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Piotr Faliszewski AGH U niversity of Science and Technology, Krakow

The Shield that Never Was: Societies with Single-Peaked Preferences are More Open to Manipulation and Control. Piotr Faliszewski AGH U niversity of Science and Technology, Krakow. Edith Hemaspaandra Rochester Institute of Technology. Lane A. Hemaspaandra University of Rochester.

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Piotr Faliszewski AGH U niversity of Science and Technology, Krakow

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  1. The Shield that Never Was:Societies with Single-Peaked Preferences are More Open to Manipulation and Control Piotr FaliszewskiAGH University of Science and Technology, Krakow Edith HemaspaandraRochester Institute of Technology Lane A. HemaspaandraUniversity of Rochester Jörg RotheHeinrich-Heine-Universität Düsseldorf Moscow, SCW 2010

  2. Outline Introduction Computational Foundations of Social Choice, an ESF Project Elections and Single-Peaked Preferences. Thanks, Toby! Control and Manipulation Overview of Results Control: Single-Peakedness Removing NP-Hardness Shields Manipulation: Single-Peaked Preferences NP-Hardness Shields: Removing them Leaving them in Place Erecting them A Dichotomy Result for 3-Candidate Scoring Protocols A Sample Proof Sketch

  3. CFSC Project Participants • Principal Investigators: • Felix Brandt (München) • Ulle Endriss (Amsterdam) • Jeffrey Rosenschein (Jerusalem) • Jörg Rothe (Düsseldorf) • Remzi Sanver (Instanbul) • Associated Partners: • Vincent Conitzer (Duke University) • Edith Elkind (Singapore/Southampton) • Edith Hemaspaandra (Rochester) • Lane Hemaspaandra (Rochester) • Jerome Lang (Paris/Toulouse) • Jean-François Laslier (Paris) • Nicolas Maudet (Paris) AI TCS AI LOG AI TCS ECON AI ECON TCS TCS LOG TCS AI LOG ECON AI

  4. What did the Düsseldorf Group do in 2009? This is Nadja Betzler from Jena, not Magnus Roos from D’dorf. Gábor Doro Jörg Claudia

  5. What did the Düsseldorf Group do in 2009? Düsseldorf Piotr Ulle Lane Edith H. Edith E. Felix Vince Frank Magnus Doro Gábor Jörg Remzi Jean-François Jeff Nicolas The Shield that Never Was: Societies with Single-Peaked Preferences are More Open to Manipulation and Control. TARK’09 Claudia Jérôme Yann

  6. Introduction • Computational Social Choice • Applications in AI • Multiagent systems • Multicriteria decision making • Meta search-engines • Planning • Applications in social choice theory and political science • Computational barrier to prevent cheating in elections • Manipulation • Control • Bribery Computational agents can systematically analyze an election to find the optimal behavior.

  7. Introduction Using the power of NP-hardness, vulcans have created complexity shields to protect elections against many types of manipulation and procedural control. Computational agents can systematically analyze an election to find the optimal behavior.

  8. Introduction Using the power of NP-hardness, vulcans have created complexity shields to protect elections against many types of manipulation and procedural control. Computational agents can systematically analyze an election to find the optimal behavior. • Our Main Theme: • Complexity shields may evaporate in single-peaked societies

  9. Elections • An election is a pair (C,V) with • candidate set C = {c1, ..., cm}: • and a list of votes V = (v1, ..., vn): • Each vote vi is represented via its preferences over C: • Either linear orders: > > > > • Or approval vectors: (1,1,0,0,1) • An election system aggregates the preferences and outputs the set of winners. Hi, my name is v7. Hi v7, I hope you are not one of those awful people who support Mr. Smith! How will they aggregate our votes?!

  10. Election Systems • Approval (any number of candidates): Every vote is an approval vector from All candidates with the most points are winners. • Example:

  11. Election Systems • Approval (any number of candidates): Every vote is an approval vector from All candidates with the most points are winners. • Example:

  12. Election Systems • Approval (any number of candidates): Every vote is an approval vector from All candidates with the most points are winners. • Example: Winners:

  13. Election Systems • Approval (any number of candidates): Every vote is an approval vector from All candidates with the most points are winners. • Scoring protocols for m candidates are specified by scoring vectors with where each voter‘s i-th candidate gets points: • m-candidate plurality: • m-candidate j-veto: • Borda: • Plurality (any number of candidates): • Veto (any number of candidates):

  14. Single-Peaked Preferences • A collection V of votes is said to be single-peaked if there exists a linear order L over C such that each voter‘s „degree of preference“ rises to a peak and then falls (or just rises or just falls).

