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

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

3. 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

4. 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. 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

6. 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

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

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

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

12. 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

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

14. 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

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

16. 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

17. 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

18. 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

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

20. 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

21. 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

22. 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

23. 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

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

25. 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

26. 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

27. 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

28. 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

29. 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

30. 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

31. 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

32. 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

33. 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

34. 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

35. 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

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

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

38. Example S[Con]= 20 + 0 = 20 S[Lib]= 19 + 1 = 20 S[NDP]= 17 + 2 = 18 S[Gre]= 10 + 3 = 13 t 1 : Gre<NDP<Lib<Con Jeremy Clark38

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

40. 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

41. 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

42. 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

43. 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

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

45. 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

46. 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

47. Questions? Jeremy Clark47

48. 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