1. Computational Aspects of Approval Voting and Declared-Strategy Voting Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu A Dissertation Proposal 15 March 2007

2. 2 Let’s vote! 45 voters A C B sincere preferences (1st) (2nd) (3rd) 35 voters B C A 20 voters C B A

3. 3 Plurality voting A : 45 votes B : 35 votes C : 20 votes sincere ballots 45 voters A C B 35 voters B C A 20 voters C B A “zero-information” result

4. 4 Plurality voting ballots so far election state 45 voters A C B 35 voters B C A A : 45 votes B : 35 votes C : 0 votes ? 20 voters C B A

5. 5 Plurality voting B : 55 votes A : 45 votes C : 0 votes strategic ballots final election state 45 voters A C B 35 voters B C A 20 voters C B A [Gibbard ’73] [Satterthwaite ’75] insincerity!

6. 6 What is manipulation? B : 55 votes A : 45 votes C : 0 votes 45 voters A C B 35 voters B C A 20 voters C B A BUBU BVBV ballot sets election state

7. 7 Manipulation decision problem Existence of Probably Winning Coalition Ballots (EPWCB) INSTANCE: Set of alternatives A and a distinguished member a of A ; set of weighted cardinal-ratings ballots B V ; the weights of a set of ballots B U which have not been cast; probability QUESTION: Does there exist a way to cast the ballots B U so that a has at least probability of winning the election with the ballots ? My generalization of problems from the literature: [Bartholdi, Tovey & Trick ’89] [Conitzer & Sandholm ’02] [Conitzer & Sandholm ’03]

8. 8 Manipulation decision problem Existence of Probably Winning Coalition Ballots (EPWCB) INSTANCE: Set of alternatives A and a distinguished member a of A ; set of weighted cardinal-ratings ballots B V ; the weights of a set of ballots B U which have not been cast; probability QUESTION: Does there exist a way to cast the ballots B U so that a has at least probability of winning the election with the ballots ? These voters have maximum possible information –They have all the power (if they have smarts too) –If this kind of manipulation is hard, any kind is

9. 9 Manipulation decision problem Existence of Probably Winning Coalition Ballots (EPWCB) INSTANCE: Set of alternatives A and a distinguished member a of A ; set of weighted cardinal-ratings ballots B V ; the weights of a set of ballots B U which have not been cast; probability QUESTION: Does there exist a way to cast the ballots B U so that a has at least probability of winning the election with the ballots ? This problem is computationally easy (in P) for: –plurality voting [Bartholdi, Tovey & Trick ’89] –approval voting

10. 10 Manipulation decision problem Existence of Probably Winning Coalition Ballots (EPWCB) INSTANCE: Set of alternatives A and a distinguished member a of A ; set of weighted cardinal-ratings ballots B V ; the weights of a set of ballots B U which have not been cast; probability QUESTION: Does there exist a way to cast the ballots B U so that a has at least probability of winning the election with the ballots ? This problem is computationally infeasible (NP-hard) for: –Hare [Bartholdi & Orlin ’91] –Borda [Conitzer & Sandholm ’02]

11. 11 What can we do about manipulation? One approach: “tweaks” [Conitzer & Sandholm ’03] –Add an elimination round to an existing protocol –Drawback: alternative symmetry (“fairness”) is lost What if we deal with manipulation by embracing it? –Incorporate strategy into the system –Encourage sincerity as “advice” for the strategy

12. 12 Declared-Strategy Voting [Cranor & Cytron ’96] election state cardinal preferences rational strategizer ballot outcome

13. 13 Declared-Strategy Voting [Cranor & Cytron ’96] election state cardinal preferences rational strategizer ballot outcome Separates how voters feel from how they vote Levels playing field for voters of all sophistications Aim: a voter needs only to give honest preferences sincerity manipulation

14. 14 What is a declared strategy? A : 0.0 B : 0.6 C : 1.0 A : 45 B : 35 C : 0 cardinal preferences current election state declared strategy A : 0 B : 1 C : 0 voted ballot Captures thinking of a rational voter

15. 15 Can DSV be hard to manipulate? I propose to show that DSV can be made to be NP- hard to manipulate (in the EPWCB sense) if a particular declared strategy is imposed on the voters.

