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Exploring and Exploiting Clones in Elections

Exploring and Exploiting Clones in Elections. Edith Elkind Nanyang Technological University , Singapore. Piotr Faliszewski AGH Univeristy of Science and Technology, Poland. Arkadii Slinko University of Auckland New Zealand. Elections for the Scariest Monster!.

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Exploring and Exploiting Clones in Elections

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  1. Exploring and Exploiting Clones in Elections EdithElkind NanyangTechnologicalUniversity, Singapore Piotr Faliszewski AGH Univeristy of Science and Technology, Poland Arkadii SlinkoUniversity of AucklandNew Zealand

  2. Elections for the Scariest Monster! C = { , , , } Bordavoting

  3. Elections for the Scariest Monster!  18  17 R1: R2: R3:

  4. Elections for the Scariest Monster!  18  17 R1: R2: R3:

  5. Elections for the Scariest Monster!  7  6  18  17 R1: Not soeasy!Whichcandidatesto colapse? R2: R3:

  6. Elections for the Scariest Monster! R1: R2: R3:

  7. Problem 1: Discovering the True Elections We start with anelection with possibleclones: For each set of clones we havesomelikelihoodthatexactlythis set resulted from cloning We seek a clone coverthathashighestlikelihood  0.01  0.8  0.5  0.3 set  1  1  1 Allothersubsetshavelikelihood 0.

  8. Problem 1: Discovering the True Elections  0.01  0.8  1  0.5  0.3  1  1  1  0.3 Allothersubsetshavelikelihood 0.  0.5  0.8  0.8  0.8

  9. Problem 2: Decloning to Become a Winner We start with anelection with possibleclones, and with a preferredcandidate: For each set of clones we havesomelikelihoodthatexactlythis set resulted from cloning We seek a clone coverthatensuresthatourguywins.  0.01  0.8  0.5  0.3  1  1  1 Allothersubsetshavelikelihood 0.

  10. Problem 2: Decloning to Become a Winner • Whatis the complexity of the decloning problem? • Whatother problem isitlike? • Votingrules? • Plurality • K-approval • Veto • Maximin • Borda • Copeland Control by deletingcandidates we areallowed to deletesomeof the clones Problem 1: Discovering thetrueelection! Independnce of irrelevantclones: The score of a candidateremainsconstantirrespectivehowothercandidatesareclones

  11. Problem 2: Decloning to Become a Winner Independnce of irrelevantclones: The score of a candidateremainsconstantirrespectivehowothercandidatesareclones Algorithm for rulessatisfying IIC: • Letp be yourpreferredcandidate • For each clone set includingp: • Decloneit • Removeall clone setsthatintersectit • Computeitsscore (afterdecloning) • For eachother clone set thatdoes not intersect • Computeitsscore (afterdecloning) • Ifhigherthanscore of p’s clone set thenremove from possible clone sets • Compute the best clone cover with given clone sets

  12. Problem 2: Decloning to Become a Winner Independnce of irrelevantclones: The score of a candidateremainsconstantirrespectivehowothercandidatesareclones Thereom. For everyvotingrulethatsatisfies IIC, the problem of decloning to become a winneris in P. Corollary. The problem of decloning to become a winneris in P for Plurality, Veto, and Maximin.

  13. Problem 2: Decloning to Become a Winner • Whatis the complexity of the decloning problem? • Whatother problem isitlike? • Votingrules? • Plurality • K-approval • Veto • Maximin • Borda • Copeland Control by deletingcandidates we areallowed to deletesomeof the clones Problem 1: Discovering thetrueelection!

  14. Problem 2: Decloning to Become a Winner • Whatis the complexity of the decloning problem? • Whatother problem isitlike? • Votingrules? • Plurality • K-approval • Veto • Maximin • Borda • Copeland Control by deletingcandidates we areallowed to deletesomeof the clones NP-completenessproofsendedupbeingquitesimple… after we understood clone structures

  15. Problem 3: What Clone StructuresCanArise in Elections? R1: R2: R3: C(R1, R2, R3) = {{ }, { }, { }, { }, { } { }, { }, { }, { }, { }, { },{ },{ },{ }}

  16. Problem 3: What Clone StructuresCanArise in Elections? C(R1, R2, R3) = {{ }, { }, { }, { }, { } { }, { }, { }, { }, { }, { },{ },{ },{ }} Question 1:Isthere a profile thatimplementsthis clone structure? Question 2:Whatproperties do clone structureshave? Question 3: How to represent clone structures? Question 4: How manyvoters do youneed for a given clone structure? We provideanaxiomaticcharacterizationof possible clone structures.

