Junta Distributions and the Average-Case Complexity of Manipulating Elections A presentation by Jeremy Clark Ariel D. Procaccia Jeffrey S. Rosenschein

Outline Introduction Manipulability Design Goals Paper Theorems Preliminaries Junta Distribution Proof of Theorems Concluding Remarks Jeremy Clark2

Introduction This paper considers the computational complexity of manipulating an election outcome A manipulatable election is one where the addition of a set number of votes will change the election outcome to a preferred outcome Jeremy Clark3

Manipulability The ability to manipulate an election depends on the current results (whether exactly known or not) and the weight of the votes at the manipulator’s disposal Given these, we can form a decisional problem Jeremy Clark4

5 Manipulation can be constructive or destructive Constructive: make a candidate win Destructive: make a candidate lose Constructive is equivalent to multiple destructive manipulations: one for each candidate ahead of your preferred candidate

In real elections Strategic voting (destructive) You are a Liberal and a federalist in a Quebec riding. Current polls have the Bloc in first, Conservatives in second, and the Liberals trailing far behind. A manipulative vote: vote Conservative to prevent the Bloc from winning Jeremy Clark6

In real (US) elections Gerrymandering (Constructive) You are a Democrat in charge of election zoning. The Republicans beat you marginally in two neighbouring districts. You restructure the districts by packing Democratic voters in one of the regions. Jeremy Clark7

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Goal Design a voting system such that manipulability is impossible Jeremy Clark9

Goal Design a voting system such that manipulability is impossible Gibbard-Satterthwaite Theorem: Any deterministic, non-dictatorial voting system contain manipulatable instances Jeremy Clark10

Goal Design a voting system such that manipulability is intractable Jeremy Clark11

Goal Design a voting system such that manipulability is intractable Lots of interesting systems where manipulability is NP-Hard However is worst-time complexity the right metric? Jeremy Clark12

Goal Design a voting system such that manipulability is average-case intractable Jeremy Clark13

Goal Design a voting system such that manipulability is average-case intractable This paper examines average-case complexity on manipulation problems It proves that general classes of NP-hard manipulation problems are polynomial in the average-case Jeremy Clark14

Outline Introduction Manipulability Design Goals Paper Theorems Preliminaries Junta Distribution Proof of Theorems Concluding Remarks Jeremy Clark15

Preliminaries Election has m candidates Election has n+N voters: n manipulatable voters and N non-manipulatable voters Voters can have different weights (reduces to a voter having multiple votes) Jeremy Clark16

Preliminaries A vote is an ordered list of candidates that gives  i points to the i th candidate. A scoring protocol,  =, is a vector of scores for each position where  i ≥  i+1. 1.Plurality: 2.Veto: 3.Borda: Jeremy Clark17

Preliminaries A voting protocol uses multiple contests, each decided with a scoring protocol For example, Exhaustive Ballot is an iterated plurality protocol where a candidate with over 50% of the vote wins. If no candidate wins, then the last place candidate is eliminated and the election is rerun. Others include Copeland, Maximin, and STV Jeremy Clark18

Sensitive Scoring Protocol In sensitive scoring protocols,  m =0 and  m-1 >  m → Jeremy Clark19

Manipulation Problems Individual Manipulation (IM): Given knowledge of all other votes, can I cast my vote for my preferred candidate such that she wins? Note: ties are considered losses P-Time in most scoring protocols (can be hard in voting protocols with unbounded candidates) Jeremy Clark20

Manipulation Problems Coalitional-Weighted-Manipulations (CWM): Given knowledge of all other votes, can I cast a set of votes for my preferred candidate such that she wins? NP-Hard in sensitive scoring protocols with just 3 candidates. Why? You are increasing the score of more than one candidate. Jeremy Clark21

Manipulation Problems Score-CWM (SCWM): Given the tally of all other candidates, can I cast a set of votes for my preferred candidate such that she wins? Assumptions: Weights are linear in precision Output is a linear (decisional) Score determination is linear/P-time Jeremy Clark22

Junta Distribution Hardness: instances are full-sized and hard Balance: both yes and no instances exist Dichotomy: instances can be impossible or have non-negligible probability. Ignore negligible cases Jeremy Clark23

Junta Distribution Symmetry: instance is unbiased toward any candidate Refinement: Manipulation fails if all manipulative votes are identical Jeremy Clark24

Theorem Let P be a sensitive scoring protocol. If m=O(1) then P, with candidates C={p,c 1,c 2,…,c m-1 }, is susceptible to SCWM. Jeremy Clark25

Theorem Let P be a sensitive scoring protocol. If m=O(1) then P, with candidates C={p,c 1,c 2,…,c m-1 }, is susceptible to SCWM.  m-1 >  m =0 such as Borda but not Plurality Jeremy Clark26

Theorem Let P be a sensitive scoring protocol. If m=O(1) then P, with candidates C={p,c 1,c 2,…,c m-1 }, is susceptible to SCWM. Fixed number of candidates Jeremy Clark27