16. 16 Favorite vs. compromise, revisited ballots so far election state 45 voters A C B 35 voters B C A A : 45 votes B : 35 votes C : 0 votes ? 20 voters C B A

17. 17 Approval voting [Ottewell ’77] [Weber ’77] [Brams & Fishburn ’78] strategic ballots 45 voters A C B 35 voters B C A 20 voters C B A B : 55 votes A : 45 votes C : 20 votes final election state insincerity avoided

18. 18 Themes of research Approval voting systems Susceptibility to insincere manipulation –encouraging sincere ballots Effectiveness of various strategies Internalizing insincerity –separating manipulation from the voter Complexity issues –complexity of manipulation –complexity of calculating the outcome

19. 19 Strands of proposed research number of alternatives outcome Area of research k = 1an approval rating Voters approve or disapprove a single alternative. What is the equilibrium approval rating? k > 1m = 1 winner Voters elect a winner by approval voting. What DSV-style approval strategies are most effective? k > 1m ≥ 1 winners Voters elect a set of alternatives with approval ballots. Which set most satisfies the least satisfied voter? [Brams, Kilgour & Sanver ’04]

20. 20 Strands of proposed research number of alternatives outcome Area of research k = 1an approval rating Voters approve or disapprove a single alternative. What is the equilibrium approval rating? k > 1m = 1 winner Voters elect a winner by approval voting. What DSV-style approval strategies are most effective? k > 1m ≥ 1 winners Voters elect a set of alternatives with approval ballots. Which set most satisfies the least satisfied voter? [Brams, Kilgour & Sanver ’04]

21. 21 Strands of proposed research number of alternatives outcome Area of research k = 1an approval rating Voters approve or disapprove a single alternative. What is the equilibrium approval rating? k > 1m = 1 winner Voters elect a winner by approval voting. What DSV-style approval strategies are most effective? k > 1m ≥ 1 winners Voters elect a set of alternatives with approval ballots. Which set most satisfies the least satisfied voter? [Brams, Kilgour & Sanver ’04]

22. 22 Approval ratings Voters are asked about one alternative: Approve or disapprove? –like a Presidential approval rating –typically, average is reported Why not allow votes between 0 (full disapproval) and 1 (full approval) and then average them? –like metacritic.com Let’s see what happens when voters are strategic

23. 23 One approach: Average outcome:

24. 24 One approach: Average outcome:

25. 25 Another approach: Median outcome:

26. 26 Another approach: Median outcome:

27. 27 Another approach: Median Nonmanipulable –voter i cannot obtain a better result by voting –if, increasing will not change –if, decreasing will not change Allows tyranny by a majority – –no concession to the 0-voters

28. 28 Average with Declared-Strategy Voting? election state cardinal preferences rational strategizer ballot outcome So Median is far from ideal—what now? –try using Average protocol in DSV context But what’s the rational Average strategy?

29. 29 Rational Average strategy For, voter i should choose to move outcome as close to as possible Choosing would give Optimal vote is After voter i uses this strategy, one of these is true: – and – – and

30. 30 Multiple equilibria are possible outcome in each case: Multiple equilibria always have same average (proof in written proposal)

31. 31 An equilibrium always exists? At equilibrium, must satisfy I propose to prove that, given a vector, at least one equilibrium exists. If an equilibrium always exists, then average at equilibrium can be defined as a function,. Applying to instead of gives a new system, Average-approval-rating DSV.

32. 32 Average-approval-rating DSV outcome:

33. 33 Average-approval-rating DSV outcome:

34. 34 AAR DSV could be manipulated if some voter i could gain an outcome closer to ideal by voting insincerely ( ). I propose to show that Average-approval-rating DSV cannot be manipulated by insincere voters. Average-approval-rating DSV

35. 35 AAR DSV could be manipulated if some voter i could gain an outcome closer to ideal by voting insincerely ( ). I propose to show that Average-approval-rating DSV cannot be manipulated by insincere voters. Intuitively, if, increasing will not change. Average-approval-rating DSV

36. 36 Higher-dimensional outcome space What if votes and outcomes exist in dimensions? Example: If dimensions are independent, Average, Median and Average-approval-rating DSV can operate independently on each dimension –Results from one dimension transfer

37. 37 Higher-dimensional outcome space But what if the dimensions are not independent? –say, outcome space is a disk in the plane: A generalization of Median: the Fermat-Weber point [Weber ’29] –minimizes sum of Euclidean distances between outcome point and voted points –F-W point is computationally infeasible to calculate exactly [Bajaj ’88] (but approximation is easy [Vardi ’01]) –cannot be manipulated by moving a voted point directly away from the F-W point [Small ’90]

38. 38 Higher-dimensional outcome space Average-approval-rating DSV can be generalized –optimal strategy moves the result as close to sincere ideal as possible (by Euclidean distance) I propose to find the optimal strategy for Average in the case and determine whether the resulting DSV system is rotationally invariant and/or nonmanipulable by insincere voters.

