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Jörg Rothe

Overview of C omputational F oundations of S ocial C hoice GASICS Workshop Aachen, October 2009. Jörg Rothe. Computational Social Choice. What is computational social choice? A new interdisciplinary field of study at the interface of social choice theory and computer science

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Jörg Rothe

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  1. Overview of ComputationalFoundations of Social ChoiceGASICS WorkshopAachen, October 2009 Jörg Rothe

  2. Computational Social Choice • What is computational social choice? • A new interdisciplinary field of study at the interface of social choice theory and computer science • What is social choice theory? • Social choice theory studies the aggregation of individual preferences • Key concepts • Preference relation: typically transitive and complete • Set of preference relations over a given set of alternatives : • Social welfare function • Social choice function • Social choice correspondence

  3. Computer Science Social Choice Theory Computational Social Choice Computational Social Choice • Bidirectional transfer • Computer science ➠ Social choice • Apply complexity theory, algorithms, learning theory to problems of social choice • Social choice ➠ Computer science • Import concepts from social choice to solve questions arising in AI (e.g., in societies of autonomous software agents), webpage ranking, or collaborative filtering

  4. Theorem (Arrow, 1951): There is no nondictatorial social welfare function satisfying Pareto-optimality and independence of irrelevant alternatives. I‘llshowyousomesolutionconcepts!

  5. 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

  6. Aims & Objectives • Social choice and theoretical computer science • To deepen our understanding of algorithmic and complexity-theoretic issues in social choice • Social choice and logic • To develop logic-based languages for modeling and reasoning about social choice problems and preference structures • Social choice and artificial intelligence • To apply established techniques from AI, such as preference elicitation and learning, to problems of social choice

  7. The Community • Where do we meet? • International Workshop on Computational Social Choice (COMSOC), coordinated by Ulle Endriss & JérômeLang) • 1st COMSOC, Amsterdam, 6-8 December 2006 • 2nd COMSOC, Liverpool, 2-5 September 2008 • 3rd COMSOC, Düsseldorf, 13-16 September 2010 • Dagstuhl Seminars • Computational Issues in Social Choice, 21-26 October 2007 • Computational Foundations of Social Choice, 7-12 March 2010 • Where do we publish? • Conferences: AAAI, AAMAS, IJCAI, SODA, TARK, WINE, ... • Journals: AIJ, IC, JACM, JAIR, MSS, SCW, TCS, TOCS, ... • MLQ special issue (edited by Paul Goldberg and Jörg Rothe): “Logic and Complexity within Computational Social Choice”

  8. Main Topics • Computational aspects of evaluating voting rules • Theorem (Bartholdi et al., 1989): There is no social welfare function that is neutral, consistent, Condorcet, and efficiently computable (unless P=NP). • Other issues: efficient algorithms, approximation, exact computational complexity, etc. • Computational hardness of manipulation • Theorem (Bartholdi et al., 1989): There is a social welfare function that is easy to compute, but not efficiently manipulable (unless P=NP). • Moreover, this function is neutral, Condorcet, Pareto-optimal, etc. • Other issues: few alternatives, weighted voting, typical-case, approximation, heuristics, other types of manipulation (control, bribery, ...), etc.

  9. Main Topics (cont.) • Computational aspects of fair division • How to fairly divide divisible goods or resources among several agents or players • e.g., cutting a cake • Indivisible goods (multiagent resource allocation) • e.g., complexity of social welfare optimization • e.g., (combinatorial) auctions and mechanism design • Social choice in combinatorial domains • Combinatorial structure gives rise to exponential growth • multiple referenda, committee election • Representation of preferences (graphical or logical) • CP-nets, weighted propositional formulas • important factors: compactness, expressiveness, computational properties I‘llshowyouhowtocut a cake!

  10. Main Topics (cont.) • Computational aspects of coalitional voting games • Voting settings are often modeled as cooperative games • e.g., weighted voting games: compact representation • Complexity of game-theoretic solution concepts • e.g., the core, the Shapley-Shubik and Banzhaf power index • Manipulation and control • e.g., false identities/splitting weight, changing threshold, adding/deleting voters • Epistemic issues in social choice • Incomplete preferences • Elicitation of preferences • Communication complexity

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

  12. What did the Düsseldorf Group do in 2009? Düsseldorf Piotr Ulle Lane Edith H. Felix Vince Edith E. Frank Magnus Doro Gábor Jörg Remzi Jean-François Jeff Nicolas The Cost of Stability in Coalitional Games. Joint with Y. Bachrach, R. Meir, D. Pasechnik, M. Zuckerman. To appear at SAGT’09; extended abstract: AAMAS’09 Claudia Jérôme Yann

  13. 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 Complexity of Probabilistic Lobbying. Joint with H. Fernau J. Goldsmith, N. Mattei, D. Raible. To appear at ADT’09 Claudia Jérôme Yann

  14. 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 Generalized Juntas and NP-Hard Sets. Joint with H. Spakowski. FCT’07; Theoretical Computer Science 2009 Claudia Jérôme Yann

