Junta Distributions and the Average Case Complexity of Manipulating Elections

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

2. Lecture outline Voting Manipulation Our Approach Main Result Conclusions • 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

3. What is Voting? Voting Manipulation Our Approach Main Result Conclusions • 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.

4. Scoring Protocols Voting Manipulation Our Approach Main Result Conclusions •  = <1,…, m> where i ≥ i+1. Candidate receives i points for each voter that ranks it in i’th place. • Examples: • Plurality: <1,0,…,0> • Veto: <1,…,1,0> • Borda: <m-1,m-2,…,0> • Sensitive scoring protocols = scoring protocols where m-1>m=0. • Including Veto and Borda.

5. Why lie? Voting Manipulation Our Approach Main Result Conclusions • 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? 21 3 11 10 10 4 3

6. Motivation Voting Manipulation Our Approach Main Result Conclusions • 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?

7. Avg. Case Analysis Voting Manipulation Our Approach Main Result Conclusions • Traditional average case complexity theory seems inappropriate for our purposes. • Distributional problem = <L,>; L is a decision problem,  is a distribution over the possible inputs. • Algorithm ALG is a heuristic polynomial time algorithm for <L,> if ALG runs in poly time, and p s.t. Prx[ALG(x)≠L(x)]≤1/p(|x|).

8. Junta Distributions Voting Manipulation Our Approach Main Result Conclusions • 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. Zero prob. Low prob. High prob. Easy instance Hard instance

9. Susceptibility to manipulation Voting Manipulation Our Approach Main Result Conclusions • A mechanism is susceptible to a manipulation M if there exists a junta dist. , s.t. there exists a heuristic poly time algorithm for <M, >. • Theorem: Let P be a sensitive scoring protocol. Then P is susceptible to coalitional manipulation when m=O(1).

10. A junta distribution Voting Manipulation Our Approach Main Result Conclusions • 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, 1W]. • * is a junta distribution. • * is intuitively appealing.

11. A heuristic poly time alg Voting Manipulation Our Approach Main Result Conclusions • 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 a b p 4 4 3.9 3 3.1 3.1 3 2.9 2.6 2.1 2 2.1 2.1 2 1.6 1 1 1 0 T 0.5 0.5 1 T 0.5 1

12. Remarks and future research Voting Manipulation Our Approach Main Result Conclusions • 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?