Projektseminar Computational Social Choice -Eine Einführung-

1. ProjektseminarComputationalSocial Choice -Eine Einführung- Jörg Rothe & Lena Schend SS 2012, HHU Düsseldorf 4. April 2012

2. Introduction Social Choice Theory • votingtheory • preferenceaggregation • judgmentaggregation Computer 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 Social Choice Theory • votingtheory • preferenceaggregation • judgmentaggregation ComputationalSocialChoice Computer Science • artificialintelligence • algorithm design • computationalcomplexitytheory - worst-case/average-casecomplexity - optimization, etc. • computationalbarrierstoprevent • manipulation • control • bribery 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 worst-casecomplexityshieldsenough? • Or do theyevaporate on "typicalelections"?

6. NP-HardnessShields Evaporating? approximation NP-hardness shields juntadistributions experimental analysis single-peakedelectorates

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

8. Voting Systems ApprovalVoting (AV) • votesareapprovalvectors in

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

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

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

12. Voting Systems PairwiseComparison v1: > > > v3: > > > v2: > > > v4: > > > Condorcet: beats all othercandidatesstrictly Copeland : 1pointforvictorypointsfortie Maximin: maximumofthe worstpairwisecomparison

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

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

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

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

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

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

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

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

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

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

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

24. War on Manipulation I like Spock but I don‘twanthimtobethecaptain!! Copeland : winner v1: > > > v3: > > > v2: > > > v4: > > >

25. 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

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

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

28. NP-HardnessShields Evaporating? approximation NP-hardness shields juntadistributions experimental analysis single-peakedelectorates

29. Junta Distributions ofProcacciaand Rosenschein (JAAMAS 2007) areomittedhere, astheyare a rathertechnicalconcept.

30. NP-HardnessShields Evaporating? approximation NP-hardness shields juntadistributions experimental analysis single-peakedelectorates

31. Experiments Manipulation • testing (heuristic) algorithmsformanipulationproblemathand on givenelections • sample real elections • generaterandomelections: Impartial Culture (IC) Polya-Eggenberger (PE) • votersvoteindependently • all preferencesareequallylikely • votersarehighlycorrelated • v1 v2 v3 ... Walsh (IJCAI 2009; ECAI 2010)

32. Experiments Manipulation • testing (heuristic) algorithmsformanipulationproblemathand on givenelections • sample real elections • generaterandomelections: Impartial Culture (IC) Polya-Eggenberger (PE) • votersvoteindependently • all preferencesareequallylikely • votersarehighlycorrelated • v1 v2 v3 ... Walsh (IJCAI 2009; ECAI 2010)

33. Experiments Manipulation • testing (heuristic) algorithmsformanipulationproblemathand on givenelections • sample real elections • generaterandomelections: Impartial Culture (IC) Polya-Eggenberger (PE) • votersvoteindependently • all preferencesareequallylikely • votersarehighlycorrelated • v1 v2 v3 ... Walsh (IJCAI 2009; ECAI 2010)

34. Experiments Manipulation • testing (heuristic) algorithmsformanipulationproblemathand on givenelections • sample real elections • generaterandomelections: Impartial Culture (IC) Polya-Eggenberger (PE) • votersvoteindependently • all preferencesareequallylikely • votersarehighlycorrelated • v1 v2 v3 ... Walsh (IJCAI 2009; ECAI 2010)

35. Experiments Manipulation • testing (heuristic) algorithmsformanipulationproblemathand on givenelections • sample real elections • generaterandomelections: Impartial Culture (IC) Polya-Eggenberger (PE) • votersvoteindependently • all preferencesareequallylikely • votersarehighlycorrelated • v1 v2 v3 ... Walsh (IJCAI 2009; ECAI 2010)

36. Experiments Manipulation • testing (heuristic) algorithmsformanipulationproblemathand on givenelections • sample real elections • generaterandomelections: Impartial Culture (IC) Polya-Eggenberger (PE) • votersvoteindependently • all preferencesareequallylikely • votersarehighlycorrelated • v1 v2 v3 ... Walsh (IJCAI 2009; ECAI 2010)

37. Experiments Manipulation • Resultsfor STV • Single Manipulation: • forupto 128 candidates/votersmanipulationhaslowcomputationalcosts (for all voterdistributions) • chanceofsuccessfulmanipulationdecreaseswithincreasingnumberofnonmanipulativevoters • Coalitional Manipulation: • larger coalitionsaremorelikelytobesuccessful • again: computationalcostsarelowforupto 128 candidates/voters • Resultsfor Veto (weighted) • ifmanipulators‘ weightsaretoobig/small => trivial • even in criticalregion: computationalcostsarelow • onlycorrelatedvotersincreasecomputationalcosts Walsh (IJCAI 2009; ECAI 2010)

38. NP-HardnessShields Evaporating? approximation NP-hardness shields juntadistributions experimental analysis single-peakedelectorates

39. Approximating Manipulation Before: Is manipulationpossible? ?

40. Approximating Manipulation Before: Is manipulationpossible? Now: Howmanymanipulatorsareneeded? (min!) Approximation Algorithms: • efficientalgorithms • do not always find optimal solution • canbeanalyzedboththeoreticallyandexperimentally ? ?

41. ApproximatingBorda 3x > > > > > > 2x > > > > > > Bordawinner manipulatorsprefer

42. ApproximatingBorda AlgorithmforBorda-CCUM : "Reverse" m1 > > > > > > Zuckerman, Procaccia & Rosenschein (ArtificialIntelligence 2009)

43. ApproximatingBorda AlgorithmforBorda-CCUM : "Reverse" m1 > > > > > > Zuckerman, Procaccia & Rosenschein (ArtificialIntelligence 2009)

44. ApproximatingBorda AlgorithmforBorda-CCUM : "Reverse" m1 > > > > > > m2 > > > > > > Zuckerman, Procaccia & Rosenschein (ArtificialIntelligence 2009)

45. ApproximatingBorda AlgorithmforBorda-CCUM : "Reverse" m1 > > > > > > m2 > > > > > > Zuckerman, Procaccia & Rosenschein (ArtificialIntelligence 2009)

46. ApproximatingBorda AlgorithmforBorda-CCUM : "Reverse" m1 > > > > > > m2 > > > > > > m3 > > > > > > Zuckerman, Procaccia & Rosenschein (ArtificialIntelligence 2009)

47. ApproximatingBorda AlgorithmforBorda-CCUM : "Reverse" m1 > > > > > > m2 > > > > > > m3 > > > > > > Zuckerman, Procaccia & Rosenschein (ArtificialIntelligence 2009)

48. ApproximatingBorda AlgorithmforBorda-CCUM : "Reverse" m1 > > > > > > m2 > > > > > > m3 > > > > > > m4 > > > > > > Zuckerman, Procaccia & Rosenschein (ArtificialIntelligence 2009)

49. ApproximatingBorda AlgorithmforBorda-CCUM : "Reverse" m1 > > > > > > m2 > > > > > > m3 > > > > > > m4 > > > > > > Zuckerman, Procaccia & Rosenschein (ArtificialIntelligence 2009)

50. ApproximatingBorda AlgorithmforBorda-CCUM : "Reverse" m1 > > > > > > m2 > > > > > > m3 > > > > > > m4 > > > > > > m5 > > > > > > Zuckerman, Procaccia & Rosenschein (ArtificialIntelligence 2009)