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Computational Aspects of Approval Voting and Declared-Strategy Voting A Dissertation Proposal 15 March 2007 Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu Let’s vote! 45 voters 35 voters 20 voters A B C (1st) sincere preferences C C B (2nd) B A A (3rd) 2 Plurality voting 45 voters 35 voters 20 voters A B C sincere ballots C C B B A A A: 45 votes “zero-information” result B: 35 votes C: 20 votes 3 Plurality voting 45 voters 35 voters 20 voters A B C ballots ? so far C C B B A A A: 45 votes election state B: 35 votes C: 0 votes 4 Plurality voting 45 voters 35 voters 20 voters A B C strategic ballots C C B insincerity! B A A B: 55 votes final [Gibbard ’73] election A: 45 votes [Satterthwaite ’75] state C: 0 votes 5 What is manipulation? 45 voters 35 voters 20 voters A B C ballot sets C C B BV B A A BU B: 55 votes election state A: 45 votes C: 0 votes 6 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 BV; the weights of a set of ballots BU which have not been cast; probability 0 1 QUESTION: Does there exist a way to cast the ballots BU so that a has at least probability of winning the election with the ballots BV BU? • My generalization of problems from the literature: [Bartholdi, Tovey & Trick ’89] [Conitzer & Sandholm ’02] [Conitzer & Sandholm ’03] 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 BV; the weights of a set of ballots BU which have not been cast; probability 0 1 QUESTION: Does there exist a way to cast the ballots BU so that a has at least probability of winning the election with the ballots BV BU? • 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 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 BV; the weights of a set of ballots BU which have not been cast; probability 0 1 QUESTION: Does there exist a way to cast the ballots BU so that a has at least probability of winning the election with the ballots BV BU? • This problem is computationally easy (in P) for: – plurality voting [Bartholdi, Tovey & Trick ’89] – approval voting 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 BV; the weights of a set of ballots BU which have not been cast; probability 0 1 QUESTION: Does there exist a way to cast the ballots BU so that a has at least probability of winning the election with the ballots BV BU? • This problem is computationally infeasible (NP-hard) for: – Hare [Bartholdi & Orlin ’91] – Borda [Conitzer & Sandholm ’02] 10 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 11 Declared-Strategy Voting [Cranor & Cytron ’96] cardinal rational preferences strategizer ballot election outcome state 12 Declared-Strategy Voting [Cranor & Cytron ’96] sincerity manipulation cardinal rational preferences strategizer ballot election outcome state • 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 13 What is a declared strategy? A: 0.0 cardinal B: 0.6 preferences C: 1.0 A: 0 declared voted strategy B: 1 ballot current A: 45 C: 0 election B: 35 state C: 0 • Captures thinking of a rational voter 14 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. 15 Favorite vs. compromise, revisited 45 voters 35 voters 20 voters A B C ballots ? so far C C B B A A A: 45 votes election state B: 35 votes C: 0 votes 16 Approval voting [Ottewell ’77] [Weber ’77] [Brams & Fishburn ’78] 45 voters 35 voters 20 voters A B C insincerity strategic avoided ballots C C B B A A B: 55 votes final election A: 45 votes state C: 20 votes 17 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 18 Strands of proposed research number of outcome Area of research alternatives k=1 an approval Voters approve or disapprove a rating single alternative. What is the equilibrium approval rating? k>1 m=1 Voters elect a winner by approval winner voting. What DSV-style approval strategies are most effective? k>1 m≥1 Voters elect a set of alternatives winners with approval ballots. Which set most satisfies the least satisfied voter? [Brams, Kilgour & Sanver ’04] 19 Strands of proposed research number of outcome Area of research alternatives k=1 an approval Voters approve or disapprove a rating single alternative. What is the equilibrium approval rating? k>1 m=1 Voters elect a winner by approval winner voting. What DSV-style approval strategies are most effective? k>1 m≥1 Voters elect a set of alternatives winners with approval ballots. Which set most satisfies the least satisfied voter? [Brams, Kilgour & Sanver ’04] 20 Strands of proposed research number of outcome Area of research alternatives k=1 an approval Voters approve or disapprove a rating single alternative. What is the equilibrium approval rating? k>1 m=1 Voters elect a winner by approval winner voting. What DSV-style approval strategies are most effective? k>1 m≥1 Voters elect a set of alternatives winners with approval ballots. Which set most satisfies the least satisfied voter? [Brams, Kilgour & Sanver ’04] 21 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 22 One approach: Average r 0, .1, .2, .6, .9 v 0, .1, .2, .6, .9 outcome: f avg (v ) .36 .36 0 1 23 One approach: