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We vote, but do we elect whom we really want?

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. So much goes wrong in this area! So many mysteries!!. Aggregation rule.

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We vote, but do we elect whom we really want?

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  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 So much goes wrong in this area! So many mysteries!! Aggregation rule So, what goes wrong with voting indicates what goes wrong elsewhere in the social sciences in particular economics, business, engineering, etc. What math can offer: Beyond ad hoc approaches, goal should be to find systematic approaches where ideas transfer to other areas.

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

  3. J C de Borda, 1770 Class ranking 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 Seven different election outcomes! Beverage example: Recently solved by Mathematics Problem: Which method is best? i.e., respects voters wishes

  4. 1/2 P A C = (1, ½, 0) C B A A B But, 7 outcomes? Procedure line (1-s) Plurality + s Antiplurality Plot election tallies Actual elections (2, 1, 0) (1, s, 0) Converse Positional rules Normalize weights Normalize election tally 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 Procedure hull 2 A B C D 2 A D C B 2 C B D A 3 D B C A Using different weights, 18 different strict (no ties) elections rankings. With ties, about 35 different election outcomes! A wins Vote for one (1, 0, 0,0): B wins Vote for two (1, 1, 0, 0): C wins Vote for three (1, 1, 1, 0): For about 85% of examples, ranking changes with procedure D wins Borda, (3,2,1,0): OK, so something goes wrong. But how likely is all of this? In general, for n candidates, can have (n-1)((n-1)!) strict rankings! Saari and Tataru, Economic Theory, 1999

  6. Solved in 2000 How do we explain all positional differences? Symmetry is the key! 4 Wine>Beer> Milk, 1 Milk>Wine>Beer Find if ties really are ties! (Systematic rather than ad hoc) 5 Milk>Wine>Beer, 5 Beer>Wine>Milk Here we have Z2 orbits Bob: 20 votes, Sue: 27 votes Cancel votes in pairs: Sue wins Me: A B C Lillian: C B A Candidate: A B C Me: 1 0 0 Bias against B! Lillian: 0 0 1 Candidate: A B C Total: 1 0 1 Me: 1 1 0 Bias for B! Lillian: 0 1 1 Candidate: A B C Total: 1 2 1 Me: 2 1 0 Lillian: 0 1 2 A tie!! Total: 2 2 2 Only the Borda Count Including the beverage example Source of all problems with positional methods

  7. Source of all cycles; voting, statistics, etc. For a price ..... Mathematics? Ranking Wheel 10 A>B>C>D>E>F 10 B>C>D>E>F>A 10 C>D>E>F>A>B Everyone prefers C to D to E to F Rotate -60 degrees A F B E C D 1 6 2 5 3 4 6 5 1 4 2 3 Symmetry: Z6 orbit 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! Reversal + ranking wheel: Explains all three candidate problems! D E C B A F D C B A A>B>C>D>E>F F B>C>D>E>F>A No candidate is favored: each is in first, second, ... once. C>D>E>F>A>B etc. Fred wins by a landslide!! Yet, pairwise elections are cycles! lost information!! Consensus?

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

  9. 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? Borda is in variety; minimizes what can go wrong 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. There exists a profile whereby for each subset of candidates, the specified ranking is the actual ranking! Namely, there exists a proper algebraic variety of weights so that if weights not in variety, then anything can happen

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

  11. Problem resolved! http://www.math.uci.edu/~dsaari 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!

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

  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. D E C B A F 10 A>B>C>D>E>F 10 B>C>D>E>F>A 10 C>D>E>F>A>B D C B A Everyone prefers C, D, E, to F F wins with 2/3 vote!! A landslide victory!! Decision by consensus: Mathematician’s take Why? What characterizes all problems?

  14. 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?

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