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Overview of Computational Foundations of Social Choice GASICS Workshop Aachen, October 2009 Jörg Rothe

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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

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3 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 Social Choice Theory Computer Science Computational Social Choice

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4 Game TheorySocial Choice Theory precursors Cournot (1801-1877) Borel (1871-1956) Llull (1232-1316) Condorcet (1743-1794) Borda (1733-1799) Dodgson / Carroll (1832-1898) early positive results 2-Player zero-sum games: security level (Minimax Theorem, v. Neumann, 1928) Voting among 2 alternatives: majority rule (May’s Theorem, 1952) seminal monograph Theory of Games and Economic Behavior (v. Neumann & Morgenstern, 1944) Social Choice and Individual Values (Arrow, 1951) Various “solution concepts” recent trend“Algorithmic Game Theory”“Computational Social Choice” Theorem (Arrow, 1951): There is no nondictatorial social welfare function satisfying Pareto-optimality and independence of irrelevant alternatives. Theorem (Arrow, 1951): There is no nondictatorial social welfare function satisfying Pareto-optimality and independence of irrelevant alternatives. I‘ll show you some solution concepts!

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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-Fran ç ois Laslier (Paris) Nicolas Maudet (Paris) AI TCS AI LOG AI TCS ECON AI ECON TCS TCS LOG TCS AI LOG ECON AI

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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

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7 The Community Where do we meet? International Workshop on Computational Social Choice (COMSOC), coordinated by Ulle Endriss & Jérôme Lang) 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”

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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.

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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‘ll show you how to cut a cake!

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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

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

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12 What did the Düsseldorf Group do in 2009? Magnus Jörg Gábor Frank Doro Edith E. Claudia Düsseldorf Felix Ulle Jeff Piotr Remzi Edith H. Lane Vince Jérôme Yann Nicolas Jean- François 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

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13 What did the Düsseldorf Group do in 2009? Magnus Jörg Gábor Frank Doro Claudia Düsseldorf Felix Ulle Jeff Piotr Remzi Edith H. Lane Vince Edith E. Jérôme Yann Nicolas Jean- François The Complexity of Probabilistic Lobbying. Joint with H. Fernau J. Goldsmith, N. Mattei, D. Raible. To appear at ADT’09

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14 What did the Düsseldorf Group do in 2009? Magnus Jörg Gábor Frank Doro Claudia Düsseldorf Felix Ulle Jeff Piotr Remzi Edith H. Lane Vince Edith E. Jérôme Yann Nicolas Jean- François Generalized Juntas and NP-Hard Sets. Joint with H. Spakowski. FCT’07; Theoretical Computer Science 2009

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15 What did the Düsseldorf Group do in 2009? Magnus Jörg Gábor Frank Doro Claudia Düsseldorf Felix Ulle Jeff Piotr Remzi Edith H. Lane Vince Edith E. Jérôme Yann Nicolas Jean- François Frequency of Correctness versus Average Polynomial Time. Joint with H. Spakowski. FCT’07; Information Processing Letters 2009

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16 What did the Düsseldorf Group do in 2009? Magnus Jörg Gábor Frank Doro Claudia Düsseldorf Felix Ulle Jeff Piotr Remzi Edith H. Lane Vince Edith E. Jérôme Yann Nicolas Jean- François Satisfiability Parsimoniously Reduces to the Tantrix(TM) Rotation Puzzle Problem. MCU’07; Fundamenta Informaticae 2009

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17 What did the Düsseldorf Group do in 2009? Magnus Jörg Gábor Frank Doro Claudia Düsseldorf Felix Ulle Jeff Piotr Remzi Edith H. Lane Vince Edith E. Jérôme Yann Nicolas Jean- François The Three-Color and Two-Color Tantrix(TM) Rotation Puzzle Problems are NP-Complete Via Parsimonious Reductions. LATA’08; Information & Computation 2009

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18 What did the Düsseldorf Group do in 2009? Magnus Jörg Gábor Frank Doro Claudia Düsseldorf Felix Ulle Jeff Piotr Remzi Edith H. Lane Vince Edith E. Jérôme Yann Nicolas Jean- François The Complexity of Computing Minimal Unidirectional Covering Sets. Joint with F. Fischer and J. Hoffman. Submitted

