How “impossible” is it to design a Voting rule?

1. How “impossible” is it to design a Voting rule? Angelina Vidali University of Athens

2. ( ) f g N 1 u u u ( ) ( ) = b n 1 = > n u ; : : : ; u a u i ; : : : ; i i f g A a a = 1 m ; : : : ; The setting individuals (voters) alternatives Voter i prefers a to b a b c a preference profile voteri

3. f f ¤ U U A U : : ! ! Voting rule vs Social welfare function • A Voting rule is a mapping • it chooses the “winning alternative” • A Social welfare function is a mapping • it chooses the “socialranking” set of all (strict) preference profilesu set of alternatives set of all (weak) preference profilesu Arrow ‘50s Gibbard-Satterthwaite ‘70s

4. Condorcet rule • a beats b in a pairwise election if the majority of voters prefers a to b • a is a Condorcet winner if a beats any other candidate in a pairwise election 4

5. the Condorcet paradox a c c c b b b a a c a b c a a b voter 1 voter 2 b b c a c voter 3 5

6. Some voting rules • Plurality vote: a single winner is chosen by having more votes than any other individual representative. • Borda vote: The Kth-ranked alternative gets a score of N-K. All scores are summed and the candidate with the highest total score wins. • Instant Runoff Voting:If no candidate receives a majority of first preference rankings, the candidate with the fewest number of votes is eliminated and that candidate's votes redistributed to the voters' next preferences among the remaining candidates. This process is repeated until one candidate has a majority of votes.

7. 0 ( ( ) ) ( ( ) ) f f > u u u u u i i i ¡ i ; KKE ΠΑΣΟΚ A manipulable voting rule • If individual i reports a false preference profile instead of his true preference the outcome of the elections is better for him. If I vote ΚΚΕ the outcome f(u) will be either ΝΔ or ΠΑΣΟΚ if I vote ΠΑΣΟΚ the outcome will be ΠΑΣΟΚ. It is better for me to report ΠΑΣΟΚ. ΝΔ non-manipulable=strategyproof voter i

8. KKE KKE KKE KKE KKE ΝΔ ΝΔ ΝΔ ΝΔ ΝΔ ΠΑΣΟΚ ΠΑΣΟΚ ΠΑΣΟΚ ΠΑΣΟΚ ΠΑΣΟΚ The voter who lies determines the winner in a tie! • Ties is the most tricky part… voter 1 voter 2 voter 3 voter 4 voter 5

9. ( ( ) ) ( ) f f l l d A U ¸ 2 2 u u u a o r a a a n u i i A dictator The dictators top alternative is the outcome of the elections.

10. ( ( ) ) ( ( ) ) f f ¼ u ¼ u = Neutral • The names of the candidates don’t matter. i.e. f commutes with permutations of [m] Example: • b is the winner if all voters a b in their preference profiles • then a is the winner.

11. 0 0 1 1 ¢ ¢ ¢ ¢ ¢ ¢ a a 1 1 B B C C . . . . a a 2 B B C C . . B B C C u = a a 3 B B C C B B C C a a 2 3 @ @ A A a a 4 4 Monotonicity“If our preference for the government increases it is reelected” • Let fbe a strategyproof voting rule, f(u)=a. • As long as, for all voters, the alternatives that were worse than a in u, remain worse in v the allocation remains the same. as long as the red elements stay below a the outcome remains a …even if one of the black elements moves below a

12. 0 1 . . . . . . a a . . . B C . . B C b . . a a B C . . u = B C B . . C b b . . a B C . . @ A . . . b b . . . . . . Pareto Optimality • “If everybody prefers a to b then b is not elected.” Pareto Optimality follows from Monotonicity

13. 3 ¸ n Gibbard (‘73)-Satterthwaite (‘75) theorem • If the number of alternatives • then a voting rule that is strategyproof and onto is dictatorial. • follows from Arrow’s impossibility theorem (1951) using the correspondence between: Independence of Irrelevant Alternatives and strategyproofness

