1 / 48

Junta Distributions and the Average-Case Complexity of Manipulating Elections

Ariel D. Procaccia Jeffrey S. Rosenschein. Junta Distributions and the Average-Case Complexity of Manipulating Elections. A presentation by Jeremy Clark. Outline. Introduction Manipulability Design Goals Paper Theorems Preliminaries Junta Distribution Proof of Theorems

naida
Download Presentation

Junta Distributions and the Average-Case Complexity of Manipulating Elections

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

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

  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

  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

  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

  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.

  8. Goal Design a voting system such that manipulability is impossible

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

  10. Goal Design a voting system such that manipulability is intractable

  11. 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?

  12. Goal Design a voting system such that manipulability is average-case intractable

  13. 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

  14. Outline Introduction • Manipulability • Design Goals Paper Theorems • Preliminaries • Junta Distribution • Proof of Theorems Concluding Remarks

  15. 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)

  16. Preliminaries A vote is an ordered list of candidates that gives i points to the ith candidate. A scoring protocol,  = <1, …, m>, is a vector of scores for each position where i≥ i+1. • Plurality: <1, 0, … , 0, 0> • Veto: <1, 1, … , 1, 0> • Borda: <m-1, m-2, … , 2, 1, 0>

  17. 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

  18. Sensitive Scoring Protocol In sensitive scoring protocols, m=0 and m-1 > m <3,2,1,0> <1,0,0,0> <3,3,3,3> → <0,0,0,0> <4,3,2,1> →<3,2,1,0>

  19. 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)

  20. 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.

  21. 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

  22. 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

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

  24. Theorem Let P be a sensitive scoring protocol. If m=O(1) then P, with candidates C={p,c1,c2,…,cm-1}, is susceptible to SCWM. Jeremy Clark 25

  25. Theorem Let P be a sensitive scoring protocol. If m=O(1) then P, with candidates C={p,c1,c2,…,cm-1}, is susceptible to SCWM. m-1>m=0 such as Borda but not Plurality Jeremy Clark 26

  26. Theorem Let P be a sensitive scoring protocol. If m=O(1) then P, with candidates C={p,c1,c2,…,cm-1}, is susceptible to SCWM. Fixed number of candidates Jeremy Clark 27

  27. Theorem Let P be a sensitive scoring protocol. If m=O(1) then P, with candidates C={p,c1,c2,…,cm-1}, is susceptible to SCWM. p is candidate to manipulate, ci are others Jeremy Clark 28

  28. Theorem Let P be a sensitive scoring protocol. If m=O(1) then P, with candidates C={p,c1,c2,…,cm-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 Clark 29

  29. 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 Clark 30

  30. 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 Clark 31

  31. 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 Clark 32

  32. Proposition 2 Let P be a sensitive scoring protocol. Then *is a junta distribution for SCWM in P with C={p,c1,c2,…,cm-1} and m=O(1). Where * is the following distribution: Independently randomly choose w(v) from [0,1] (with discrete precision). Independently randomly choose S[ci] from [W,(m-1)W]. Jeremy Clark 33

  33. 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 Clark 34

  34. Greedy Algorithm Sort candidates from lowest score to highest Choose p as first choice, and rest in sorted order Recalculate scores and repeat for each vote When finished, return true iff p has highest score Jeremy Clark 35

  35. Example Borda: <3,2,1,0>, n=5 S[Con] = 20 S[Lib] = 19 S[NDP] = 17 S[Gre] = 10  p Jeremy Clark 36

  36. Example S[Con] = 20 S[Lib] = 19 S[NDP] = 17 S[Gre] = 10 t1 : Gre<NDP<Lib<Con Jeremy Clark 37

  37. Example S[Con] = 20 + 0 = 20 S[Lib] = 19 + 1 = 20 S[NDP] = 17 + 2 = 18 S[Gre] = 10 + 3 = 13 t1 : Gre<NDP<Lib<Con Jeremy Clark 38

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

  39. 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 t1 : Gre<NDP<Lib<Con t2 : Gre<NDP<Lib<Con t3 : Gre<NDP<Con<Lib t4 : Gre<Con<Lib<NDP t5 : Gre<Lib<NDP<Con Jeremy Clark 40

  40. 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 Clark 41

  41. Theorem Let P be a sensitive scoring protocol. If m=O(1) then P, with candidates C={p,c1,c2,…,cm-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 Clark 42

  42. 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 Clark 43

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

  44. 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

  45. 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 Clark 46

  46. Questions?

  47. 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 Clark 48

More Related