  15. Single-Peaked Preferences • A collection V of votes is said to be single-peaked if there exists a linear order L over C such that each voter‘s „degree of preference“ rises to a peak and then falls (or just rises or just falls). A voter‘s preference curve on galactic taxes low galactic taxes high galactic taxes

  16. Single-Peaked Preferences • A collection V of votes is said to be single-peaked if there exists a linear order L over C such that each voter‘s „degree of preference“ rises to a peak and then falls (or just rises or just falls). A voter‘s > > > preference curve on galactic taxes low galactic taxes high galactic taxes Single-peaked preference consistent with linear order of candidates

  17. Single-Peaked Preferences • A collection V of votes is said to be single-peaked if there exists a linear order L over C such that each voter‘s „degree of preference“ rises to a peak and then falls (or just rises or just falls). A voter‘s > > > preference curve on galactic taxes low galactic taxes high galactic taxes Preference that is inconsistent with linear order of candidates

  18. Single-Peaked Preferences • A collection V of votes is said to be single-peaked if there exists a linear order L over C such that each voter‘s „degree of preference“ rises to a peak and then falls (or just rises or just falls). • If each vote vi in V is a linear order >i over C, this means that for each triple of candidates c, d, and e: (c L d L e or e L d L c) implies that for each i, if c >i d then d >i e.

  19. Single-Peaked Preferences • A collection V of votes is said to be single-peaked if there exists a linear order L over C such that each voter‘s „degree of preference“ rises to a peak and then falls (or just rises or just falls). • If each vote vi in V is a linear order >i over C, this means that for each triple of candidates c, d, and e: (c L d L e or e L d L c) implies that for each i, if c >i d then d >i e. • Bartholdi & Trick (1986); Escoffier, Lang & Öztürk (2008): Given a collection V of linear orders over C, in polynomial time we can produce a linear order L witnessing V‘s single-peakedness or can determine that V is not single-peaked.

  20. Single-Peaked Preferences • A collection V of votes is said to be single-peaked if there exists a linear order L over C such that each voter‘s „degree of preference“ rises to a peak and then falls (or just rises or just falls). • If each vote vi in V is an approval vector over C, this means that for each triple of candidates c, d, and e: c L d L e implies that for each i, if vi approves of both c and e then vi approves of d.

  21. Single-Peaked Preferences • A collection V ofvotesissaidtobesingle-peakedifthereexists a linear order L over C such thateach voter‘s „degreeofpreference“ risesto a peakandthen falls (or just risesor just falls). • Ifeachvotevi in V is an approvalvectorover C, thismeansthatforeachtripleofcandidates c, d, and e: c L d L e impliesthatforeach i, ifviapprovesofboth c and e thenviapprovesof d. • Fulkerson & Gross (1965); Booth & Lueker (1976):Given a collection V ofapprovalvectorsover C, in polynomial time wecanproduce a linear order L witnessingV‘ssingle-peakednessorcandeterminethat V is not single-peaked.

  22. Control and Manipulation • Thebadguy wants to make someone win (constructive) or prevent someone from winning (destructive). • Thebadguy knows everybody else’s votes. • In control,thechair modifiesan election‘sstructure by: • Adding candidates (limited/unlimitednumber) • Deleting candidates • Partition of candidates with/without runoff • Adding/deleting voters • Partition of voters • In manipulation, a coalition of agents change their votes to obtain their desired effect. • Bothnonmanipulatorsandmanipulatorsareweighted. • In thesingle-peakedcase, bothnonmanipulatorsandmanipulatorsaresingle-peakedw.r.t. the same order L. • See Bartholdi, Tovey & Trick (1989; 1992), Conitzer, Sandholm & Lang (2007), Hemaspaandra, Hemaspaandra & Rothe (2007).

