Speaker: Ariel Procaccia Joint work with: Michael Zuckerman, Jeff Rosenschein Hebrew University of Jerusalem

 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.

 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. 3

 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. Voter 1Voter 2Voter 3 a b c a c b b c a

 Often it is in the voters’ interest to reveal false preferences.  May lead to the election of a socially bad candidate. 5 pab

 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.

 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.

 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. A second, problematic property. Voter 1Voter 2 Voter 3 a b b c a d c b d a d c

 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?

 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.

 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.

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 Theorem: Let W be the list of weights for the manipulators. 1. If there is no manipulation, the greedy alg will return false. 2. If there is a manipulation, then for the same instance with weights W+{w 1,...,w k }, where  w i  max W, the alg will return true.  In particular, can add one manipulator with weight max W.

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Algorithm fails Manipulation exists All instances

 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.

 Similar results for three other voting rules: Maximin, Plurality with runoff, Veto.