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19 What did the Düsseldorf Group do in 2009? Magnus Jörg Gábor Frank Doro Claudia Düsseldorf Felix Ulle Jeff Piotr Remzi Edith H. Lane Vince Edith E. Jérôme Yann Nicolas Jean- François Degrees of Guaranteed Envy-Freeness in Finite Bounded Cake-Cutting Protocols. WINE’09

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20 What did the Düsseldorf Group do in 2009? Magnus Jörg Gábor Frank Doro Claudia Düsseldorf Felix Ulle Jeff Piotr Remzi Edith H. Lane Vince Edith E. Jérôme Yann Nicolas Jean- François Complexity of Social Welfare Optimization in Multiagent Resource Allocation. Submitted

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21 What did the Düsseldorf Group do in 2009? Magnus Jörg Gábor Frank Doro Claudia Düsseldorf Felix Ulle Jeff Piotr Remzi Edith H. Lane Vince Edith E. Jérôme Yann Nicolas Jean- François Hybrid Elections Broaden Complexity-Theoretic Resistance to Control. IJCAI’07; Mathematical Logic Quarterly 2009

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22 What did the Düsseldorf Group do in 2009? Magnus Jörg Gábor Frank Doro Claudia Düsseldorf Felix Ulle Jeff Piotr Remzi Edith H. Lane Vince Edith E. Jérôme Yann Nicolas Jean- François Llull and Copeland Voting Computationally Resist Bribery and Constructive Control. AAAI’07; AAIM’08; Journal of Artificial Intelligence Research 2009

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23 What did the Düsseldorf Group do in 2009? Magnus Jörg Gábor Frank Doro Claudia Düsseldorf Felix Ulle Jeff Piotr Remzi Edith H. Lane Vince Edith E. Jérôme Yann Nicolas Jean- François 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

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24 What did the Düsseldorf Group do in 2009? Magnus Jörg Gábor Frank Doro Claudia Düsseldorf Felix Ulle Jeff Piotr Remzi Edith H. Lane Vince Edith E. Jérôme Yann Nicolas Jean- François Control Complexity in Fallback Voting. To appear at CATS’10

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25 What did the Düsseldorf Group do in 2009? Magnus Jörg Gábor Frank Doro Claudia Düsseldorf Felix Ulle Jeff Piotr Remzi Edith H. Lane Vince Edith E. Jérôme Yann Nicolas Jean- François 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

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26 What did the Düsseldorf Group do in 2009? Magnus Jörg Gábor Frank Doro Claudia Düsseldorf Felix Ulle Jeff Piotr Remzi Edith H. Lane Vince Edith E. Nicolas Jérôme Yann Jean- François Computational Aspects of Approval Voting. To appear in „Handbook of Approval Voting“ J.-F. Laslier & R. Sanver, eds., Springer

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

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

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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

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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.

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

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

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Elections Candidates and voters: C = {c 1,..., c m } V = {v 1,..., v n } Each voter v i 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 v 7, I hope you are not one of those awful people who support c 3 ! Hi, my name is v 7. How will they aggregate our votes?!

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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.

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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).

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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

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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 Preferences Single-peaked preference consistent with linear order of candidates

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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 Preferences Preference that is inconsistent with linear order of candidates

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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 v i 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.

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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 v i 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.

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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 v i 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 v i approves of both c and e then v i approves of d.

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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 v i 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 v i approves of both c and e then v i approves of d. Theorem 1: Given a collection V of approval vectors 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.

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Control and Manipulation The bad guy wants to make someone win (constructive) or prevent someone from winning (destructive). The bad guy knows everybody else’s votes. In control, the chair modifies an election‘s structure by: Adding candidates (limited/unlimited number) 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. Both nonmanipulators and manipulators are weighted. In the single-peaked case, both nonmanipulators and manipulators are single-peaked w.r.t. the same order L. See Bartholdi, Tovey & Trick (1989; 1992), Conitzer, Sandholm & Lang (2007), Hemaspaandra, Hemaspaandra & Rothe (2007).

