When Are Elections with Few Candidates Hard to Manipulate V. Conitzer , T. Sandholm , and J. Lang

1. When Are Elections with Few Candidates Hard to ManipulateV. Conitzer, T. Sandholm, and J. Lang Subhash Arja CS 286r October 29, 2008

2. Motivation • Avoid coalitional manipulation by a group of weighted voters in regular voting protocols. • Study constructive manipulation and destructive manipulation • Find the exact number of candidates that makes manipulation hard • Expand this result to manipulation by an individual in un-weighted and uncertain setting.

3. Outline • Prior Work • Background information • Voting protocols • Conditions • Proof of easiness for constant number of candidates • Proof of NP-hard through reduction of Partition problem • Extend to case of destructive manipulation and weighted voters • Conclusion and future work

4. Applications • Political elections • Surveys • Shareholder meetings • Allocation of goods and resources • Any application where the goal to achieve optimal preference aggregation.

5. Prior Work • Number of candidates and voters was unbounded • Method 1: Voters’ preferences are restricted • Problem: protocol designer cannot guarantee that the agents’ preferences fall within restrictions • Method 2: randomization approach • Problem: too much noise, could introduce manipulation possibilities • Method 3: make it hard to manipulate so agents will be unlikely to succeed.

6. Voting Protocols • Positional scoring protocols • α = (α1, α2,…, αm) where α1 ≥ α2 ≥ … ≥ αm • For each voter, a candidate receives α1 if it is ranked first by the voter, etc. The score of a candidate is the total number of points he receives. • Examples: Borda, plurality, veto • Maximin • For two distinct candidates, i and j, N(i,j) = number of voters who prefer i to j. The score of i is s(i) = minj≠iN(i,j) • Copeland • For two distinct candidates, i and j, let C(i,j) = 1 if N(i,j) > N(j,i), C(i,j) = 0 if N(i,j) = N(j,i), and C(i,j) = -1 if N(i,j) < N(j,i) • Score of candidate: s(i) = ∑j≠iC(i,j)

7. Voting Protocols • Single Transferable Vote (STV) • Series of m-1 rounds. • In each round, candidate with the lowest number of voters ranking it first among the remaining candidates is eliminated. • Votes for that candidate transfer to the next remaining candidate. • Plurality with Run-off • All candidates except two with the highest plurality scores. • Scores are transferred to the winners and second round determines the winner.

8. Voting Protocols • Cup • Balanced binary tree with each leaf representing each candidate. • Each non-leaf node is assigned the winner of its children. • In randomized version, assignment of candidates to leaves is chosen at random after voters have voted.

9. Conditions • Complete information – hardness results directly imply hardness for the incomplete information setting • Coalitional Manipulation – individuals have a small effect on the outcome • Weighted voters – the case of un-weighted voters is easy. • Constructive and Destructive

10. Easiness Results • Plurality protocol: constructive manipulation can be solved in polynomial time (any number of candidates) • Proof: manipulators check if p will win if they vote for p. Otherwise, they cannot make p win. • Cup protocol: constructive manipulation can be solved in polynomial time for any number of candidates • Proof: key claim is that a candidate can win a sub-election iff it can win one of its children and beat the potential winners of the sibling child. • Coalition ranks all candidates in p’s half above those in h’s half. h = potential winner of other half. • When p and h win their halves, p will defeat h in the final.

11. Easiness Results • Copeland protocol: easy if there are 3 candidates and all manipulators vote identically • Proof: involves four cases • Weights of manipulators’ votes are greater than weights for nonmanipulators votes for both candidates other than p. • Any configuration of votes where p ranks first wins the election. • Weights of manipulators’ votes are equal to that of nonmanipulators for one candidate and greater for the other candidate (K > Ds(a,p) and K = Ds(b,p) ) • It is proven that all manipulators must vote in the order (p,a,b)

