Ties Matter: Complexity of Voting Manipulation Revisited

1. Ties Matter:Complexity of Voting Manipulation Revisited Edith Elkind(Nanyang Technological University, Singapore) based on joint work with Svetlana Obraztsova(NTU/PDMI)andNoam Hazon(CMU)

2. Synopsis • We will talk about voting • and, in particular, voting manipulation • We will focus on a frequently neglected aspect of voting: tie-breaking rules • We will show that ties matter

3. Setup • An election is given by • a set of candidates C, |C| = m • a list of voters V = {1, ..., n} • for each voter i in V, a preference order Ri • each Ri is a total order over C • a voting rule F: • for each list of voters’ preference orders, F outputs a candidate in C

4. Examples of Voting Rules (1/2) • Scoring rules: • any vector s = (s1, ..., sm) defines a scoring rule Fs: • each candidate receives si points from each voter who ranks him in positioni • a candidate’s score is his total # of points • the candidate with the highest score wins • Examples: • Plurality: (1, 0, ..., 0) • Borda: (m-1, m-2, ..., 2, 1, 0) 7 5 2 1 0 a b c d e

5. Examples of Voting Rules (2/2) • Copeland: • for a, b  C, we say that a beats bin a pairwise election if more than half of the voters rank a above b • the score of a candidate cis # of pairwise elections cwins- # of pairwise elections closes • Maximin: • for a, b  C, let S(a, b) = # of voters who prefer a to b • the score of a candidate c is minaC\{c}S(c, a) • the number of votes c gets against his toughest opponent

6. Applications • Political elections • Hiring new faculty • Prizes • Decision-making in multi-agent systems • voting over joint plans

7. Manipulation • A voting rule is manipulable if there exists a preference profile s.t. some voter has an incentive to lie about their preferences • i prefers F(R1, ..., R’i, ..., Rn) to F(R1, ..., Ri, ..., Rn) • Gibbard’73, Satterthwaite’75: for |C|>2, any non-dictatorial voting rule is manipulable • But maybe manipulations are hard to compute? • Bartholdi, Tovey, Trick’89: given a profile, one can find a beneficial manipulation in poly-time for most voting rules • Plurality, Borda, Copeland, maximin

8. A Complication • The common voting “rules” are voting correspondences: several candidates may have the top score • Ties need to be broken • BTT’89 assumes that ties are broken in favor of manipulator • The algorithm extends to any lexicographic tie-breaking rule • i.e., one that uses a priority order over C • What if the tie-breaking rule is not lexicographic?

9. This Work • We consider two types of tie-breaking rules: • randomized tie-breaking: • the winner is selected from the tied candidatesuniformly at random • the manipulator assigns utilities to candidates, maximizes his E[utility] • arbitrary poly-time tie-breaking • tie-breaking rule is given by an oracle • Question: do easiness results of BTT’89 still hold under these tie-breaking rules?

10. Results: Randomized Tie-Breaking, Scoring Rules • Theorem: for any scoring rulethe manipulation problem is poly-time solvable under randomized tie-breaking What is the best outcome where winners have t points, for each feasiblet? t manipulator’s vote non-manipulator’ votes

11. Results: Randomized Tie-Breaking, Maximin, “Special” Utilities • Theorem: for Maximin, if the manipulator’s utility is given by u(p)=1, u(c)=0 for c ≠ p, then the manipulation problem is poly-time solvable under randomized tie-breaking • Proof sketch: • let s(c) denote c’s score before we vote • our vote changes each score by at most 1: • c’s score goes up iff c appears beforeeach of its toughest opponents • if s(c) > s(p)+1 for some c ≠ p, we lose; suppose this is not the case • rank p first

12. Maximin Proof, Continued • cis good if s(c) < s(p) • c is bad if s(c) = s(p) • c is ugly if s(c) = s(p)+1 • G= directed graph with vertex set Cs.t. there is an edge from a to b iff a is b’s toughest opponent • Goal: sort G so that each ugly vertex and as many bad vertices as possible have an incoming edge • can be done in poly-time

13. Results: Randomized Tie-Breaking, Maximin, General Utilities • Theorem: for Maximinthe manipulation problem is NP-hard under randomizedtie-breaking • even if u(d)=0, u(c)=1for c ≠ d • Proof idea: • set up the instance so that d necessarily wins • need to maximize the number of winners • i.e., sort G so that as few vertices as possible have an incoming edge • reduction from Feedback Vertex Set

14. Results: Randomized Tie-Breaking, Copeland, General Utilities • Theorem: for Copelandthe manipulation problem is NP-hard under randomizedtie-breaking • even if u(c)  {0, 1} for all c  C • Reduction from Independent Set • extends to a hardness of approximation result :assuming P ≠ NP, no poly-time algorithm can find a manipulative vote such that the manipulator’s utility is within a constant factor from optimal

15. Arbitrary Poly-Time Tie-Breaking • Theorem: there exists a poly-time computable tie-breaking rule Ts.t. its combinations with Borda, Copeland and maximin are NP-hard to manipulate • T depends on the set of tied candidates only • if we allow tie-breaking rules that depend on the rest of the profile, even Plurality is NP-hard to manipulate • Proof idea: • the winning set encodes a Boolean formulafand a truth assignmenta for f • the manipulator can affect a, but not f • the tie-breaking rule checks if asatisfiesf

16. Conclusions and Future Work TIES MATTER! • Approximation algorithms/inapproximability results for Maximin? • Complexity of control/bribery/coalitional manipulation under randomized tie-breaking?