  17. AxiomaticCharacterization A – alternative set F – a family of A subsets F is a clone structureif and onlyif: A1 {a} ∈ F for each a ∈ A A2 ∅ ∉ F, A ∈ F

  18. AxiomaticCharacterization A – alternative set F – a family of A subsets F is a clone structureif and onlyif: A1 {a} ∈ F for each a ∈ A A2 ∅ ∉ F, A ∈ F A3If C1 and C2are in F and C1 ⋂ C2 ≠∅ thenC1 ⋂ C2 and C1 ⋃ C2are in F

  19. AxiomaticCharacterization C1 ⋈ C2: C1 ⋂ C2 ≠∅ and C1 - C2 ≠∅, C2 - C1≠∅ A – alternative set F – a family of A subsets F is a clone structureif and onlyif: A1 {a} ∈ F for each a ∈ A A2 ∅ ∉ F, A ∈ F A3If C1 and C2are in F and C1 ⋂ C2 ≠∅ thenC1 ⋂ C2 and C1 ⋃ C2are in F A4If C1 and C2are in F and C1 ⋈ C2thenC1 - C2 and C2 - C1are in F

  20. AxiomaticCharacterization A – alternative set F – a family of A subsets F is a clone structureif and onlyif: A1 {a} ∈ F for each a ∈ A A2 ∅ ∉ F, A ∈ F A3If C1 and C2are in F and C1 ⋂ C2 ≠∅ thenC1 ⋂ C2 and C1 ⋃ C2are in F A4If C1 and C2are in F and C1 ⋈ C2thenC1 - C2 and C2 - C1are in F A5Eachmember of F hasat most twominimalsupersets in F.

  21. AxiomaticCharacterization A – alternative set F – a family of A subsets F is a clone structureif and onlyif: A1 {a} ∈ F for each a ∈ A A2 ∅ ∉ F, A ∈ F A3If C1 and C2are in F and C1 ⋂ C2 ≠∅ thenC1 ⋂ C2 and C1 ⋃ C2are in F A4If C1 and C2are in F and C1 ⋈ C2thenC1 - C2 and C2 - C1are in F A5Eachmember of F hasat most twominimalsupersets in F. A6 F is „acyclic”

  22. Proof Idea for the Characterization • Thereareonlytwobasictypes of clone structures • Both satisfyouraxioms, bothcompose induction (a) a string of sausages (b) a fatsausage

  23. Clone StructureRepresentations • How to convenientlyrepresent the aboveclone structure?

  24. Clone StructureRepresentations X X = { , , , , , , , , } X

  25. Clone StructureRepresentations Y Z X = { , , , , , , , , } Y = { , , , }, Z = { , , } X Y Z

  26. Clone StructureRepresentations Y X = { , , , , , , , , } Y = { , , , }, Z = { , , } X Y Z

  27. Clone StructureRepresentations U X = { , , , , , , , , } Y = { , , , }, Z = { , , } U = { , } X Y Z U

  28. Clone StructureRepresentations X = { , , , , , , , , } Y = { , , , }, Z = { , , } U = { , } X Y Z U

  29. Problem 3: What Clone StructuresCanArise in Elections? C(R1, R2, R3) = {{ }, { }, { }, { }, { } { }, { }, { }, { }, { }, { },{ },{ },{ }} Question 1:Isthere a profile thatimplementsthis clone structure? Question 2:Whatproperties do clone structureshave? Question 3: How to represent clone structures? Question 4: How manyvoters do youneed for a given clone structure?

  30. How Many VotersNeeded to Represent a Clone Structure? Strings of sausages Fatsausages a b c d a b c d a b c a > b > c > d a > b > c > d c > a > d > b a > b > c a > c > b b > a > c A single votersuffices Twovoterssuffice … The onlyfatsausagethatneedsthreevoters!

