Strategic Behavior in Multi-Winner Elections A follow-up on previous work by Ariel Procaccia, Aviv Zohar and Jeffrey S. Rosenschein Reshef Meir The School of Engineering and Computer Science Hebrew University, Jerusalem, Israel

Lecture content definitions previous work problems left open by previous work more definitions 3½ theorems and proofs results and summary

Voting A way of aggregating the preferences of multiple agents regarding a set of possible alternatives C={c 1,…,c m } Each agent is assumed to have a ranking of the alternatives (or perhaps even a utility function) A voting protocol / scheme takes these preferences and selects one of the alternatives Procaccia et al. ‘07

Single Winner Elections Only one candidate is elected Many well-known election mechanisms that essentially boil down to plurality/majority –The candidate with the most votes gets elected –The difference between schemes is usually in how votes are distributed, and in the number of election rounds Procaccia et al. ‘07

Example: Single non-Transferable Vote Each voter votes for a single specific candidate Votes are counted and the candidate with the most votes is elected Procaccia et al. ‘07

More voting schemes BLOC - the voter selects L preferred candidates, and gives 1 point to each Approval - the voter approves/disapproves each candidate independently Cumulative - the voter distributes a ”pool” of L points between the candidates

Multi-Winner Elections The elected assembly is of size k This means that if we have m candidates, there are possible assemblies Procaccia et al. ‘07

Manipulation A voter is said to manipulate an election if it votes strategically. It misreports its preferences to achieve a better result. In single-winner elections, the manipulator simply attempts to change the winner In multi-winner elections there can be several forms of manipulation: –Can a voter make a specific candidate win? –Can a voter make some candidate lose? –Can a voter make a set of candidates win? –…–… Procaccia et al. ‘07

Manipulation We use a general formulation (single manipulator): Given –a set of candidates C –a set of voters V that have already voted –and a utility function of the manipulator u:C  Z we ask whether the manipulator can, by some vote, get some group of winners with value of at least t Procaccia et al. ‘07

Example of Manipulation (SNTV) 1000 votes 900 votes U(Kermit)=50 U(Gonzo)=20 U(Miss Piggy)=70 Elmo prefers Miss Piggy, but is better off voting for Kermit! Winners Procaccia et al. ‘07

The Gibbard-Satterthwaite Impossibility Result One of several impossibility results regarding voting No voting rule with 3 or more alternatives in which voters give a full ranking over alternatives has all 3 properties: 1.Every alternative can be selected (with some preference profiles of the voters) 2.Non-dictatorial 3.No voter can manipulate the vote Procaccia et al. ‘07

The Complexity of Manipulation Alternative One way to address the problem of manipulability of reasonable voting schemes: find schemes that are computationally hard to manipulate What we would like to see are proofs that voting is always or usually hard to manipulate We present here analysis of the worst case of the problem –Not enough to convince of hardness of manipulation in all cases, but it gives some bounds Procaccia et al. ‘07

Control (add voters) This is another form of manipulation This time not by voters, but by someone that can register new voters (a Chairman) Given: –Candidates C –A set V of registered voters (with their known votes) –A set V' of unregistered voters (also with known preferences) –A utility function of the manipulator u:C  Z choose a subset of voters V'' from V' of size at most r that will be registered to vote, and will get a better result Procaccia et al. ‘07

Results from Procaccia et al.

Other problems Other voting rules –scoring rules Other types of manipulation Other types of control

Other types of Manipulation Can a voter: Force a distinguished candidate to win? Force a distinguished candidate to loose? Force a subset of candidates to win/lose? All these are special cases of the general case with boolean utility function

Other types of Control Control by removing Voters –The chairman is allowed to remove at most r voters from V Control by adding / removing Candidates –The chairman is allowed to remove at most r candidates from C, or add at most r from C’ –We assume the chairman knows the voters’ preferences over ALL candidates

Results so far ???

Theorem 1 Manipulation under Cumulative voting with boolean utility function is in P.

Observations After voting, each candidate has a final score There is some “threshold”, such that all (and only) candidates with score above the threshold are elected The threshold is dynamic, and may change as the Manipulator distributes his score

Algorithm Sort all candidates with decreasing score D  all candidates with utility 1 Find j* = max( j : |{1…j-1}  D| + k + 1- j  t, and j not in D) threshold  score(j*) Return TRUE iff :

Proof idea In order to be included in the winner set, there is some “bad” candidate j*, that all t “good” candidates have to win Since all “good” candidates have the utility 1, the manipulator should focus on giving his score to those that have the best chances We check if it is possible to bring the t good candidates with highest score, to pass the threshold (i.e. win j*)

Illustration j* = 5 k = 6 t = 3 Manipulation possible iff L 

Theorem 2 Control by adding candidates under Approval is in P.

Observations Each candidate c in C,C’ has a fixed number of votes: score(c) It is only effective to add candidates that will actually be winners If we can solve for adding exactly r candidates, then we can do it with ≤ r

Algorithm Sort C and C’ by decreasing score threshold  score(c k+1-r ) Return the best r candidates from C’, whose score is above threshold

“proof” by illustration k=8 20 =Thresh(3) r = 3

Theorem 3 Control by removing candidates under SNTV is in NPc, even in a single winner setting.

Min Set-Cover Definition: in the problem of Min Set- Cover, we are given a set U = {1…m}, a family F of subsets of U, and integers s,r. we ask whether it is possible to choose (at least) s subsets from F, such that their union covers at most r points in U.

Min Set-Cover Reduction from CLIQUE –Each vertex is an element –Each edge is a set –Can we find sets, which cover no more than r vertices? Min Set-Cover is NP-hard

Proof I will show a reduction from Min Set-Cover The reduction lies on representing the Voting problem in a “convenient” way The exact details of the constructed instance appear, but are not as important

Proof idea Given an instance of Min Set-Cover: –Add a candidate for each element in U –Add a distinguished candidate c’ –Add a voter for each set in F: this is the set of candidates that this voter prefers over c’

Proof idea Each voter represents a set of candidates that are preferred over c’ Show that: c’ wins iff at least s voters vote for him r candidates that are preferred by s voters are equivalent to r elements covered by s sets. Denote by R the subset of r selected elements/candidates c’ gets 1 vote for each set from F contained in R

Proof idea Each voter represents a set of candidates that are preferred over c’ c’ gets 1 vote for each set from F contained in R Thus: –c’ wins iff at least s voters vote for him –iff we choose s subsets contained in r elements Control is possible iff there is Min Set-Cover

Illustration c’

Formal construction Let (U,F,t,s) an instance of MIN Set-Cover r := t C1 := U C2 := {c(1),…,c(r+1)} C := C1  C2  {c’}

Formal construction For each set f in F, add a voter v(f) to V1, which votes: f > c’ > C1\f > C2 For each candidate c in C2, add n+s-1 voters to V2, which vote: c > C2\{c} > others Add n voters to V3, all vote: c’ > others

Formal proof outline Claim I: c’ will win iff at least s voters from V1 will vote for him. Claim II: s voters from V1 will vote for c’ iff there are at least s subsets in F, that cover at most r elements in U. proof: each v(f) in V1 votes for c’ iff all candidates in f are removed.

Results Black – from Procaccia et al. Blue – from my paper

Summary Different voting schemes may be more resistant to some strategic behaviors, but more prone to others Each problem can be represented in different ways. We need to find “convenient” representations for our purposes