14. Independence of Irrelevant Alternatives The social relative ranking of two alternatives a, b depends only on their relative ranking. If one candidate dies the choice to be made among the set S of surviving candidates should be independent of the the preferences of individuals for candidates not in S. [See: Social Choice and Individual Values, K.J. Arrow p.26]

15. see: www.scorevoting.net a voting method that violates IIA • The alternative with the highest weighted sum of votes wins. At first a is chosen a:4+4+2=10 b:7 c:8 d:6 but if b leaves: a:10 c:10 d:7 we get a tie between a and c weight 4 3 2 1 a a c d a b b c d c d b voter 1 voter 2 voter 3

16. 0 0 1 1 b b a a b b a c v u = = @ @ A A a c c c The proof of G-S theorem forn=2 voters So a wins in both u and v. voter 1 becomes a dictator for a. c cannot win (Pareto Optimality) Assume a wins (w.l.o.g.) Then b wins (monotonicity) cdc c cannot win (Pareto Optimal.) Suppose b wins

17. j n ( ) j A A A 1 · \ 1 2 ; The proof of G-S theorem forn=2 voters (2) • Repeat for every pair of alternatives {a,b} • A1={x| player 1 is a dictator for x} • A2={y| player 2 is a dictator for y} • because: if it had two distinct elements then one of them should belong to A1 or A2. • Finally some Ai= all the element belong to Aj and j is the dictator.

18. Towards a Quantitative version Gibbard-Satherwaite theorem: “Every non-trivial (=non dictatorial) voting rule is strategically vulnerable.” • How often? • For what fraction of profiles does such a manipulation exist?

19. Impartial culture assumption • Voters vote independently and randomly • We draw independently and uniformly a random ranking for each voter • possible rankings for voter i: m! • P(each ranking)= 1/m!

20. 0 u i Manipulation power of a voter: Mi(f) The manipulation power Mi( f ), of voter ion the socialchoice function f, is the probability that : if voter i reports a chosen uniformly at random this is a profitable manipulation of f for voter i . $i$ What is the probability I can gain something by just drawing one of the m alternatives randomly and reporting this instead of my true preference? individual i

21. ( ) f M [ ( 0 ) ( ) ] f f P < · ² > u u u ² i i ¡ i ; ε-strategyproof = The manipulation power Mi( f ), of voter ion the socialchoice function f, satisfies

22. ( ( ) ) [ ( ) ( ) ] f f ± f 6 ¢ ¢ P > g g u g u = = U 2 u ; ; δ-far from dictarorship • The distance between two functions f,g is • f is δ-far from dictatorship, if for any dictatorship g

23. 1 ( ) ­ n ² ² Quantitative version of G-S theorem[Friedgut-Kalai-Nisan FOCS’08] For every >0 if f is a voting rule for n voters is • neutral • among 3 alternatives • -far from dictatorship, then one of the voters has a non-negligible manipulation power of .

24. Quantitative version of G-S theorem[Xia-Conitzer EC’08] • a list of assumptions… • homogeneity • anonymity • non-imposition • a canceling out condition • there exists a stable profile that is still stable after one given alternative is uniformly moved to different positions • (they argue that many known voting rules satisfy them) • for arbitrarily many alternatives and players

25. 8 1 6 m ² 1 < ² 9 3 2 m ² Quantitative version of G-S theorem2 voters[Dobzinski-Proccacia WINE’08] For if a voting rule f for 2 voters is • Pareto optimal (annoying condition!) • among at least 3 alternatives • with manipulation power < then f is -far from dictatorship. no neutrality assumption here! m can also be greater than 3

26. Some open problems • Quantitative version of G-S theorem for more than 2 voters (with less conditions!) • What about the impartial culture assumption? is it plausible? • Find quantitative versions of known mechanism design results: straptegyproof ε-strategyproof

27. Endnote "Most systems are not going to work badly all ofthe time, all I proved is that all can work badly attimes." K. J. Arrow …or do they work badly most of the time???