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Outline Introduction Elections and Single-Peaked Preferences 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

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Control Results: Approval Theorem 2: For the single-peaked case, approval voting is vulnerable to constructive control by adding voters and constructive control by deleting voters, in the unique-winner and the nonunique-winner model, for the standard and the succinct input model.

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Control Results: Approval Theorem 2: For the single-peaked case, approval voting is vulnerable to constructive control by adding voters and constructive control by deleting voters, in the unique-winner and the nonunique-winner model, for the standard and the succinct input model. For comparison: Among all types of control by adding/deleting candidates/voters, the above two cases are the only two resistances in the general case. (Hemaspaandra, Hemaspaandra & Rothe, AAAI’05; Artificial Intelligence 2007 )

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Control Results: Plurality Theorem 3: For the single-peaked case, plurality voting is vulnerable to constructive and destructive control by adding candidates, adding an unlimited number of candidates, and deleting candidates in the unique-winner and the nonunique-winner model.

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Control Results: Plurality Theorem 3: For the single-peaked case, plurality voting is vulnerable to constructive and destructive control by adding candidates, adding an unlimited number of candidates, and deleting candidates in the unique-winner and the nonunique-winner model. For comparison: For each of these six types of candidate control plurality voting is resistant in the general case, but is vulnerable to the four types of control involving adding/deleting voters. (Bartholdi, Tovey & Trick, 1992; Hemaspaandra, Hemaspaandra & Rothe, 2007)

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Outline Introduction Elections and Single-Peaked Preferences 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

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Manipulation: Removing NP-Hardness Shields Theorem 4: For the single-peaked case, the constructive coalition weighted manipulation problem (in both the unique-winner and the nonunique-winner model) for each of the following election systems is in P: The scoring protocol, i.e., 3-candidate Borda. Each of the scoring protocols,. Veto.

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Manipulation: Removing NP-Hardness Shields Theorem 4: For the single-peaked case, the constructive coalition weighted manipulation problem (in both the unique-winner and the nonunique-winner model) for each of the following election systems is in P: The scoring protocol, i.e., 3-candidate Borda. Each of the scoring protocols,. Veto. For comparison: 3-candidate Borda, Veto, and the „ “ cases of,, are NP-complete in the general case (and the rest is in P). (Hemaspaandra & Hemaspaandra, 2007; Procaccia & Rosenschein, 2007; Conitzer, Sandholm & Lang, 2007).

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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.

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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 ).

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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.

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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.

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

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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. General case Single-peaked case

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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.

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Outline Introduction Elections and Single-Peaked Preferences 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

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A Sample Proof Sketch Theorem 2: For the single-peaked case, approval voting is vulnerable to constructive control by adding voters and constructive control by deleting voters, in the unique-winner and the nonunique-winner model, for the standard and the succinct input model.

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A Sample Proof Sketch Theorem 2: For the single-peaked case, approval voting is vulnerable to constructive control by adding voters and constructive control by deleting voters, in the unique-winner and the nonunique-winner model, for the standard and the succinct input model. We focus on: constructive control by adding voters in the unique-winner model for the succinct input model.

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A Sample Proof Sketch Theorem 2: For the single-peaked case, approval voting is vulnerable to constructive control by adding voters and constructive control by deleting voters, in the unique-winner and the nonunique-winner model, for the standard and the succinct input model. We focus on: constructive control by adding voters in the unique-winner model for the succinct input model. We give a poly-time algorithm that, given collections V and V‘ of votes over candidate set C and single-peaked w.r.t. order L, a candidate p in C, and an addition limit k, decides if by adding at most k votes from V‘ we can make p the unique winner.

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A Sample Proof Sketch 1 1 4 7 3 9 5 2 number of approvals from voters in V for candidates that are votes in V‘ that can be added (with multiplicities)

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A Sample Proof Sketch 1 1 4 7 3 9 5 2 number of approvals from voters in V for candidates that are votes in V‘ that can be added (with multiplicities) Which vote types from V‘ should we add? Especially if they are incomparable?

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

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A Sample Proof Sketch 1 1 4 7 3 9 5 2 number of approvals from voters in V for candidates that are votes in V‘ that can be added (with multiplicities) Why are F, C, B, c, f, and j dangerous but the remaining candidates can be ignored?

