How Hard Is It To Manipulate Voting?

1. How Hard Is It To Manipulate Voting? Edith Elkind, U. of Warwick Helger Lipmaa, Tartu U.

2. Short Bio • High school diploma, School № 15, Tallinn, 1993 • M.Sc., Moscow State University, Department of Mathematics, 1998 • Ph.D, Princeton University, 2005 • Now: Postdoctoral researcher, U. of Warwick, UK Research interests: algorithmic game theory, voting, algorithms, complexity…

3. Bib Info • Small Coalitions Cannot Manipulate Voting, Financial Cryptography 05 • Hybrid Voting Protocols And Hardness of Manipulation, ISAAC 05, to appear

4. What Is Manipulation? • In a small country far, far away there is an election coming up…

5. Manipulation: Example • 99 voters, 3 candidates (Red, Blue, Green). • 49 voters: R > B > G. • 48 voters: B > R > G. • 2 voters (Edith and Helger): G > B > R. • Aggregation rule: Plurality • each voter casts a vote for one candidate. • the candidate with the largest number of votes wins. • draws are resolved by a coin toss.

6. What Will Edith and Helger Do? R: 49 votes B: 48 votes G> B>R If I vote forG,Rwill getelected, so I’d rather vote forB If Edith and Helger vote B > G > R, they can guarantee that B is elected

7. Why Manipulation Is Bad • Aggregation rules are designed with certain social welfare criteria in mind. • Misrepresentation of preferences results in a suboptimal choice w.r.t. these criteria. • Also, election results do not reflect true distribution of preferences in the society: • maybe, in fact, in 2000 20% of the U.S. population prefered Nader to Gore to Bush?

8. 2nd round R > B 49 votes B wins B > R 50 votes What If We Change Aggregation Rule? • Single Transferable Vote: • This time, Edith and Helger are better off voting honestly, but this will not always be the case… • Other popular voting schemes (Borda, Copeland) suffer from the same problem 1st round R > B > G 49 votes B > R > G 48 votes G > B > R 2 votes

9. Formal Setup • n voters • m candidates c1, …, cm • Preference of a voter i: a permutation pi of c1, …, cm (best to worst). • Aggregation rule S: p1, …, pn → cj.

10. Voting Schemes: Examples • Borda: a candidate gets • m points for each voter who ranks him 1st, • m-1 point for each voter who ranks him 2nd, etc. • Copeland: • candidate that wins the largest # of pairwise elections • Maximin: • c’s score against d: # of voters that prefer c to d; • c’s # of points: min score in any pairwise election. • many, many others…

11. Voting Schemes: Properties • Pareto-optimality: if everyone prefers a to b, b does not win • Condorcet-consistency: if there is a candidate that wins every pairwise election, this candidate wins • Majority: if there is a candidate that is ranked first by a majority of voters, this candidate wins • Monotonicity: it is impossible to cause a winning candidate to lose by moving it up in one’s vote Arrow’s theorem: there is no perfect scheme

12. Manipulation: Definition • A voter i can manipulate a voting scheme S if there is • a preference vector p = (p1,…,pi, …,pn) • a permutation pi’ s.t. S(p1,…,pi’, …,pn) >iS(p). Theorem (Gibbard-Satterthwaite, 1971): every non-dictatorial aggregation rule with ≥3 candidates is manipulable.

13. How Do We Get Around The Impossibility Result? • We cannot make manipulation impossible… • But we can try to make it hard! • How do you manipulate Plurality? • vote for your favorite candidate among those tied for the top position. • How do you manipulate Borda? • rank your favorite feasible candidate highest, move his competitors to the bottom of your vote. • How do you manipulate STV? • try all m! possible ballots…

14. What Is Known? • 2nd order Copeland is NP-hard to manipulate (Bartholdi, Tovey, Trick 1989) • STV is NP-hard to manipulate (Bartholdi, Orlin 1991) • these rules may not reflect the welfare goals (why is there so many voting rules out there?) • Want a universal method to turn any voting protocol into a hard-to-manipulate one.

15. P r e r o u n d Alice Bob Do most voters prefer Alice to Bob? Alice Diana Ernest Original protocol Adding a Preround (Conitzer-Sandholm’03) Carl Diana Ernest Frank • Retains some of the flavor of the original protocol. • Is NP-hard to manipulate for manybase protocols. • Still, the outcome may be very different from the • original protocol…

16. A C G E F H B D C G A Binary Cup Do most voters prefer A to B? R1 F R2 C F R3 C Binary Cup itself is easy to manipulate.

