1. 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, Krakow Jörg Rothe Heinrich-Heine-Universität Düsseldorf Lane A. Hemaspaandra University of Rochester Edith Hemaspaandra Rochester Institute of Technology Moscow, SCW 2010 1

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 2

3. 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-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

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

5. 5 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

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. 6

7. 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. 7

8. 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 7

9. Elections  An election is a pair (C,V) with candidate set C = {c 1,..., c m }: and a list of votes V = (v 1,..., v n ):  Each vote v i 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 v 7, I hope you are not one of those awful people who support Mr. Smith! Hi, my name is v 7. How will they aggregate our votes?! 8

10. Election Systems  Approval (any number of candidates): Every vote is an approval vector from All candidates with the most points are winners.  Example: v1v1 11001 v2v2 01100 v3v3 11001 v4v4 00010 v5v5 10011 v6v6 10001 9

11. Election Systems  Approval (any number of candidates): Every vote is an approval vector from All candidates with the most points are winners.  Example: v1v1 11001 v2v2 01100 v3v3 11001 v4v4 00010 v5v5 10011 v6v6 10001 ∑ 43124 9

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: v1v1 11001 v2v2 01100 v3v3 11001 v4v4 00010 v5v5 10011 v6v6 10001 ∑ 43124 9

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): 9

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). 10

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 10

16.  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 10

17.  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 10

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 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. 10

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 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. 10

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 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. 10

21. 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.  Fulkerson & Gross (1965); Booth & Lueker (1976): 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. 10

22. 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). 11

23. 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 12

24. 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. 13

25. 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 either candidates or voters, the above two cases are the only two resistances in the general case. (Hemaspaandra, Hemaspaandra & Rothe, AAAI’05; Artificial Intelligence 2007) 13

26. 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: 13 Approval Voting (general case) constructivedestructive Adding Candidates (limited) ImmuneVulnerable Adding Candidates (unlimited) ImmuneVulnerable Deleting Candidates VulnerableImmune Adding Voters ResistantVulnerable Deleting Voters ResistantVulnerable

27. 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. 14

28. 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) 14

29. 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: 14 Plurality (general case) constructivedestructive Adding Candidates (limited) Resistant Adding Candidates (unlimited) Resistant Deleting Candidates Resistant Adding Voters Vulnerable Deleting Voters Vulnerable

30. 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 15

31. 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. 16

32. 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). 16

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. 17

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 ). 18

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. 19

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. 19

37. Manipulation: Erecting NP-Hardness Shields  Can restricting to single-peaked preferences ever erect a complexity shield? General case Single-peaked case 20

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. General case Single-peaked case 20

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. 21

40. 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 22

41. 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. 23

42. 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. 23

43. 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 give a poly-time algorithm that, given collections V and W of votes over candidate set C and single-peaked w.r.t. order L, a designated candidate p in C, and an addition limit k, decides if by adding at most k votes from W we can make p the unique winner. 23

44. 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 W that can be added (with multiplicities) 24

45. 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 W that can be added (with multiplicities) Which vote types from W should we add? Especially if they are incomparable? 24

46. 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 W that can be added (with multiplicities) We‘ll handle this by a „smart greedy“ algorithm. 24

47. 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 W that can be added (with multiplicities) Why are F, C, B, c, f, and j dangerous but the remaining candidates can be ignored? 24

48. 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 W that can be added (with multiplicities) First, each added vote will be an interval including p. So drop all others. 24

49. A Sample Proof Sketch 1 1 4 7 3 2 number of approvals from voters in V for candidates that are votes in W that can be added (with multiplicities) First, each added vote will be an interval including p. So drop all others. 24

50. A Sample Proof Sketch 1 1 4 7 3 2 number of approvals from voters in V for candidates that are votes in W that can be added (with multiplicities) Now, if adding votes from W causes p to beat c then p must also beat a and b. 24

51. A Sample Proof Sketch 1 1 4 7 3 2 number of approvals from voters in V for candidates that are votes in W that can be added (with multiplicities) Thus, c is a dangerous rival for p but a and b can safely be ignored. 24

52. A Sample Proof Sketch 1 1 4 7 3 2 number of approvals from voters in V for candidates that are votes in W that can be added (with multiplicities) Likewise, f is dangerous but d and e can safely be ignored. 24

53. A Sample Proof Sketch 1 1 4 7 3 2 number of approvals from voters in V for candidates that are votes in W that can be added (with multiplicities) Likewise, j is dangerous but g, h, and i can safely be ignored. 24

54. A Sample Proof Sketch 1 1 4 7 3 2 number of approvals from voters in V for candidates that are votes in W that can be added (with multiplicities) Hey, why do you do that step by step? Just say j is dangerous and ignore a, …, i. 24

55. A Sample Proof Sketch 1 1 4 7 3 2 number of approvals from voters in V for candidates that are votes in W that can be added (with multiplicities) No! Look what happens if we add 6 votes of the type with multiplicity 7! 24

56. A Sample Proof Sketch 1 1 4 1 3 2 votes in W that can be added (with multiplicities) No! Look what happens if we add 6 votes of the type with multiplicity 7! 24

57. A Sample Proof Sketch 1 1 4 7 3 2 number of approvals from voters in V for candidates that are votes in W that can be added (with multiplicities) OK, that‘s not illogical. But how does your „smart greedy“ algorithm work? 24

58. Smart Greedy Algorithm  OK, first I need more space for that! 25

59. 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 W whose right endpoints fall in [p,c) 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! 25

60. Smart Greedy Algorithm 1 1 4 7 3 2 votes in W that can be added (with multiplicities) 25

61. Smart Greedy Algorithm 1 1 2 votes in B that can be added (with multiplicities) 25

62. Smart Greedy Algorithm 1 0 2 votes in B that can be added (with multiplicities) 25

63. Smart Greedy Algorithm 1 First rival defeated 1 votes in B that can be added (with multiplicities) 25

64. 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 W whose right endpoints fall in [p,c) 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. 25

65. 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. 26

66. 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? 26

67. COMSOC 2010 Third International Workshop on Computational Social Choice Düsseldorf, Germany, September 13–16, 2010 Important Dates Paper submission deadline: May 15, 2010 Notification of authors: July 1, 2010 Camera-ready copies due: July 15, 2010 Early registration deadline: July 15, 2010 LogICCC Tutorial day: September 13, 2010 Workshop dates: September 14–16, 2010 28

68. Thank you! Stop it! No more questions please! 29