  23. Outline Introduction ComputationalFoundationsofSocialChoice, an ESF Project Electionsand Single-PeakedPreferences. Thanks, Toby! Controland Manipulation OverviewofResults Control: Single-PeakednessRemoving NP-Hardness Shields Manipulation: Single-PeakedPreferences NP-Hardness Shields: Removingthem Leavingthem in Place Erectingthem A DichotomyResultfor 3-Candidate ScoringProtocols A Sample Proof Sketch

  24. Control Results: Approval Theorem 2: Forthesingle-peakedcase, approvalvotingis vulnerable to constructivecontrolbyaddingvotersand constructivecontrolbydeletingvoters, in theunique-winnerandthenonunique-winner model, forthestandardandthesuccinctinput model.

  25. Control Results: Approval Theorem 2: Forthesingle-peakedcase, approvalvotingis vulnerable to constructivecontrolbyaddingvotersand constructivecontrolbydeletingvoters, in theunique-winnerandthenonunique-winner model, forthestandardandthesuccinctinput model. Forcomparison: Among all typesofcontrolbyadding/deletingeithercandidatesorvoters, theabovetwocasesaretheonlytworesistances in thegeneralcase. (Hemaspaandra, Hemaspaandra & Rothe, AAAI’05; ArtificialIntelligence 2007)

  26. Control Results: Approval Theorem 2: Forthesingle-peakedcase, approvalvotingis vulnerable to constructivecontrolbyaddingvotersand constructivecontrolbydeletingvoters, in theunique-winnerandthenonunique-winner model, forthestandardandthesuccinctinput model. Forcomparison:

  27. Control Results: Plurality Theorem 3: Forthesingle-peakedcase, pluralityvotingis vulnerable toconstructiveanddestructivecontrolby addingcandidates, adding an unlimitednumberofcandidates, and deletingcandidates in theunique-winnerandthenonunique-winner model.

  28. Control Results: Plurality Theorem 3: Forthesingle-peakedcase, pluralityvotingis vulnerable toconstructiveanddestructivecontrolby addingcandidates, adding an unlimitednumberofcandidates, and deletingcandidates in theunique-winnerandthenonunique-winner model. Forcomparison: Foreachofthesesixtypesofcandidatecontrolpluralityvotingisresistant in thegeneralcase, but is vulnerable tothefourtypesofcontrolinvolvingadding/deletingvoters. (Bartholdi, Tovey & Trick, 1992; Hemaspaandra, Hemaspaandra & Rothe, 2007)

  29. Control Results: Plurality Theorem 3: Forthesingle-peakedcase, pluralityvotingis vulnerable toconstructiveanddestructivecontrolby addingcandidates, adding an unlimitednumberofcandidates, and deletingcandidates in theunique-winnerandthenonunique-winner model. Forcomparison:

  30. Outline Introduction ComputationalFoundationsofSocialChoice, an ESF Project Electionsand Single-PeakedPreferences. Thanks, Toby! Controland Manipulation OverviewofResults Control: Single-PeakednessRemoving NP-Hardness Shields Manipulation: Single-PeakedPreferences NP-Hardness Shields: Removingthem Leavingthem in Place Erectingthem A DichotomyResultfor 3-Candidate ScoringProtocols A Sample Proof Sketch

  31. Manipulation: Removing NP-Hardness Shields Theorem 4: Forthesingle-peakedcase, theconstructivecoalitionweightedmanipulationproblem (in boththeunique-winnerandthenonunique-winner model) foreachofthefollowingelectionsystemsis in P: The scoringprotocol , i.e., 3-candidate Borda. Eachofthescoringprotocols , . Veto.

  32. Manipulation: Removing NP-Hardness Shields Theorem 4: Forthesingle-peakedcase, theconstructivecoalitionweightedmanipulationproblem (in boththeunique-winnerandthenonunique-winner model) foreachofthefollowingelectionsystemsis in P: The scoringprotocol , i.e., 3-candidate Borda. Eachofthescoringprotocols , . Veto. Forcomparison: 3-candidate Borda, Veto, andthe „ “ casesof , , are NP-complete in thegeneralcase (andtherestis in P). (Hemaspaandra & Hemaspaandra, 2007; Procaccia & Rosenschein, 2007; Conitzer, Sandholm & Lang, 2007).

  33. Manipulation: Removing NP-Hardness Shields Theorem 5: For the single-peaked case, the constructive coalition weighted manipulation problem for m-candidate 3-veto is in P for m in {3,4,6,7,8,…} and is resistant (indeed, NP-complete) for m=5 candidates.

