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

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.

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

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

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.

Voting Protocols Positional scoring protocols – α = (α 1, α 2,…, α m ) where α 1 ≥ α 2 ≥ … ≥ α m – For each voter, a candidate receives α 1 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) = min j≠i N(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≠i C(i,j)

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.

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.

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

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.

Easiness Results Copeland protocol: easy if there are 3 candidates and all manipulators vote identically – Proof: involves four cases 1.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. 2.Weights of manipulators’ votes are equal to that of nonmanipulators for one candidate and greater for the other candidate (K > D s (a,p) and K = D s (b,p) ) – It is proven that all manipulators must vote in the order (p,a,b)

Easiness Results 3.Same case as above but reverse “a” and “b” 4.Weights of manipulators’ votes are less than weights of nonmanipulators’ votes. (K < D s (a,p) and K < D s (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

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.

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 S 1 and S 2 where sum(S 1 ) = sum(S 2 ) – 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.

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.

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

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.

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 k i there is a vote of weight 2k i Plurality with runoff: with 3 candidates, manipulation is NP-complete – Proof: coincides with the STV protocol for 3 candidates.

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.

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

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.

Conclusions # of candidates234,5,6≥ 7 BordaPNP-complete VetoPNP-complete STVPNP-complete Plurality w/ runoffPNP-complete CopelandPPNP-complete MaximinPPNP-complete Randomized CupPPPNP-complete Regular cupPPPP pluralityPPPP Adopted from V. Conitzer, et al. Constructive CW-Manipulation

Conclusions Adopted from V. Conitzer, et al. Destructive CW-Manipulation Number of Candidates23 STVPNP-complete Plurality with runoffPNP-complete BordaPP VetoPP CopelandPP MaximinPP Regular cupPP PluralityPP

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.

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.