Theorem Let P be a sensitive scoring protocol. If m=O(1) then P, with candidates C={p,c 1,c 2,…,c m-1 }, is susceptible to SCWM. p is candidate to manipulate, c i are others Jeremy Clark28

Theorem Let P be a sensitive scoring protocol. If m=O(1) then P, with candidates C={p,c 1,c 2,…,c m-1 }, is susceptible to SCWM. There exists a heuristic polynomial time algorithm A to solve decisional problem M with a junta distribution  over set of inputs to M Jeremy Clark29

Proposition 1 Let P be a sensitive scoring protocol. Then CWM in P is NP-Hard (with m  3) Sketch of proof: CWM  P Partition Jeremy Clark30

Proposition 1 Partition: given a set of integers that sum to 2K, does there exist a subset that sums to K? Let m=3. Set n~2K. Structure N such that CWM is true iff exactly K vote p>a>b and K vote p>b>a. If, say, K+1 vote p>a>b and K-1 vote p>b>a, then CWM is false. Jeremy Clark31

Corollary Let P be a sensitive scoring protocol. Then SCWM in P is NP-Hard (with m  3) Sketch: If CWM is NP-Hard, then SCWM is as well as partitioning does not depend on generating tally from votes Jeremy Clark32

Proposition 2 Let P be a sensitive scoring protocol. Then  * is a junta distribution for SCWM in P with C={p,c 1,c 2,…,c m-1 } and m=O(1). Where  * is the following distribution: 1.Independently randomly choose w(v) from [0,1] (with discrete precision). 2.Independently randomly choose S[c i ] from [W,(m-1)W]. Jeremy Clark33

Is this Junta? Hard? Yes Balance? Authors calculate bounds using Chernoff’s bounds Dichotomy? First discrete step is non-negligible Symmetry? Invariant to candidates Refinement? 2nd ranked candidate will at least tie p Jeremy Clark34

Greedy Algorithm 1.Sort candidates from lowest score to highest 2.Choose p as first choice, and rest in sorted order 3.Recalculate scores and repeat for each vote 4.When finished, return true iff p has highest score Jeremy Clark35

Example Borda:, n=5 S[Con]= 20 S[Lib]= 19 S[NDP]= 17 S[Gre]= 10  p Jeremy Clark36

Example S[Con]= 20 S[Lib]= 19 S[NDP]= 17 S[Gre]= 10 t 1 : Gre<NDP<Lib<Con Jeremy Clark37

Example S[Con]= = 20 S[Lib]= = 20 S[NDP]= = 18 S[Gre]= = 13 t 1 : Gre<NDP<Lib<Con Jeremy Clark38

Example S[Con]= 20 S[Lib]= 20 S[NDP]= 18 S[Gre]= 13 Jeremy Clark39

Example S[Con]= 20, 20, 20, 21, 23, 23 S[Lib]= 19, 20, 21, 21, 22, 24 S[NDP]= 17, 18, 20, 22, 22, 23 S[Gre]= 10, 13, 16, 19, 22, 25 t 1 : Gre<NDP<Lib<Con t 2 : Gre<NDP<Lib<Con t 3 : Gre<NDP<Con<Lib t 4 : Gre<Con<Lib<NDP t 5 : Gre<Lib<NDP<Con Jeremy Clark40

Greedy Properties Greedy is P-time Greedy never issues false positives Greedy does issue false negatives, however these are bounded to Pr[err]  1/p(n) Therefore Greedy is deterministic heuristic polynomial time Jeremy Clark41

Theorem Let P be a sensitive scoring protocol. If m=O(1) then P, with candidates C={p,c 1,c 2,…,c m-1 }, is susceptible to SCWM. There exists a heuristic polynomial time algorithm A to solve decisional problem M with a junta distribution  over set of inputs to M  Jeremy Clark42

Theorem 2 The paper contains a second theorem, related to the first, regarding uncertainty about the other votes We are allowed to sample the distribution of the other votes Essentially, we try every (m+1)! orders of candidates and sample the distribution Jeremy Clark43

Outline Introduction Manipulability Design Goals Paper Theorems Preliminaries Junta Distribution Proof of Theorems Concluding Remarks Jeremy Clark44

Conclusions Complexity is best considered in the average-case, not worst-case Manipulation problems have been demonstrated to be worst-case intractable and average-case tractable This is bad news if it generalizes to any NP-Hard manipulation problem Jeremy Clark45

There is still hope These results are for scoring protocols. Voting protocols may offer intractable manipulation. Large number of candidates may increase average case complexity (intuitively seems the case with Theorem 2: (m+1)! grows very fast) Junta distributions may be too permissible to easy instances Jeremy Clark46

Questions? Jeremy Clark47

Discussion What if we make manipulability as easy as possible and let voters adapt to voting strategically? What happens with (non-sensitive) cardinal voting schemes instead of ordinal ones, such as range voting? Jeremy Clark48