  31. How Many VotersNeeded to Represent a Clone Structure? X Y X with Y in place of b a 1 2 3 4 c a b c 12 3 4 a > b > c b > a > c 1 > 2 > 3 > 4 4 > 2 > 3 > 1 a > 1 > 2 > 3 > 4 > c 4 > 2 > 3 > 1 > a > c

  32. How Many VotersNeeded to Represent a Clone Structure? X Y X with Y in place of b a 1 2 3 4 c a b c 12 3 4 a > b > c b > a > c 1 > 2 > 3 > 4 4 > 2 > 3 > 1 a > 1 > 2 > 3 > 4 > c 4 > 2 > 3 > 1 > a > c 1 > 3 > 2 > 4 > a > c

  33. How Many VotersNeeded to Represent a Clone Structure? X Y X with Y in place of b a 1 2 3 4 c a b c 12 3 4 a > b > c b > a > c 1 > 2 > 3 > 4 4 > 2 > 3 > 1 a > 1 > 2 > 3 > 4 > c 4 > 2 > 3 > 1 > a > c 1 > 3 > 2 > 4 > a > c Theorem. For every clone structure F overalternative set A, therearethreeorders R1, R2, R3thatjointlygenerate F.

  34. Problem 2: Decloning to Become a Winner • Whatis the complexity of the decloning problem? • Whatother problem isitlike? • Votingrules? • Plurality • K-approval • Veto • Maximin • Borda • Copeland Control by deletingcandidates we areallowed to deletesomeof the clones

  35. Problem 4: Decloning to DiscoverHiddenStructure We start with anelection; Perhaps the electionssatisfied: • Single-peakedness? • Single-crossingness? But clonesdestroyed the structure? Goal: Declone as little as possible to discover single-peakednessor single-crossingness.

  36. Clonesin Single-PeakedElections Single-peakednessmodelsvotes in naturalelections Def.Anelection (A,R) is single-peaked with respect to an order > if for all c, d, e in A suchthat c > d > e (or e > d > c) and allRiitholdsthat: c Ri d ⇒ c Ri e

  37. Clonesin Single-PeakedElections Single-peakednessmodelsvotes in naturalelections Def.Anelection (A,R) is single-peaked with respect to an order > if for all c, d, e in A suchthat c > d > e (or e > d > c) and allRiitholdsthat: c Ri d ⇒ c Ri e Profile losessingle-peakednessdue to cloning

  38. DecloningToward Single-Peakedness • Decloning a clone set in (A,R) • Operation of contracting a clone-set into a single candidate • We havea polynomial-timealgorithmthatfinds a decloning of a preference profile suchthat: • The profile becomes single-peaked • Maximum number of candidatesremain in the election

  39. DecloningToward Single-Peakedness • Decloning • Operation of contracting a clone-set into a single candidate • We have a polynomial-timealgorithmthatfinds a decloning of a preference profile suchthat: • The profile becomes single-peaked • Maximum number of candidatesremain in the election

  40. DecloningToward Single-Peakedness • Decloning • Operation of contracting a clone-set into a single candidate • We have a polynomial-timealgorithmthatfinds a decloning of a preference profile suchthat: • The profile becomes single-peaked • Maximum number of candidatesremain in the election

  41. DecloningToward Single-Peakedness • Decloning • Operation of contracting a clone-set into a single candidate • We have a polynomial-timealgorithmthatfinds a decloning of a preference profile suchthat: • The profile becomes single-peaked • Maximum number of candidatesremain in the election

  42. Characterizing Single-Peaked Clone Structures • It would be interesting to knowwhatclonesstructurescan be implemented by single-peakedprofiles • Not all clone structurescan be! • However, all clone structureswhosetreerepresentationcontains P-nodesonlycan be implemented • Work in progress!

  43. Clones in Single-CrossingElections Single-CrossingPreferences a > b > c > d > e b > a > c > d > e b > c > a > d > e c > b > a > e > d c > b > e > a > d

  44. Clones in Single-CrossingElections Single-CrossingPreferences a >b > c > d > e b > a > c > d > e b> c > a > d > e c > b > a > e > d c > b > e > a > d Every clone structurecan be implemented. Decloningtowardsingle-crossingpreferencesisNP-complete. Unless the order of votersisfixed; thenitis in P.

  45. Conclusions • Clone structures form aninterestingmathematicalobject • Clonescan be used in variousways to manipulateelections; understanding clone structureshelps in thisrespect. • Clonescanspoil single-peakedness of anelection; decloningtoward single-peakednesscan be a usefulpreprocessing step when holding anelection. ThankYou!

  46. COMSOC-2012 in Kraków, Poland

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