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

2. Outline • Background: • Intro to voting. • Hardness of manipulation. • Coalitional manipulation. • A greedy algorithm. • New results: characterization of alg’s window of error. • Implications w.r.t. approximation.

3. Voting: motivation • Agents have to reach a consensus regarding a preferred alternative in a shared environment. • Examples: • Joint plans. • Beliefs. • Recommendations. • Voting theory gives a well studied framework for preference aggregation.

4. 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 c c a b

5. Manipulation • Often it is in the voters’ interest to reveal false preferences. • May lead to the election of a socially bad candidate. 2 4 0 1 3 2 4 0 1 3 2 4 0 1 3

6. The path less taken • Theorem (Gibbard-Satterthwaite): any nondictatorial voting rule is manipulable. • Circumvent Gibbard-Satterthwaite by: • Mechanism design. • Single-peaked preferences. • [Bartholdi et al. SC&W 89]: Computational hardness to the rescue! • [Bartholdi and Orlin SC&W 91]: STV is NP-hard to manipulate. • A lot of recent work.

7. 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.

8. “Average-case” complexity of manipulation • 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?

9. 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 other limitations.

10. 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.

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

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

13. 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.

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

15. Window of Error

16. Unweighted case • Conjecture: unweighted coalitional manipulation (CCUM) is NP-complete in Borda. • 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 manipulators need additional max W. • Corollary: Approximation of CCUO in Borda to additive 1.

17. Important Closing Remark • Similar results for three other voting rules: Maximin, Plurality with runoff, Veto.

18. Thank You!