39. 39 Strands of proposed research number of alternatives outcome Area of research k = 1an approval rating Voters approve or disapprove a single alternative. What is the equilibrium approval rating? k > 1m = 1 winner Voters elect a winner by approval voting. What DSV-style approval strategies are most effective? k > 1m ≥ 1 winners Voters elect a set of alternatives with approval ballots. Which set most satisfies the least satisfied voter? [Brams, Kilgour & Sanver ’04]

40. 40 Approval strategies for DSV Rational plurality strategy has been well explored [Cranor & Cytron, ’96] But what about approval strategy? If each alternative’s probability of winning is known, optimal strategy can be computed [Merrill ’88] But what about in a DSV context? –have only a vote total for each alternative Let’s look at several approval strategies and approaches to evaluating their effectiveness

41. 41 DSV-style approval strategies Strategy Z [Merrill ’88] : –Approve alternatives with higher-than-average cardinal preference (zero-information strategy) Z recommends:

42. 42 DSV-style approval strategies Strategy T [Ossipoff ’02] : –Approve favorite of top two vote-getters, plus all liked more T recommends:

43. 43 DSV-style approval strategies Strategy J [Brams & Fishburn ’83] : –Use strategy Z if it distinguishes between top two vote- getters; otherwise use strategy T J recommends:

44. 44 DSV-style approval strategies Strategy A: –Approve all preferred to top vote-getter, plus top vote- getter if preferred to second-highest vote-getter But how to evaluate these strategies? A recommends:

45. 45 Election-state-evaluation approaches Evaluate a declared strategy by evaluating the election states that are immediately obtained Calculate expected value of an election state by estimating each alternative’s probability of eventually winning How to calculate those probabilities?

46. 46 Election-state-evaluation: Merrill metric Estimate an alternative’s probability of winning to be proportional to its current vote total raised to some power x [Merrill ’88]

47. 47 Strategy comparison using the Merrill metric Current election state Focal voter’s preferences [1, 0, 0] (strategies A & T) [1, 0, 0] (A & T) [0, 1, 0] (A & T) [0, 1, 1] (A); [0, 1, 0] (T) [1, 0, 1] (A & T) [0, 1, 1] (A & T)

48. 48 Strategy comparison using the Merrill metric Current election state Focal voter’s preferences [0, 1, 1] (A) [0, 1, 0] (T) expected values of possible next election states:

49. 49 Strategy comparison using the Merrill metric Current election state Focal voter’s preferences so T is better than A only when: or, equivalently:

50. 50 Strategy comparison using the Merrill metric Current election state Focal voter’s preferences so T is better than A only when: or, equivalently: Intuitively, T does better than A only when: s 1 and s 2 are relatively close x is relatively small p 3 is relatively close to p 1 compared to p 2

51. 51 Strategy comparison using the Merrill metric Current election state Focal voter’s preferences Corollaries: –When x is taken to infinity and, strategy A dominates strategy T –When, strategy A dominates strategy T T is better than A only when:

52. 52 Approval strategy evaluation I propose to extend this 3-alternative result to strategy pairs A vs. J, T vs. J and A vs. Z. I propose to extend this result to strategy pairs A vs. T and A vs. J in the 4-alternative case.

53. 53 Further result for strategy A More generally, it is true that if –the election state is free of ties and near-ties: –and the focal voter’s cardinal preferences are tie-free: when –and the Merrill-metric exponent x is taken to infinity then strategy A dominates all other strategies according to the Merrill metric (proof in written proposal)

54. 54 Election-state-evaluation: Branching-probabilities metric Estimate an alternative’s probability of winning by looking ahead Assume that the probability that alternative a is approved on each future ballot is equal to the proportion of already-voted ballots that approve a

55. 55 Approval strategy evaluation I propose to extend the Merrill-metric results to strategy pairs A vs. T, A vs. J, T vs. J and A vs. Z in the 3-alternative case using the branching- probabilities metric. I propose to determine whether strategy A dominates all others in the near-tie-free case using the branching-probabilities metric as the number of future ballots goes to infinity.