  34. Manipulation: Removing NP-Hardness Shields Theorem 5: For the single-peaked case, the constructive coalition weighted manipulation problem for m-candidate 3-veto is in P for m in {3,4,6,7,8,…} and is resistant (indeed, NP-complete) for m=5 candidates. For comparison: m-candidate 3-veto is in P for m in {3,4} and is resistant (indeed, NP-complete) for five or more candidates. (Hemaspaandra & Hemaspaandra; Journal of Computer and System Sciences 2007).

  35. Manipulation: Leaving them in Place Theorem 6: For the single-peaked case, the constructive coalition weighted manipulation problem (in both the unique-winner and the nonunique-winner model) is resistant (indeed, NP-complete) for the scoring protocol and the scoring protocol , i.e., 4-candidate Borda.

  36. Manipulation: Leaving them in Place Theorem 6: For the single-peaked case, the constructive coalition weighted manipulation problem (in both the unique-winner and the nonunique-winner model) is resistant (indeed, NP-complete) for the scoring protocol and the scoring protocol , i.e., 4-candidate Borda. For comparison: These problems are known to be NP-complete also in the general case. (Hemaspaandra & Hemaspaandra, 2007) These results are particularly inspired by Walsh (2007) who proved the same for Single Transferable Voting.

  37. Manipulation: Erecting NP-Hardness Shields Can restricting to single-peaked preferences ever erect a complexity shield? Single-peaked case General case

  38. Manipulation: Erecting NP-Hardness Shields Can restricting to single-peaked preferences ever erect a complexity shield? Theorem 7: There exists an election system, whose votes are approval vectors, for which constructive size-3-coalition unweighted manipulation is in P for the general case but is NP-complete in the single-peaked case. Single-peaked case General case

  39. Manipulation: A Dichotomy Result Theorem 8: Consider a 3-candidate scoring protocol For the single-peaked case, the constructive coalition weighted manipulation problem (in both the unique-winner and the nonunique-winner model) is resistant (indeed, NP-complete) when and is in P otherwise.

  40. Outline Introduction ComputationalFoundationsofSocialChoice, an ESF Project Electionsand Single-PeakedPreferences. Thanks, Toby! Controland Manipulation OverviewofResults Control: Single-PeakednessRemoving NP-Hardness Shields Manipulation: Single-PeakedPreferences NP-Hardness Shields: Removingthem Leavingthem in Place Erectingthem A DichotomyResultfor 3-Candidate ScoringProtocols A Sample Proof Sketch

  41. A Sample Proof Sketch Theorem 2: Forthesingle-peakedcase, approvalvotingis vulnerable to constructivecontrolbyaddingvotersand constructivecontrolbydeletingvoters, in theunique-winnerandthenonunique-winner model, forthestandardandthesuccinctinput model.

  42. A Sample Proof Sketch Theorem 2: Forthesingle-peakedcase, approvalvotingis vulnerable to constructivecontrolbyaddingvotersand constructivecontrolbydeletingvoters, in theunique-winnerandthenonunique-winner model, forthestandardandthesuccinctinput model. Wefocus on: constructivecontrolbyaddingvoters in theunique-winner model forthesuccinctinput model.

  43. A Sample Proof Sketch Theorem 2: Forthesingle-peakedcase, approvalvotingis vulnerable to constructivecontrolbyaddingvotersand constructivecontrolbydeletingvoters, in theunique-winnerandthenonunique-winner model, forthestandardandthesuccinctinput model. Wegive a poly-time algorithmthat, given collections V and W ofvotesovercandidateset C andsingle-peakedw.r.t. order L, a designatedcandidate p in C, and an additionlimit k, decidesifbyaddingatmost k votesfrom W wecanmake p theuniquewinner.

  44. A Sample Proof Sketch number of approvals from voters in V for candidates that are 1 2 votes in W that can be added (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 Which vote types from W should we add? Especially if they are incomparable? 2 votes in W that can be added (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 We‘ll handle this by a „smart greedy“ algorithm. 2 votes in W that can be added (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 Whyare F, C, B, c, f, and j dangerous but theremainingcandidatescanbeignored? 2 votes in W that can be added (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 W that can be added (with multiplicities) 4 7 3 1 9 5

  49. 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 W that can be added (with multiplicities) 4 7 3 1

  50. A Sample Proof Sketch number of approvals from voters in V for candidates that are 1 Now, if adding votes from W causes p to beat c then p must also beat a and b. 2 votes in W that can be added (with multiplicities) 4 7 3 1

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