Algorithms for the Coalitional Manipulation Problem

1. Algorithms for the Coalitional Manipulation Problem Speaker: Ariel Procaccia Joint work with: Michael Zuckerman, Jeff Rosenschein Hebrew University of Jerusalem (To appear in SODA 2008)

2. Outline • Background on coalitional manipulation problem: • Worst-case. • Average-case. • A greedy algorithm. • New results: characterization of this alg’s and others’ windows of error. • Implications w.r.t. approximation in unweighted case.

3. Notations • Set of voters V={1,...,n}. • Set of Candidates C={a,b,c...}; |C|=m. • Voters (strictly) rank the candidates. • Preference profile: a vector of rankings. • Voting rule: maps preference profiles to candidates. • Plurality. • Borda. a b a b c b c a c

4. The path less taken • Gibbard-Satterthwaite: nondictatorial voting rules,  settings s.t. a voter gains by lying. • Circumvent Gibbard-Satterthwaite by: • Mechanism design. • Single-peaked preferences. • [Bartholdi et al. SC&W 89]: Computational hardness to the rescue! • [Bartholdi et al. SC&W 91]: STV is NP-hard to manipulate. • A lot of recent work.

5. Coalitional manipulation • A coalition of manipulators cooperates in order to make pC win the election. • Votes are weighted. • Formulation as decision problem (CCWM): • Instance: a set of weighted votes which have been cast, the weights of the manipulators, pC. • Question: Can p win the election? • Conitzer et al. [JACM 07]: NP-hard for a variety of voting rules, even when m is constant.

6. “Average-case” complexity of manipulation a c d d b • Worst-case hardness is not a strong guarantee. • Is there a voting rule which is hard to manipulate on a large fraction of the instances? • Apparently not? • Conitzer and Sandholm [AAAI 06]: Instance can be manipulated efficiently if: • Weakly monotone. • Manipulators can make either of exactly two candidates win. b c b a c a d

7. Junta Distributions • Procaccia and Rosenschein [JAIR 07]: Junta distributions are hard. Susceptibility to manipulation if can manipulate with high prob. w.r.t. a Junta distribution. • Scoring rules are susceptible; very loose bound on the error window of a greedy algorithm. • Only scoring rules and constant m.

8. The greedy algorithm • Reminder: in Borda, each voter awards m-k points to candidate ranked k. • Reminder: CCWM • Instance: a set of weighted votes which have been cast, the weights of the manipulators, pC. • Question: Can p win the election? • Greedy algorithm for coalitional manipulation [Procaccia and Rosenschein, JAIR 07]: each manipulator ranks p first, and the other candidates by inverse score.

9. Example: Algorithm is correct 40 0 10 20 30 10 5 5 40 0 10 20 30 40 0 10 20 30

10. Example: Algorithm is wrong 40 0 10 20 30 10 5 40 0 10 20 30 40 0 10 20 30

11. Theorem: Borda • Theorem: Let W be the list of weights for the manipulators. • If there is no manipulation, the greedy alg will return false. • If there is a manipulation, then for the same instance with weights W+{w1,...,wk}, where wi  max W, the alg will return true. • In particular, can add one manipulator with weight max W.

12. Example for the theorem 40 0 10 20 30 10 10 5 40 0 10 20 30 40 0 10 20 30

13. Window of Error

14. Theorem: Maximin • P(a,b):= {iN: a >i b} • The score of aC is minbP(a,b). • Maximin elects candidate with maximal score. • Generalized Greedy algorithm: each manipulator ranks p first, and the other candidates in a way which minimizes their score including manipulator’s vote. • Theorem: Let W be the list of weights for the manipulators. • If there is no manipulation, the greedy alg will return false. • If manipulation, then for the same instance with two copies of W the alg will return true.

15. Theorem: Plurality w. Runoff • a beats b in a pairwise election if majority prefers a to b. • Plurality with runoff: first round eliminates all candidates except two w. highest plurality score; then pairwise election. • Theorem: There is an algs.t. • If there is no manipulation, the alg will return false. • If manipulation, then for the same instance with weights W+{w1,...,wk}, where wi  u, the alg will return true. • Running time is poly in (max W)/(u+1).

16. Discussion: approximation • Unweighted coalitional manipulation is in P for m=O(1). • Conjecture: unweighted coalitional manipulation (CCUM) is NP-complete in Borda and Maximin. • CCUO: given (unweighted) votes of truthful voters, how many manipulators are needed to make p win? • Theorem (saw earlier): Let W be the list of weights. In Borda need additional max W, in Maximin need two copies. • Corollary: • Approximation of CCUO in Borda to additive 1. • 2-Approximation of CCUO in Maximin.

17. Discussion • Theorem (saw earlier): Let W be the list of weights. In Plurality w. runoff need additional u, running time poly in(max W)/(u+1). • Corollary: CCUM in Plurality w. Runoff is in P. • Theorem: CCUM in Veto is in P. • Contrast: CCWM in Plurality w. Runoff and Veto is NP-hard, even when m=3. • Other voting rules: • Copeland: not monotone in weights. • STV: no score, hard even for a single manipulator.

18. Thank You!