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  1. Manipulation andControlforApprovalVotingandOther Voting Systems Jörg Rothe Oxford Meeting for COST Action IC1205 on ComputationalSocial Choice April 16, 2013

  2. Introduction SocialChoice Theory • votingtheory • preferenceaggregation • judgmentaggregation TheoreticalComputer Science • artificialintelligence • algorithm design • computationalcomplexitytheory - worst-case/average-casecomplexity - optimization, etc. • voting in multiagentsystems • multi-criteriadecisionmaking • metasearch, etc. ... Software agentscansystematicallyanalyzeelectionsto find optimal strategies

  3. Introduction SocialChoice Theory • votingtheory • preferenceaggregation • judgmentaggregation TheoreticalComputer Science • artificialintelligence • algorithm design • computationalcomplexitytheory - worst-case/average-casecomplexity - optimization, etc. • computationalbarrierstoprevent • manipulation • control • bribery • ComputationalSocialChoice Software agentscansystematicallyanalyzeelectionsto find optimal strategies

  4. Computational Social Choice Withthe power of NP-hardnessvulcanshaveconstructedcomplexityshieldstoprotectelectionsagainstmanytypesofmanipulationandcontrol.

  5. Computational Social Choice Withthe power of NP-hardnessvulcanshaveconstructedcomplexityshieldstoprotectelectionsagainstmanytypesofmanipulationandcontrol. • Question: • Are NP-hardnesscomplexityshieldsenough? • Or do theyevaporatefor single-peakedelectorates?

  6. NP-HardnessShields toProtectElections Elections & Voting Systems NP-hardness shields Manipulation & Control Manipulation & Control in Single-peakedElectorates Proof Sketch: CCAV in Approval

  7. NP-HardnessShields toProtectElections Elections & Voting Systems NP-hardness shields Manipulation & Control Manipulation & Control in Single-peakedElectorates Proof Sketch: CCAV in Approval

  8. Elections • An electionis a pair (C,V) with • a finite setCofcandidates: • a finite listVofvoters. • VotersarerepresentedbytheirpreferencesoverC: • eitherbylinear orders: > > > • orbyapprovalvectors: (1,1,0,1) • Votingsystem: determineswinnersfromthepreferences

  9. Voting Systems ApprovalVoting (AV) • votesareapprovalvectors in

  10. Voting Systems ApprovalVoting (AV) • votesareapprovalvectors in • winners: all candidateswiththemostapprovals

  11. Voting Systems ApprovalVoting (AV) • votesareapprovalvectors in • winners: all candidateswiththemostapprovals winners:

  12. Voting Systems PositionalScoring Rules (formcandidates) • definedbyscoringvectorwith • eachvotergivespointstothecandidate on positioni • winners: all candidateswithmaximum score Borda: PluralityVoting (PV): k-Approval (m-k-Veto): Veto (Anti-Plurality):

  13. Voting Systems PairwiseComparison v1: > > > v3: > > > v2: > > > v4: > > > Condorcet: beats all othercandidatesstrictly Copeland : 1pointforvictorypointsfortie Maximin: maximumofthe worstpairwisecomparison Hi, I am Ramon Llull. In 1299, Icame up with thevotingsystem that these guys nowstudy!

  14. Llull/Copeland Rule For FIFA World Championships or UEFA European Championships: Simply use = 1/3 as the tie value. Difference between the LlullandtheCopeland rule? What happens if the head-to-head contest ends with a tie? Llull: Both get 1 point Copeland0: Both get 0 points Copeland0.5: Both get half a point Copeland: Both get  points, for a rational , 0<<1

  15. Voting Systems Round-based: Single Transferable Vote (STV) v1: > > > v2: > > > v3: > > > v4: > > >

  16. Voting Systems Round-based: Single Transferable Vote (STV) v1: > > v2: > > v3: > > v4: > >

  17. Voting Systems Round-based: Single Transferable Vote (STV) v1: v2: v3: v4:

  18. Voting Systems Level-based: BucklinVoting (BV) v1: > > > v2: > > > v3: > > > v4: > > > v5: > > > • 5 voters => strictmajoritythresholdis 3

  19. Voting Systems Level-based: BucklinVoting (BV) v1: > > > v2: > > > v3: > > > v4: > > > v5: > > > • 5 voters => strictmajoritythresholdis 3

  20. Voting Systems Level-based: BucklinVoting (BV) v1: > > > v2: > > > v3: > > > v4: > > > Level 2 Bucklin v5: > > > winners: • 5 voters => strictmajoritythresholdis 3

  21. Voting Systems Level-based: FallbackVoting (FV) • combines AV and BV Candidates: v: { , } | { , } v: > | { , } • Bucklinwinnersarefallbackwinners. • IfnoBucklinwinnerexists (due todisapprovals), thenapprovalwinnerswin.

