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Analyzing Clone Dynamics in Election Processes: Strategies for Outcome Manipulation

This study delves into the complexities of managing clones within electoral systems, exploring various mechanisms that influence election outcomes. Two key problems are highlighted: identifying the most probable true elections from a cluster of candidates and strategizing on decloning to secure victory for a preferred candidate. By investigating clone structures, the research aims to unveil the underlying principles governing voter behavior and the implications of voting rules on election integrity. The findings could significantly impact electoral strategy and design.

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Analyzing Clone Dynamics in Election Processes: Strategies for Outcome Manipulation

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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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