  15. 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 Frequency of Correctness versus Average Polynomial Time. Joint with H. Spakowski. FCT’07; Information Processing Letters 2009 Claudia Jérôme Yann

  16. 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 Satisfiability Parsimoniously Reduces to the Tantrix(TM) Rotation Puzzle Problem. MCU’07; Fundamenta Informaticae 2009 Claudia Jérôme Yann

  17. 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 Three-Color and Two-Color Tantrix(TM) Rotation Puzzle Problems are NP-Complete Via Parsimonious Reductions. LATA’08; Information & Computation 2009 Claudia Jérôme Yann

  18. 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 Complexity of Computing Minimal Unidirectional Covering Sets. Joint with F. Fischer and J. Hoffman. Submitted Claudia Jérôme Yann

  19. 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 Degrees of Guaranteed Envy-Freeness in Finite Bounded Cake-Cutting Protocols. WINE’09 Claudia Jérôme Yann

  20. 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 Complexity of Social Welfare Optimization in Multiagent Resource Allocation. Submitted Claudia Jérôme Yann

  21. 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 Hybrid Elections Broaden Complexity-Theoretic Resistance to Control. IJCAI’07; Mathematical Logic Quarterly 2009 Claudia Jérôme Yann

  22. 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 Llull and Copeland Voting Computationally Resist Bribery and Constructive Control. AAAI’07; AAIM’08; Journal of Artificial Intelligence Research 2009 Claudia Jérôme Yann

  23. 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 Sincere-Strategy Preference-Based Approval Voting Fully Resists Constructive Control and Broadly Resists Destructive Control. Joint with M. Nowak. MFCS’08; Mathematical Logic Quarterly 2009 Claudia Jérôme Yann

  24. 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 Control Complexity in Fallback Voting. To appear at CATS’10 Claudia Jérôme Yann

  25. 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 A Richer Understanding of the Complexity of Election Systems. In „Fundamental Problems in Computing: Essays in Honor of Professor Daniel J. Rosenkrantz“ S. Ravi & S. Shukla, eds., Springer, 2009 Claudia Jérôme Yann

  26. 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 Computational Aspects of Approval Voting. To appear in „Handbook of Approval Voting“ J.-F. Laslier & R. Sanver, eds., Springer Claudia Jérôme Yann

  27. 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

  28. The Shield that Never Was:Societies with Single-Peaked Preferences are More Open to Manipulation and Control Piotr FaliszewskiAGH University of Science and Technology Edith HemaspaandraRochester Institute of Technology Lane A. HemaspaandraUniversity of Rochester Jörg RotheHeinrich-Heine-Universität Düsseldorf TARK XII, Palo Alto, USA, July 2009

  29. Outline Introduction Elections and Single-Peaked Preferences. Thanks, Toby! Control and Manipulation Overview of Results Control: Single-Peakedness Removing NP-Hardness Shields Manipulation: Single-Peaked Preferences Removing NP-Hardness Shields Leaving them in Place Erecting them Giving a Dichotomy for 3-Candidate Scoring Protocols A Sample Proof Sketch Single-Peaked Approval Voting is Vulnerable to Constructive Control by Adding Voters

  30. 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.

  31. 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.

  32. 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

  33. Elections • Candidates and voters: • C = {c1, ..., cm} • V = {v1, ..., vn} • Each voter vi is represented via his or her preferences over C: • Linear orders: c > e > a > b > d • Approval vectors: (0,1,1,0,1) • Election system aggregates these 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 c3! How will they aggregate our votes?!

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

  35. 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).

  36. 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

  37. 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

  38. 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

  39. 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.

  40. 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.

  41. 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.

  42. 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. • Theorem 1:Given a collection V ofapprovalvectorsover C, in polynomial time wecanproduce a linear order L witnessingV‘ssingle-peakednessorcandeterminethat V is not single-peaked.

  43. 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 candidateswith/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).

  44. Outline Introduction Electionsand Single-PeakedPreferences Controland Manipulation OverviewofResults Control: Single-PeakednessRemoving NP-Hardness Shields Manipulation: Single-PeakedPreferences Removing NP-Hardness Shields Leavingthem in Place Erectingthem Giving a Dichotomyfor 3-Candidate ScoringProtocols A Sample Proof Sketch Single-PeakedApprovalVotingis Vulnerable toConstructiveControlbyAddingVoters

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

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

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

  48. 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)

  49. Outline Introduction Electionsand Single-PeakedPreferences Controland Manipulation OverviewofResults Control: Single-PeakednessRemoving NP-Hardness Shields Manipulation: Single-PeakedPreferences Removing NP-Hardness Shields Leavingthem in Place Erectingthem Giving a Dichotomyfor 3-Candidate ScoringProtocols A Sample Proof Sketch Single-PeakedApprovalVotingis Vulnerable toConstructiveControlbyAddingVoters

  50. 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.

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