  22. NP-HardnessShields toProtectElections Elections & Voting Systems NP-hardness shields Manipulation & Control Manipulation & Control in Single-peakedElectorates Proof Sketch: CCAV in Approval

  23. War on ElectoralControl AV winners: "chair": knows all preferences

  24. War on ElectoralControl AV winner: "chair": knows all preferences andcanchangethestructure of an election

  25. War on ElectoralControl AV winner: "chair": knows all preferences andcanchangethestructure Other typesofcontrol: of an election • adding/partitioningvoters • deleting/adding/partitioningcandidates

  26. NP-HardnessShields forControl Resistance = NP-hardness,Vulnerability = P, Immunity, andSusceptibility

  27. NP-HardnessShields forControl

  28. References: Control • J. Bartholdi, C. Tovey, and M. Trick: HowHard isittoControl an Election?Mathematicaland Computer Modelling, 1992. • E. Hemaspaandra , L. Hemaspaandra, and J. Rothe: Anyone but Him: The ComplexityofPrecluding an Alternative. ArtificialIntelligence, 2007. (AAAI-2005) • P. Faliszewski, E. Hemaspaandra , L. Hemaspaandra, and J. Rothe: Llulland Copeland VotingComputationallyResistBriberyandConstructiveControl. Journal ofArtificialIntelligence Research, 2009.(AAAI-2007; AAIM-2008) • G. Erdélyi, M. Nowak, and J. Rothe: SP-AV FullyResistsConstructiveControlandBroadlyResistsDestructiveControl. MathematicalLogic Quarterly, 2009. (MFCS-2008) • G. ErdélyiandJ. Rothe: ControlComplexity in FallbackVoting. Proceedingsof CATS-2010. • G. Erdélyi, L. Piras, and J. Rothe: The ComplexityofVoter Partition in BucklinandFallbackVoting: SolvingThree Open Problems. ProceedingsofAAMAS-2011.

  29. War on Manipulation I like Spock but I don‘twanthimtobethecaptain!! Copeland : winner v1: > > > v3: > > > v2: > > > v4: > > > assumption: .v4knowstheother voters‘ votes v4 lies tomakehis mostpreferred candidatewin

  30. War on Manipulation I like Spock but I don‘twanthimtobethecaptain!! Copeland : winners v1: > > > v3: > > > v2: > > > v4: > > > Here: unweightedvoters, singlemanipulator . Other types: - coalitionalmanipulation - weightedvoters

  31. NP-Hardness Shields for Manipulation Results due toConitzer, Sandholm, Lang (J.ACM 2007)

  32. NP-HardnessShields toProtectElections Elections & Voting Systems NP-hardness shields Manipulation & Control Manipulation & Control in Single-peakedElectorates Proof Sketch: CCAV in Approval

  33. Single-Peaked Preferences • A collection V ofvotesissaidtobesingle-peakedifthereexists a linear order L over C such thateachvoter‘s „degreeofpreference“ risesto a peakandthen falls (or just risesor just falls). A voter‘s preference curve on galactic taxes lowgalactictaxes high galactictaxes

  34. Single-Peaked Preferences • A collection V ofvotesissaidtobesingle-peakedifthereexists a linear order L over C such thateachvoter‘s „degreeofpreference“ risesto a peakandthen falls (or just risesor just falls). A voter‘s > > > preference curve on galactic taxes lowgalactictaxes high galactictaxes Single-peakedpreferenceconsistentwith linear orderofcandidates

  35. Single-PeakedPreferences • A collection V ofvotesissaidtobesingle-peakedifthereexists a linear order L over C such thateachvoter‘s „degreeofpreference“ risesto a peakandthen falls (or just risesor just falls). A voter‘s > > > preference curve on galactic taxes lowgalactictaxes high galactictaxes Preference thatisinconsistentwiththis linear orderofcandidates

  36. Single-PeakedPreferences • A collection V ofvotesissaidtobesingle-peakedifthereexists a linear order L over C such thateachvoter‘s „degreeofpreference“ risesto a peakandthen falls (or just risesor just falls). • Ifeachvotevi in V is a linear order >iover C, thismeansthatforeachtripleofcandidates c, d, and e: (c L d L e or e L d L c) impliesthatforeach i, if c >i d then d >i e.

