1. Junta Distributions and the Average Case Complexity of Manipulating Elections A. D. Procaccia & J. S. Rosenschein

2. Lecture outline Introduction to voting Scoring protocols Manipulation Our approach Junta distributions Main Result: Manipulating scoring protocols is NP-hard but easy in the average-case Closing remarks and future research VotingManipulationOur ApproachMain ResultConclusions

3. In multiagent environments, different agents may have diverse preferences. A common scheme for preference aggregation is voting. Agents reveal their preferences by ranking m candidates. Winner determined by a voting protocol. Plurality. VotingManipulationOur ApproachMain ResultConclusions What is Voting?

4.  = where  i ≥  i+1. Candidate receives  i points for each voter that ranks it in i’th place. Examples: Plurality: Veto: Borda: Sensitive scoring protocols = scoring protocols where  m-1 >  m =0. Including Veto and Borda. VotingManipulationOur ApproachMain ResultConclusions Scoring Protocols

5. Selfish agents may prefer to reveal their intentions untruthfully. This is problematic, since the outcome may be one that is socially undesirable. COALITIONAL MANIPULATION. Given: a set S of weighted votes, a set T of manipulators’ weights, and a candidate p. Can the votes in T be cast so that p wins? 10 3 2 1 3 11 4 VotingManipulationOur ApproachMain ResultConclusions Why lie?

6. Motivation Voting protocol is non-dictatorial   elections where an agent is better off voting untruthfully. Bounded rationality comes to the rescue! Individual manipulation of some protocols is NP- hard. We prove: coalitional manipulation of sensitive scoring protocols is NP-hard, even when m=3. A weak guarantee of resistance to manipulation. Given a reasonable dist., how hard is it to manipulate? VotingManipulationOur ApproachMain ResultConclusions

7. Traditional average case complexity theory seems inappropriate for our purposes. Distributional problem = ; L is a decision problem,  is a distribution over the possible inputs. Algorithm ALG is a heuristic polynomial time algorithm for if ALG runs in poly time, and  p s.t. Pr x  [ALG(x)≠L(x)]≤1/p(|x|). VotingManipulationOur ApproachMain ResultConclusions Avg. Case Analysis

8. Junta Distributions If an algorithm succeeds in deciding instances drawn from a junta distribution, it will also succeed with most reasonable distributions. Properties: Hardness: Still enough hard instances. Dichotomy: Instances are either probable or impossible. Easy instance Hard instance Zero prob. Low prob. High prob. VotingManipulationOur ApproachMain ResultConclusions

9. A mechanism is susceptible to a manipulation M if there exists a junta dist. , s.t. there exists a heuristic poly time algorithm for. Theorem: Let P be a sensitive scoring protocol. Then P is susceptible to coalitional manipulation when m=O(1). VotingManipulationOur ApproachMain ResultConclusions Susceptibility to manipulation

10. Sampling algorithm for  *: All v in T: randomly choose w(v) in [0,1]. Total weight is W. All candidates≠p: randomly choose initial score in [(  1 -  2 )W,  1 W].  * is a junta distribution.  * is intuitively appealing. VotingManipulationOur ApproachMain ResultConclusions A junta distribution

11. Greedy algorithm: each voter in T ranks p first, and the other candidates in an order inversely proportional to their current score. a b p 2.9 2.1 1 1 2.6 3.1 3.9 2 2 4 4 T 0.50.51 2.1 1.6 1 1 T 0.51 2.1 3.1 3 3 4 3 2 1 0 VotingManipulationOur ApproachMain ResultConclusions A heuristic poly time alg

12. Remarks and future research Paper contains additional results on individual manipulation with uncertainty about others’ votes. Starting point for studying average case complexity of manipulating different protocols and mechanisms. Can be regarded as an impossibility result; which protocols are average-case hard to manipulate? VotingManipulationOur ApproachMain ResultConclusions