56. 56 Strands of proposed research number of alternatives outcome Area of research k = 1an approval rating Voters approve or disapprove a single alternative. What is the equilibrium approval rating? k > 1m = 1 winner Voters elect a winner by approval voting. What DSV-style approval strategies are most effective? k > 1m ≥ 1 winners Voters elect a set of alternatives with approval ballots. Which set most satisfies the least satisfied voter? [Brams, Kilgour & Sanver ’04]

57. 57 Electing a committee from approval ballots 11110 00011 00111 0000110111 01111 What’s the best committee of size m = 2? approves of candidates 4 and 5 k = 5 candidates n = 6 ballots

58. 58 Sum of Hamming distances 11110 00011 00111 0000110111 0111111000 45 24 43 sum = 22 m = 2 winners

59. 59 Fixed-size minisum 11110 00011 00111 0000110111 0111100011 Minisum elects winner set with smallest sumscore Easy to compute (pick candidates with most approvals) 21 40 21 sum = 10 m = 2 winners

60. 60 Maximum Hamming distance 11110 00011 00111 0000110111 0111100011 21 40 21 sum = 10 max = 4 m = 2 winners

61. 61 Fixed-size minimax Minimax elects winner set with smallest maxscore Harder to compute? 11110 00011 00111 0000110111 0111100110 21 22 23 sum = 12 max = 3 m = 2 winners [Brams, Kilgour & Sanver ’04]

62. 62 Complexity Endogenous minimax = EM = BSM(0, k) Bounded-size minimax = BSM(m 1, m 2 ) Fixed-size minimax = FSM(m) = BSM(m, m) NP-hard [Frances & Litman ’97] NP-hard (generalization of EM) ?

63. 63 Complexity Endogenous minimax = EM = BSM(0, k) Bounded-size minimax = BSM(m 1, m 2 ) Fixed-size minimax = FSM(m) = BSM(m, m) NP-hard [Frances & Litman ’97] NP-hard (generalization of EM) NP-hard (proof in written proposal)

64. 64 Approximability Endogenous minimax = EM = BSM(0, k) Bounded-size minimax = BSM(m 1, m 2 ) Fixed-size minimax = FSM(m) = BSM(m, m) has a PTAS* [Li, Ma & Wang ’99] no known PTAS * Polynomial-Time Approximation Scheme: algorithm with approx. ratio 1 + ε that runs in time polynomial in the input and exponential in 1/ε

65. 65 Approximability Endogenous minimax = EM = BSM(0, k) Bounded-size minimax = BSM(m 1, m 2 ) Fixed-size minimax = FSM(m) = BSM(m, m) has a PTAS* [Li, Ma & Wang ’99] no known PTAS; has a 3-approx. (proof in written proposal) no known PTAS; has a 3-approx. (proof in written proposal) * Polynomial-Time Approximation Scheme: algorithm with approx. ratio 1 + ε that runs in time polynomial in the input and exponential in 1/ε

66. 66 Approximating FSM 00111 00001 10111 01111 00011 11110 00111 m = 2 winners choose a ballot arbitrarily

67. 67 Approximating FSM 00111 00001 10111 01111 00011 11110 0010100111 coerce to size m m = 2 winners choose a ballot arbitrarily outcome = m-completed ballot

68. 68 Approximation ratio ≤ 3 00111 00001 10111 01111 00011 11110 00110 2 2 1 3 2 2 ≤ OPT optimal FSM set OPT = optimal maxscore

69. 69 Approximation ratio ≤ 3 00111 00001 10111 01111 00011 11110 0011000111 2 2 1 3 2 2 1 ≤ OPT optimal FSM set chosen ballot OPT = optimal maxscore

70. 70 Approximation ratio ≤ 3 00111 00001 10111 01111 00011 11110 001100011100011 2 2 1 3 2 2 11 ≤ OPT ≤ 3·OPT optimal FSM set chosen ballot m-completed ballot OPT = optimal maxscore (by triangle inequality)

71. 71 Better in practice? So far, we can guarantee a winner set no more than 3 times as bad as the optimal. –Nice in theory... How can we do better in practice? –Try local search

72. 72 Local search approach for FSM 1.Start with some c  {0,1} k of weight m 01001 4

73. 73 Local search approach for FSM 1.Start with some c  {0,1} k of weight m 2.In c, swap up to r 0-bits with 1-bits in such a way that minimizes the maxscore of the result 01001 1100010001 01100 0101000011 00101 4 44 4 5 4 4

74. 74 Local search approach for FSM 1.Start with some c  {0,1} k of weight m 2.In c, swap up to r 0-bits with 1-bits in such a way that minimizes the maxscore of the result 01010 4

75. 75 Local search approach for FSM 1.Start with some c  {0,1} k of weight m 2.In c, swap up to r 0-bits with 1-bits in such a way that minimizes the maxscore of the result 01010 4