  37. Single-Peaked Preferences • A collection V ofvotesissaidtobesingle-peakedifthereexists a linear order L over C such thateachvoter‘s „degreeofpreference“ risesto a peakandthen falls (or just risesor just falls). • Ifeachvotevi in V is a linear order >iover C, thismeansthatforeachtripleofcandidates c, d, and e: (c L d L e or e L d L c) impliesthatforeach i, if c >i d then d >i e. • Bartholdi & Trick (1986); Escoffier, Lang & Öztürk (2008): Given a collection V of linear ordersover C, in polynomial time wecanproduce a linear order L witnessingV‘s single-peakednessorcandeterminethat V is not single-peaked.

  38. Single-PeakedApprovalVectors • A collection V ofvotesissaidtobesingle-peakedifthereexists a linear order L over C such thateachvoter‘s „degreeofpreference“ risesto a peakandthen falls (or just risesor just falls).

  39. Removing NP-hardnessshields: 3-candidate Borda veto everyscoringprotocolfor -candidate 3-veto, Leavingthem in place: STV (Walsh, AAAI-2007) 4-candidate Borda 5-candidate 3-veto Erecting NP-hardnessshields: Artificialelectionsystemwithapprovalvotes, for size-3-coalition unweightedmanipulation Results due toFaliszewski, Hemaspaandra, Hemaspaandra& Rothe (Information & Computation 2011) ConstructiveCoalitionalWeighted Manipulation General Single-peaked

  40. Removing NP-hardnessshields: Approval Constructivecontrolbyaddingvoters Constructivecontrolbydeletingvoters Plurality constructivecontrolbyaddingcandidates destructivecontrolbyaddingcandidates constructivecontrolbydeletingcandidates destructivecontrolbydeletingcandidates Results due toFaliszewski, Hemaspaandra, Hemaspaandra& Rothe (2011) Brandt, Brill, Hemaspaandra & Hemaspaandra (AAAI 2010)achievedsimilarresults forothervotingsystemsaswell (e.g., forsystemssatisfyingtheweak Condorcet criterion) and also forconstructivecontrolbypartitionofvoters. Controlfor Single-PeakedElectorates General Single-peaked

  41. Removing NP-hardnessshields: Approval Constructivecontrolbyaddingvoters Constructivecontrolbydeletingvoters Plurality constructivecontrolbyaddingcandidates destructivecontrolbyaddingcandidates constructivecontrolbydeletingcandidates destructivecontrolbydeletingcandidates Results due toFaliszewski, Hemaspaandra, Hemaspaandra& Rothe (2011) Brandt, Brill, Hemaspaandra & Hemaspaandra (AAAI 2010)achievedsimilarresults forothervotingsystemsaswell (e.g., forsystemssatisfyingtheweak Condorcet criterion) and also forconstructivecontrolbypartitionofvoters. Controlfor Single-PeakedElectorates General Single-peaked

  42. NP-HardnessShields toProtectElections Elections & Voting Systems NP-hardness shields Manipulation & Control Manipulation & Control in Single-peakedElectorates Proof Sketch: CCAV in Approval

  43. A Sample Proof Sketch number of approvals from voters in V for candidates that are 1 2 votes in Wthatcanbeadded (with multiplicities) 4 7 3 1 9 5

  44. A Sample Proof Sketch number of approvals from voters in V for candidates that are 1 Whichvotetypesfrom Wshouldweadd? Especiallyiftheyareincomparable? 2 votes in Wthatcanbeadded (with multiplicities) 4 7 3 1 9 5

  45. A Sample Proof Sketch number of approvals from voters in V for candidates that are 1 We‘ll handle this by a „smart greedy“ algorithm. 2 votes in Wthatcanbeadded (with multiplicities) 4 7 3 1 9 5

  46. A Sample Proof Sketch number of approvals from voters in V for candidates that are 1 Whyare F, C, B, c, f, and j dangerous but theremainingcandidatescanbeignored? 2 votes in Wthatcanbeadded (with multiplicities) 4 7 3 1 9 5

  47. A Sample Proof Sketch number of approvals from voters in V for candidates that are 1 First, each added vote will be an interval including p. So drop all others. 2 votes in Wthatcanbeadded (with multiplicities) 4 7 3 1 9 5

  48. A Sample Proof Sketch number of approvals from voters in V for candidates that are 1 First, each added vote will be an interval including p. So drop all others. 2 votes in Wthatcanbeadded (with multiplicities) 4 7 3 1

  49. A Sample Proof Sketch number of approvals from voters in V for candidates that are 1 Now, ifaddingvotesfrom Wcauses p tobeat c then p must also beat a and b. 2 votes in Wthatcanbeadded (with multiplicities) 4 7 3 1

  50. A Sample Proof Sketch number of approvals from voters in V for candidates that are 1 Thus, c is a dangerousrivalfor p but a and b cansafelybeignored. 2 votes in Wthatcanbeadded (with multiplicities) 4 7 3 1