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. Business decisions Party time! Wash, Boston Milk,Wine,Beer 6 Milw. 5 Boston, Beer,Wash, Wine,Milw Milk Plurality Milk-6, Beer-5, Wine-4 Pairwise Wine, Beer, Milk Runoff election Beer? Wash, Boston, Milw 4 Wine, Beer, Milk Milk Wine Milk Beer Beer 6 9 6 9 5 Wine 10 Why? That is the basic issue addressed today Rather than voter preferences, an election outcome can 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 Beverage example:Seven different election outcomes! Problem: Which method is best? i.e., respects voters wishes Recently solved by Mathematics C But, 7 outcomes? Procedure line (1-s) Plurality + s Antiplurality Plot election tallies Actual elections A 1/2 (2, 1, 0) = (1, ½, 0) B (1, s, 0) C Converse Positional rules Normalize weights Normalize election tally A Goal: find systematic approach A P B 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 2ABCD 2CBDA 2 ADCB 3 D B CA Vote for one (1, 0, 0,0): A wins Vote for two (1, 1, 0, 0): B wins Using different weights, 18 different strict (no ties) elections rankings. With ties, about 35 different election outcomes! Vote for three (1, 1, 1, 0): C wins For about 85% of examples, OK, so something goes with wrong. ranking changes procedure D wins Borda, (3,2,1,0): But how likely is all of In general, for n candidates, can have (n-1)((n-1)!) strict rankings! this? Saari and Tataru, Economic Theory, 1999 How do we explain all positional differences? Solved in 2000 Symmetry is 4 Wine>Beer> Milk, 1 Milk>Wine>Beer if ties really are ties! (SystematicFind rather than ad hoc) the key! 5 Milk>Wine>Beer, 5 Beer>Wine>Milk Bob: 20 votes, Sue: 27 Here we have Z2votes orbitsCancel votes in pairs: Sue wins Me: A B C Lillian: C B A Candidate: A B C Me: 1 0 0 Bias against Lillian: 0 0 1 Candidate: A B C B! 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 Total: 2 2 2 A tie!! Including the beverage example Only the Borda Count Source of all problems with positional methods Source of all cycles; voting, statistics, etc. For a price ..... Mathematics? Ranking Wheel 10 A>B>C>D>E>F Everyone prefers C to D to E to F 10 B>C>D>E>F>A 10 C>D>E>F>A>B Symmetry: Z6 orbit Rotate -60 degrees A F 56 61 12 B I will come to your group before your next election. You 32 Reversal + 45 D E C everyone in 43 tell me who you want to win. After talking to D ranking C E your group, I will design a “fair” election rule, which B C D wheel: A B candidates. Your candidate will win! includes all A>B>C>D>E>F Explains all A F F B>C>D>E>F>A three No candidate is favored: each is in C>D>E>F>A>B candidate first, second, ... once. etc. problems! Yet, pairwise elections are cycles! Fred wins by a landslide!! lost information!! Consensus? Example 3 x A>C>D>B 2 C>B>D>A 6 A>D>C>B x 5 C>D>B>A 3 B>C>D>A 2 X D>B>C>A 5 B>D>C>A 4 X D>C>B>A Now: C>B>A OUTCOME: A>B>C>D Now: D>C>B by 9: 8: 7: 2 3 4 6 6 Drop any one or any two candidates and outcome reverses! Conclusion in general holds for ALMOST ANY Weights -- except Borda Count! A mathematician’s take on all of this: OK, some examples are given. Can we find everything, all possible examples, of what could Borda is in variety; minimizes what can go wrong ever happen? Chaos! Symbolic Extends to almost all otherDynamics choices of weights 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 Borda Count! Seven candidates 50 10 Number Borda lists < Number plurality lists More than a billion times the Number of droplets of water in all oceans of the world Problem resolved! http://www.math.uci.edu/~dsaari Using mathematical symmetry Conclusion: The Borda Count is the unique choice where outcome reflects voters Only one example of where mathematics plays crucial role in views understanding problems of our society Thank you! A>B, B>C implies A>C Arrow 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 Voting rule:Pareto: Everyone has same voters have transitive ranking of a pair, then that is the societal ranking preferences Binary independence (IIA): The societal ranking of a Borda 2, 1, 0 pair depends only on the voters’ relative ranking of pair Lost info: same as with binary:And cannot see info like higher transitivity Modify!!e.g., of candidates between symmetry or #transitivity 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 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 D D E C C B B A A F 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? 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?