76. 76 Local search approach for FSM 1.Start with some c  {0,1} k of weight m 2.In c, swap up to r 0-bits with 1-bits in such a way that minimizes the maxscore of the result 3.Repeat step 2 until maxscore(c) is unchanged k times 4.Take c as the solution 01010 1100010010 01100 0100100011 00110 4 44 4 5 3 4

77. 77 Local search approach for FSM 1.Start with some c  {0,1} k of weight m 2.In c, swap up to r 0-bits with 1-bits in such a way that minimizes the maxscore of the result 3.Repeat step 2 until maxscore(c) is unchanged k times 4.Take c as the solution 00110 3

78. 78 Heuristic evaluation Parameters: –starting point of search –radius of neighborhood Ran heuristics on generated and real-world data All heuristics perform near-optimally –highest approx. ratio found: 1.2 –highest average ratio < 1.04 The fixed-size-minisum starting point performs best overall (with our 3-approx. a close second) When neighborhood radius is larger, performance improves and running time increases (maxscore of solution found) (maxscore of exact solution)

79. 79 Manipulating FSM 00110 00011 00111 0000110111 0111100011 Voters are sincere Another optimal solution: 00101 21 20 21 max = 2 m = 2 winners

80. 80 Manipulating FSM 11110 00011 00111 0000110111 0111100110 A voter manipulates and realizes ideal outcome But our 3-approximation for FSM is nonmanipulable 21 22 23 00110 0 max = 3 m = 2 winners

81. 81 Fixed-size Minimax contributions BSM and FSM are NP-hard Both can be approximated with ratio 3 Polynomial-time local search heuristics perform well in practice –some retain ratio-3 guarantee Exact FSM can be manipulated Our 3-approximation for FSM is nonmanipulable

82. 82 Progress so far Area of researchState of progress Approval rating Completed: rational Average strategy, equality of average at equilibria To do: equilibrium always exists, nonmanipulability of AAR DSV, analysis of Average in planar disk DSV-style approval strategies Completed: comparison of A and T in 3-alt. case, domination of A as To do: comparisons of other pairs, analysis using branching-probabilities metric Fixed-size minimax Completed: NP-hardness proof, 3-approximation, heuristic evaluation, manipulability analysis

83. 83 Fin Thanks to –my adviser, Ron Cytron –Morgan Deters and the rest of the DOC Group –co-authors Vangelis Markakis and Aranyak Mehta –my committee Questions?

84. 84 What happens at equilibrium? The optimal strategy recommends that no voter change So And –equivalently, Therefore any average at equilibrium must satisfy two equations: –(A) –(B)

85. 85 Proof: Only one equilibrium average Theorem: Proof considers two symmetric cases: –assume Each leads to a contradiction

86. 86 Proof: Only one equilibrium average case 1:, contradicting

87. 87 Proof: Only one equilibrium average Case 1 shows that Case 2 is symmetrical and shows that Therefore Therefore, given, the average at equilibrium is unique

88. 88 Specific FSM heuristics Two parameters: –where to start vector c: 1.a fixed-size-minisum solution 2.a m-completion of a ballot (3-approx.) 3.a random set of m candidates 4.a m-completion of a ballot with highest maxscore –radius of neighborhood r: 1 and 2

89. 89 Heuristic evaluation Real-world ballots from GTS 2003 council election Found exact minimax solution Ran each heuristic 5000 times Compared exact minimax solution with heuristics to find realized approximation ratios –example: 15/14 = 1.0714 maxscore of solution found = 15 maxscore of exact solution = 14 We also performed experiments using ballots generated according to random distributions (see paper)

90. 90 Average approx. ratios found radius = 1radius = 2 fixed-size minimax 1.00121.0000 3-approx. 1.00171.0000 random set 1.00571.0000 highest- maxscore 1.00591.0000 performance on GTS ’03 election data k = 24 candidates, m = 12 winners, n = 161 ballots

91. 91 Largest approx. ratios found radius = 1radius = 2 fixed-size minimax 1.07141.0000 3-approx. 1.07141.0000 random set 1.07141.0000 highest- maxscore 1.07141.0000 performance on GTS ’03 election data k = 24 candidates, m = 12 winners, n = 161 ballots

92. 92 Conclusions from all experiments All heuristics perform near-optimally –highest ratio found: 1.2 –highest average ratio < 1.04 When radius is larger, performance improves and running time increases The fixed-size-minisum starting point performs best overall (with our 3-approx. a close second)