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Clone Structures in Voters ’ Preferences

Clone Structures in Voters ’ Preferences. Edith Elkind Nanyang Technological University , Singapore. Piotr Faliszewski AGH Univeristy of Science and Technology, Poland. Arkadii Slinko University of Auckland New Zealand. Elections and Clone Structures. Example

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Clone Structures in Voters ’ Preferences

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  1. Clone Structures in Voters’ Preferences EdithElkind NanyangTechnologicalUniversity, Singapore Piotr Faliszewski AGH Univeristy of Science and Technology, Poland Arkadii SlinkoUniversity of AucklandNew Zealand

  2. Elections and Clone Structures Example R1: b > c > d > e > f > a > g > h > i R2: e > f > d > c > i > g > h > b > a R3:b >a > c >d > e > f > g >i > h 8 7 6 5 4 3 2 1 0 a 10b  17 c 18 d 17 e 17 f 14 g 7 h 3 i  4 Def.Anelectionis a pair (A,R) where A is the set of alternatives and R = (R1, …, Rn) isvoters’ preference profile. EachRiis a totallinear order over A.

  3. Elections and Clone Structures Example R1: b > c > d > e > f > a > g > h > i R2: e > f > d > c > i > g > h > b > a R3:b >a > c >d > e > f > g >i > h 8 7 6 5 4 3 2 1 0 a 10b  17 c 18 d 17 e 17 f 14 g 7 h 3 i  4 Def.Let (A,R) be anelection. A subset C of A is a clone set ifmembers of C arerankedconsecutively in allorders. C(R) is the set of allclonessets for R.

  4. Elections and Clone Structures Example R1: b > c > d > e > f > a > g > h > i R2: e > f > d > c > i > g > h > b > a R3:b >a > c >d > e > f > g >i > h C(R) = { {c, d}, {e, f}, {d, e, f}, {c, d, e, f}, {g, h, i} , {a}, {b}, {c}, {d}, {e}, {f}, {g}, {h}, {i}, {a,b,c,d,e,f,g,h,i} } 8 7 6 5 4 3 2 1 0 a 10b  17 c 18 d 17 e 17 f 14 g 7 h 3 i  4

  5. Elections and Clone Structures Example R1: b > c > d > e > f > a > g > h > i R2: e > f > d > c > i > g > h > b > a R3:b >a > c >d > e > f > g >i > h X = {c, d, e, f} Y = {g,h,i} 8 7 6 5 4 3 2 1 0 a 10b  17 c 18 d 17 e 17 f 14 g 7 h 3 i  4

  6. Elections and Clone Structures Example R1: b > X > a > Y R2: X > Y > b > a R3:b >a > X > Y X = {c, d, e, f} Y = {g,h,i} 321 0 a 3b  7 X  6 Y  2 Previously a member of X was winning! Questions Whichsetsare clone structures? How to represent clone structures? How to exploit clone structures?

  7. OurResults • Anaxiomaticcharacterizationof clone structure • Compact representations of clone structures • A polynomial-timealgorithm for decloningtoward single-peakedelections • Preliminary results on characterizingsingle-peakedelections PQ-trees Part 1 voterrepresentation Part 2

  8. 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 a b c d e

  9. 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 a b c d e f

  10. 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 a b c d e f

  11. 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. g h i a b c de f

  12. 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” g f h e a d b c

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

  14. OurResults • Anaxiomaticcharacterizationof clone structure • Compact representations of clone structures • A polynomial-timealgorithm for decloningtoward single-peakedelections • Preliminary results on characterizingsingle-peakedelections PQ-trees Part 1 voterrepresentation Part 2

  15. Clone StructureRepresentations b c d e f a g h i • How to convenientlyrepresent the aboveclone structure?

  16. Clone StructureRepresentations b c d e f a g h i X X = {a, b, c, d, e, f, g, h, i} X

  17. Clone StructureRepresentations b c d e f a g h i b Y a Z X = {a, b, c, d, e, f, g, h, i} Y = {c,d, e, f}, Z = {g, h, i} X b Y a Z

  18. Clone StructureRepresentations b c d e f a g h i b Y a g h i X = {a, b, c, d, e, f, g, h, i} Y = {c,d, e, f}, Z = {g, h, i} X b Y a Z g h i

  19. Clone StructureRepresentations b c d e f a g h i b c d U a g h i X = {a, b, c, d, e, f, g, h, i} Y = {c,d, e, f}, Z = {g, h, i} U = {e, f} X b Y a Z g h i c d U

  20. Clone StructureRepresentations b c d e f a g h i b c d e f a g h i X = {a, b, c, d, e, f, g, h, i} Y = {c,d, e, f}, Z = {g, h, i} U = {e, f} X b Y a Z g h i c d U e f

  21. Clone StructureRepresentations b c d e f a g h i b c d e f a g h i X = {a, b, c, d, e, f, g, h, i} Y = {c,d, e, f}, Z = {g, h, i} U = {e, f} X b Y a Z g h i c d U e f P-node – fatsausage Q-node – string of sausage

  22. 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!

  23. 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

  24. 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

  25. 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 1 > 3 > 2 > 4 > a > c Theorem. For every clone structure F overalternative set A, therearethreeorders R1, R2, R3thatjointlygenerate F.

  26. OurResults • Anaxiomaticcharacterizationof clone structure • Compact representations of clone structures • A polynomial-timealgorithm for decloningtoward single-peakedelections • Preliminary results on characterizingsingle-peakedelections PQ-trees Part 1 voterrepresentation Part 2

  27. Part 2: Clones in 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 a b c d b > c > d > a a > b > c > d c > b > a > d

  28. Part 2: Clones in 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 a b c d1 d2 b > c > d1 > d2 > a Profile losessingle-peakednessdue to cloning a > b > c > d1 > d2 c > b > a > d2 > d1

  29. 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 a b c d g h i e f

  30. 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 a b c d g h i e f

  31. 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 a b c d g h i e f

  32. 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 a b c d g h i e f

  33. 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!

  34. 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!

  35. Breaking News! IntermediatePreferences 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

  36. Breaking News! IntermediatePreferences 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 Decloningtowardintermediatepreferencesis NP-complete

  37. COMSOC-2012 in Kraków, Poland

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