17. Our Work: Hybrid Protocols • Protocols with a preround can be viewed as hybrids of BC and other protocols • how about other hybrids? • Hyb(Xk, Y): execute k steps of X, then apply Y to the remaining candidates. • Hyb(Pluralityk, Borda): eliminate k candidates with the lowest Plurality scores, then compute Borda scores w.r.t. survivors. • Observation: Hyb(Plurality1, …, Pluralitym) = STV. a b c d e 3 2 1

18. New Results • New protocols that are hard to manipulate: • Hyb(Xk, STV), Hyb(STVk, Y) (for any reasonable X, Y) • Hyb(Bordak, Plurality), Hyb(Maximink, Plurality) • Generally, Hyb(Xk, X) ≠ X (and may be much harder to manipulate) • manipulating Hyb(Bordak, Borda) is NP-hard • Extensions to utlity-based voting (voters rate candidates rather that rank them) • manipulating Hyb(HighScorek, HighScore) is NP-hard

19. s1 s3 s4 s2 G Proof Idea • Set Cover: G = {g1, …, gn}, S = {s1, …, sm}, each si is a subset of G. Is there a cover of size K, i.e, C = {si1, …, siK} s.t. each gi is contained in some sj? • Set Cover is NP-hard. • Can we reduce Set Cover to protocol manipulation? Suppose someone can manipulate Hyb (Xk, Y). We want to show that he can also find a set cover.

20. Proof Idea (continued) manipulator’s prefered candidate • Candidates: g1, …, gn, s1, …, sm, p, d1, …, dt • Voters: • ( p, … ): A ballots • for each gi • (gi, …, ): A - 1 ballot • (sj1, …, sjr, gi, … ) s.t. gi is in sj1, …, sjr: B ballots • some ballots of the form (dj, …, ) • k = m – K • s1, …, sm are tied for the last place under Xk • manipulator’s vote decides which K candidates survive the preround

21. s1 s5 s8 g1 … s3 s4 s5 g2 … g1 … … … … p … ... ... … s1 s5 s8 g1 … s3 s4 s5 g2 … g1 … … … … p … … … … p ………… d1 ………… Proof Idea (continued) • p wins iff none of gi gains votes after Xk • gi gains votes iff all sjs.t. gi is in sjare eliminated • no gi gains votes iff surviving sjcover all gi • manipulation is successful iff manipulator can find a set cover of size K

22. Worst-Case vs. Average-Case Hardness • Is 3-Coloring hard on a random graph? • likely to contain just say “no” • poly-time, works for almost all graphs • We embed a problem that is sometimes hard into some class of preference profiles • Want a reduction that • works for (almost) all preference profiles • reduces from a problem that is (almost) always hard We show how to do (2) using one-way functions; (1) is an open problem.

23. y One-Way Functions f is a one-way function (OWF) if it is • easy to compute: for any x, f(x) is polynomial time computable. • hard to invert: • x is drawn at random from {0, 1}n • we are given y = f(x) • have to guess x, or any z s.t. y = f(z) • average-case hardness: any poly-time algorithm has negligible chance of success (over the choice of x). x OWF

24. OWF: Properties • It is not known if OWF exist (implies P ≠ NP). • Multiplication of large primes is conjectured to be a OWF (factoring is believed to be hard) • OWF are widely used in cryptography • adversary’s task must be hard almost always, not just sometimes. • Can we make manipulation as hard as inverting OWF? • sort of: we embed inverting OWF into some preference profiles.

25. P r e r o u n d Alice Bob Do most voters prefer Alice to Bob? Alice Diana Ernest Original protocol Recall: Preround (CS’03) Carl Diana Ernest Frank • Retains some of the flavor of the original protocol. • Is NP-hard to manipulate for manybase protocols. • Still, the outcome may be very different from the • original protocol…

26. Alice Bob … p1 Alice ↕ Frank XOR 01101… 11101… Bob Carl … p2 Bob ↕ Ernest 11100… OWF CarlAlice … pn Carl ↕ Diana 10111… 00100… Preround schedule Our Scheme: Preround With a Twist Idea: use votes themselves to select preround schedule.

27. Key Properties • No trusted source of randomness is needed. • Manipulating this protocol is as hard as inverting the underlying one-way function • even for a large coalition of conspiring would-be manipulators • Proof proceeds by constructing a vector of honest voters’ preferences s.t. there is a unique preround schedule that makes manipulators’ protégé a winner • still a worst-case construction…

28. Average-Case Hardness? • Can we make manipulation hard with high probability over honest voters’ preferences? • Do we want to make assumptions about the distribution of preference profiles? • In real life, not all preference profiles are equally likely…

29. Hyb(BC, Plurality): Easy for Any Preround Schedule • Candidates a, b, c1, …, c2t+1, p • k+1 voters: p > a > b > C, k ≈ n/3 • k voters: a > b > p > C • n-2k-2 voters: C > p > a > b • Manipulator: C > a > b > p • Under any preround schedule • p survives • a cj survives • either a or b survives • In the final round, p competes against a or b • manipulator is better off voting a > b > p > C

30. Open Problems • Average-case hardness seems hard to achieve using a preround. Other approaches? • What is the maximum fraction of manipulators we can tolerate? • for most base protocols, we can prove security against 1/6 of all voters. Is it optimal? • Our results are for specific protocols. More general proofs?