business decisions

3. J C de Borda, 1770 Plurality: one point for first place, zero for all others Weighted: Points to first, second, third,.... Borda: Number below, so for three candidates 2, 1, 0 Beverage example: Seven different election outcomes! Problem: Which method is best? i.e., respects voters wishes Recently solved by Mathematics Class ranking

4. A C B A B C Plot election tallies Normalize election tally Positional rules Normalize weights (1, s, 0) (2, 1, 0)1/2= (1, ½, 0) (1-s) Plurality + s Antiplurality P A Actual elections Converse But, 7 outcomes? Procedure line Goal: find systematic approach

5. Good news and bad, first: How bad can it get? Three candidates: About 70% of the time, election ranking can change with weights! More candidates, more severe problems 2 A B C D 2 A D C B 2 C B D A 3 D B C A A wins B wins C wins D wins Using different weights, 18 different strict (no ties) elections rankings. With ties, about 35 different election outcomes! For about 85% of examples, ranking changes with procedure Vote for one (1, 0, 0,0): Vote for two (1, 1, 0, 0): Vote for three (1, 1, 1, 0): Borda, (3,2,1,0): OK, so something goes wrong. But how likely is all of this? Saari and Tataru, Economic Theory, 1999 In general, for n candidates, can have (n-1)((n-1)!) strict rankings! Procedure hull

6. How do we explain all positional differences? Solved in 2000 Bob: 20 votes, Sue: 27 votesCancel votes in pairs: Sue wins Me: A B C Lillian: C B ACandidate: A B C Me: 1 0 0 Lillian: 0 0 1 Total: 1 0 1 Candidate: A B C Me: 1 1 0 Lillian: 0 1 1 Total: 1 2 1 Candidate: A B C Me: 2 1 0 Lillian: 0 1 2 Total: 2 2 2 Find if ties really are ties! A tie!! Bias against B! Bias for B! Here we have Z 2 orbits Source of all problems with positional methods Only the Borda Count Including the beverage example 4 Wine>Beer> Milk, 1 Milk>Wine>Beer 5 Milk>Wine>Beer, 5 Beer>Wine>Milk Symmetry is the key! (Systematic rather than ad hoc)

7. I will come to your group before your next election. You tell me who you want to win. After talking to everyone in your group, I will design a “fair” election rule, which includes all candidates. Your candidate will win! 10 A>B>C>D>E>F 10 B>C>D>E>F>A 10 C>D>E>F>A>B D E C B A F D C B A F Mathematics? 1 6 2 5 3 4 A F B E C D Ranking Wheel A>B>C>D>E>F 6 5 1 4 2 3 Rotate -60 degrees B>C>D>E>F>A C>D>E>F>A>B etc. Symmetry: Z 6 orbit No candidate is favored: each is in first, second,... once. Source of all cycles; voting, statistics, etc. For a price..... Yet, pairwise elections are cycles! lost information!! Fred wins by a landslide!! Everyone prefers C to D to E to F Consensus? Reversal + ranking wheel: Explains all three candidate problems!

8. 3A>C>D>B2C>B>D>A 6A>D>C>B5C>D>B>A 3B>C>D>A2D>B>C>A 5B>D>C>A4D>C>B>A X OUTCOME: A>B>C>D by 9: 8: 7: 6 X Now: C>B>A x Now: D>C>B Drop any one or any two candidates and outcome reverses! Conclusion in general holds for ALMOST ANY Weights -- except Example 2 4 x 3 6 Borda Count!

9. Borda is in variety; minimizes what can go wrong Extends to almost all other choices of weights A mathematician’s take on all of this: OK, some examples are given. Can we find everything, all possible examples, of what could ever happen? Chaos! Symbolic Dynamics Theorem: For n >2 candidates, anything you can imagine can happen with the plurality vote! Namely, for each set of candidates, the set of n, the n sets of n-1, etc., etc., select a transitive ranking. Namely, there exists a proper algebraic variety of weights so that if weights not in variety, then anything can happen There exists a profile whereby for each subset of candidates, the specified ranking is the actual ranking!

10. Number of droplets of water in all oceans of the world Borda Count! Seven candidates Number plurality listsNumber Borda lists <10 50 More than a billion times the

11. Problem resolved! Using mathematical symmetry Conclusion: The Borda Count is the unique choice where outcome reflects voters views Only one example of where mathematics plays crucial role in understanding problems of our society Thank you! http://www.math.uci.edu/~dsaari

12. Arrow Inputs: Voter preferences are transitive No restrictions Output: Societal ranking is transitive Voting rule: Pareto: Everyone has same ranking of a pair, then that is the societal ranking Binary independence (IIA): The societal ranking of a pair depends only on the voters’ relative ranking of pair Conclusion: With three or more alternatives, rule is a dictatorship With Red wine, White wine, Beer, I prefer R>W. Are my preferences transitive? Cannot tell; need more information Determining societal ranking cannot use info that voters have transitive preferences Modify!! You need to know my {R, B} and {W, B} rankings! A>B, B>C implies A>C No voting rule is fair! Borda 2, 1, 0 And transitivity Dictator = EX profile restriction e.g., # of candidates between Lost info: same as with binary: cannot see info like higher symmetry or transitivity

13. For a price... I will come to your organization for your next election. You tell me who you want to win. I will talk with everyone, and then design a “fair” election procedure. Your candidate will win. 10 A>B>C>D>E>F 10 B>C>D>E>F>A 10 C>D>E>F>A>B Decision by consensus: Everyone prefers C, D, E, to F D E C B A F D C B A Mathematician’s take F wins with 2/3 vote!! A landslide victory!! Why? What characterizes all problems?

15. A mathematician’s take on all of this OK, so something goes wrong. But how likely is all of this? Saari and Tataru, Economic Theory, 1999 Instead of the plurality vote, how about using other weights to tally ballots?