1 / 30

How Hard Is It To Manipulate Voting?

How Hard Is It To Manipulate Voting?. Edith Elkind, U. of Warwick Helger Lipmaa, Tartu U. Short Bio. High school diploma , School № 15, Tallinn, 1993 M.Sc. , Moscow State University, Department of Mathematics, 1998 Ph.D , Princeton University, 2005

jolie
Download Presentation

How Hard Is It To Manipulate Voting?

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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?

More Related