12. Easiness Results • Same case as above but reverse “a” and “b” • Weights of manipulators’ votes are less than weights of nonmanipulators’ votes. (K < Ds(a,p) and K < Ds(b,p)) • p cannot be guaranteed to win, therefore there is no successful manipulation • Maximin protocol: easy with 3 candidates and all manipulators vote identically. • Proof: all manipulators set p as rank 1 and two cases involve the ranking of other two candidates. • When all of the coalition votes same way for other two candidates with p as rank 1, p will win. • Randomized cup protocol: easy if there are six candidates and all manipulators vote identically. • Proof: divide candidates into two sets – B = candidates that defeat p and G = candidates that p defeats

13. Easiness Results • Have to make sure that p doesn’t face an opponent from set B. • Rest of the proof goes over how the manipulators should choose the order of candidates within B and within G.

14. Hardness Results • Basic idea: proved P results for protocols with l candidates and now prove each is NP-complete for l+1 candidates. • Partition problem: given a set of S of integers, determine two disjoint subsets S1 and S2 where sum(S1) = sum(S2) • NP-complete problem. • Use reduction from the Partition problem to show that a protocol is NP-complete. • All the proofs use this idea with variations. • S = non-manipulators’ votes, T = weights of manipulators’ votes, and K = total weight in T.

15. Hardness Results • Any positional scoring rule other than the plurality protocol is NP-complete for 3 candidates • Proof: one half of the partition in T are (p,a,b) and the other half is (p,b,a) => this makes p the winner • This only happens in the case where the total weight of the voters voting (p,a,b) equals the total weight of the manipulators voting (p,b,a). • Copeland Protocol: manipulation is NP-complete for 4 candidates. • Proof: p wins if two other candidates tie, and the third loses. • This only happens if the combined weight of the manipulators’ votes maintain this tie => requires a partition.

16. Hardness Results • Maximin protocol: manipulation is NP-complete for 4 candidates • STV protocol: manipulation is NP-complete for 3 candidates • Plurality with Runoff: NP-complete for 3 candidates

17. Destructive Manipulation • Destructive manipulation can never be harder than constructive manipulation. • Can be done in polynomial time for veto, Borda, Copeland, and maximin protocols • Proof: each colluder places candidate h at the bottom and order the other candidates in any order. • A total of m-1 winner determinations are done to and each winner determination is in P.

18. Destructive Manipulation • STV Protocol: with 3 candidates, manipulation is NP-complete • Proof: reduce partition to case where three candidates are a, b, and h. • Show that in T for every ki there is a vote of weight 2ki • Plurality with runoff: with 3 candidates, manipulation is NP-complete • Proof: coincides with the STV protocol for 3 candidates.

19. Uncertainty about others’ votes • Only the distribution over the other voters is known • Restricted probability distributions. • Overall conclusions: • With weighted voters, whenever coalitional manipulation is hard, evaluating a candidate’s probability to win is hard when there is uncertainty. • Individual manipulation is also hard • An individual cannot find the strategically optimal vote for him to make.

20. Uncertainty about others’ votes • Approval protocol: each candidate either approves or disapproves of a candidate • Easy for constructive, non-weighted case. • In weighted case, manipulation is NP-hard

21. Un-weighted Voters • Special case of weighted voting where each vote is assigned the same weight. • General conclusion: • For every protocol that is hard in the weighted case, it is also hard in the un-weighted case.

22. Conclusions Constructive CW-Manipulation Adopted from V. Conitzer, et al.

23. Conclusions Destructive CW-Manipulation Adopted from V. Conitzer, et al.

24. Future Work • Ideal case: make all or most instances hard to manipulate. • Prove hardness of protocols that are more restricted (e.g. auctions) • “Can manipulation be made hard for most instances?” • It is too much to ask for every instance hard to manipulate. • Combine some amount of randomization with computational complexity. • Pivotal voters will not benefit or lose from the chosen candidate. • Pivotal voters could be “banished”. • Achieve a middle ground between making voting truthfully a dominant strategy and altering the definition of the voting rule.

25. Future Work • Make the voting rule itself hard to execute. • Simulations become complex and manipulation is thwarted. • Disadvantage: determining the election winner also becomes difficult.