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

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

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

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A Sample Proof Sketch 1 1 4 7 3 2 number of approvals from voters in V for candidates that are votes in V‘ that can be added (with multiplicities) Thus, c is a dangerous rival for p but a and b can safely be ignored.

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A Sample Proof Sketch 1 1 4 7 3 2 number of approvals from voters in V for candidates that are votes in V‘ that can be added (with multiplicities) Likewise, f is dangerous but d and e can safely be ignored.

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A Sample Proof Sketch 1 1 4 7 3 2 number of approvals from voters in V for candidates that are votes in V‘ that can be added (with multiplicities) Likewise, j is dangerous but g, h, and i can safely be ignored.

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A Sample Proof Sketch 1 1 4 7 3 2 number of approvals from voters in V for candidates that are votes in V‘ that can be added (with multiplicities) Hey, why do you do that step by step? Just say j is dangerous and ignore a, …, i.

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A Sample Proof Sketch 1 1 4 7 3 2 number of approvals from voters in V for candidates that are votes in V‘ that can be added (with multiplicities) No! Look what happens if we add 6 votes of the type with multiplicity 7!

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A Sample Proof Sketch 1 1 4 1 3 2 votes in V‘ that can be added (with multiplicities) No! Look what happens if we add 6 votes of the type with multiplicity 7!

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A Sample Proof Sketch 1 1 4 7 3 2 number of approvals from voters in V for candidates that are votes in V‘ that can be added (with multiplicities) OK, that‘s not illogical. But how does your „smart greedy“ algorithm work?

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Smart Greedy Algorithm OK, first I need more space for that!

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Smart Greedy Algorithm OK, first I need more space for that! In smart greedy, we eat through all dangerous rivals to the right of p starting with the leftmost: c. To become the unique winner, p must beat c. Only votes in V‘ whose right endpoints fall in can help. Let B be the set of those votes. Choose votes from B starting with the rightmost left endpoint. This is a perfectly safe strategy!

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Smart Greedy Algorithm 1 1 4 7 3 2 votes in V‘ that can be added (with multiplicities)

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Smart Greedy Algorithm 1 1 2 votes in B that can be added (with multiplicities)

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Smart Greedy Algorithm 1 0 2 votes in B that can be added (with multiplicities)

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Smart Greedy Algorithm 1 First rival defeated 1 votes in B that can be added (with multiplicities)

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Smart Greedy Algorithm OK, first I need more space for that! In smart greedy, we eat through all dangerous rivals to the right of p starting with the leftmost: c. To become the unique winner, p must beat c. Only votes in V‘ whose right endpoints fall in can help. Let B be the set of those votes. Choose votes from B starting with the rightmost left endpoint. This is a perfectly safe strategy! Iterate. If you run out of dangerous candidates on the right of p, mirror image the societal order (i.e., reverse L) and finish off the remaining dangerous candidates until you either succeed or reach the addition limit.

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Summary and Future Directions Single-peakedness removes many complexity shields against control and manipulation leaves others in place can even erect complexity shields When choosing election systems for single-peaked electorates, one must not rely on such shields.

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Summary and Future Directions Single-peakedness removes many complexity shields against control and manipulation leaves others in place can even erect complexity shields When choosing election systems for single-peaked electorates, one must not rely on such shields. Do such shield removals hold in two-dimensional (or k-dimensional) analogues of our unidimensional single-peakedness? Can our results be extended to „very nearly“ single-peaked societies?

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... and a Call for (Reading) Papers „Logic and Complexity within Computational Social Choice“ Special issue of Mathematical Logic Quarterly, August 2009 Edited by Paul Goldberg and Jörg Rothe

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... and a Call for (Writing) Papers Third International Workshop on Computational Social Choice Düsseldorf, Germany, September 13–16, 2010 Important Dates Paper submission deadline: June 1, 2010 Notification of authors: July 15, 2010 Camera-ready copies due: August 1, 2010 Early registration deadline: August 1, 2010 Tutorial day: September 13, 2010 Workshop dates: September 14–16, 2010

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