Average r 0, .1, .2, .6, .9 v 0, .1, .2, 1, .9 outcome: f avg (v ) .44 .44 0 1 24 Another approach: Median r 0, .1, .2, .6, .9 v 0, .1, .2, .6, .9 outcome: f med (v ) .2 .2 0 1 25 Another approach: Median r 0, .1, .2, .6, .9 v 0, .1, .2, 1, .9 outcome: f med (v ) .2 .2 0 1 26 Another approach: Median • Nonmanipulable – voter i cannot obtain a better result by voting vi ri – if f med (v ) vi , increasing vi will not change f med (v ) – if f med (v ) vi , decreasing vi will not change f med (v ) • Allows tyranny by a majority – v 0, 0, 0,1,1,1,1 – f med (v ) 1 – no concession to the 0-voters 27 Average with Declared-Strategy Voting? • So Median is far from ideal—what now? – try using Average protocol in DSV context cardinal rational preferences strategizer ballot election outcome state • But what’s the rational Average strategy? 28 Rational Average strategy • For 1 i n, voter i should choose vi to move outcome as close to ri as possible • Choosing vi ri n j i v j would give f avg (v ) ri • Optimal vote is vi min(max(ri n j i v j , 0),1) • After voter i uses this strategy, one of these is true: – f avg (v ) ri and vi 1 – f avg (v ) ri – f avg (v ) ri and vi 0 29 Multiple equilibria are possible r .2, .3, .5, .5, .8 v 0, 0, .5, 1, 1 v 0, 0, .6, .9, 1 v 0, 0, .75, .75, 1 outcome in each case: f avg (v ) .5 Multiple equilibria always have same average (proof in written proposal) 30 An equilibrium always exists? • At equilibrium, v must satisfy (i) vi min(max(ri n j i v j , 0),1) I propose to prove that, given a vector r , at least one equilibrium exists. • If an equilibrium always exists, then average at equilibrium can be defined as a function, f aveq (r ) . • Applying f aveq to v instead of r gives a new system, Average-approval-rating DSV. 31 Average-approval-rating DSV r 0, .1, .2, .6, .9 v 0, .1, .2, .6, .9 outcome: f aveq (v ) .4 .4 0 1 32 Average-approval-rating DSV r 0, .1, .2, .6, .9 v 0, .1, .2, 1, .9 outcome: f aveq (v ) .4 .4 0 1 33 Average-approval-rating DSV • AAR DSV could be manipulated if some voter i could gain an outcome closer to ideal by voting insincerely ( vi ri ). I propose to show that Average-approval-rating DSV cannot be manipulated by insincere voters. 34 Average-approval-rating DSV • AAR DSV could be manipulated if some voter i could gain an outcome closer to ideal by voting insincerely ( vi ri ). I propose to show that Average-approval-rating DSV cannot be manipulated by insincere voters. • Intuitively, if f aveq (v ) vi , increasing vi will not change f aveq (v ). 35 Higher-dimensional outcome space • What if votes and outcomes exist in d 1 dimensions? • Example: x, y 2 : 0 x 1 0 y 1 • If dimensions are independent, Average, Median and Average-approval-rating DSV can operate independently on each dimension – Results from one dimension transfer 36 Higher-dimensional outcome space • But what if the dimensions are not independent? – say, outcome space is a disk in the plane: x, y : x2 y2 1 2 • 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] 37 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 x, y 2 : x 2 y 2 1 case and determine whether the resulting DSV system is rotationally invariant and/or nonmanipulable by insincere voters. 38 Strands of proposed research number of outcome Area of research alternatives k=1 an approval Voters approve or disapprove a rating single alternative. What is the equilibrium approval rating? k>1 m=1 Voters elect a winner by approval winner voting. What DSV-style approval strategies are most effective? k>1 m≥1 Voters elect a set of alternatives winners with approval ballots. Which set most satisfies the least satisfied voter? [Brams, Kilgour & Sanver ’04] 39 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 40 DSV-style approval strategies • Strategy Z [Merrill ’88]: – Approve alternatives with higher-than-average cardinal preference (zero-information strategy) s [30, 25,15,10] p [0, 1, .8, .3] Z recommends: b [0, 1, 1, 0] 41 DSV-style approval strategies • Strategy T [Ossipoff ’02]: – Approve favorite of top two vote-getters, plus all liked more s [30, 25,15,10] p [0, 1, .8, .3] T recommends: b [0, 1, 0, 0] 42 DSV-style approval strategies • Strategy J [Brams & Fishburn ’83]: – Use strategy Z if it distinguishes between top two vote- getters; otherwise use strategy T s [30, 25,15,10] p [0, 1, .8, .3] J recommends: b [0, 1, 1, 0] 43 DSV-style approval strategies • Strategy A: – Approve all preferred to top vote-getter, plus top vote- getter if preferred to second-highest vote-getter s [30, 25,15,10] p [0, 1, .8, .3] A recommends: b [0, 1, 1, 1] But how to evaluate these strategies? 44 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? 45 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] x s w i i k sj j 1 46 Strategy comparison using the Merrill metric Current election state s [ s1 , s2 , s3 ] s1 s2 s3 Focal voter’s preferences p [ p1 , p2 , p3 ] p1 p2 p3 [1, 0, 0] (strategies A & T) p1 p3 p2 [1, 0, 0] (A & T) p2 p1 p3 [0, 1, 0] (A & T) p2 p3 p1 [0, 1, 1] (A); [0, 1, 0] (T) p3 p1 p2 [1, 0, 1] (A & T) p3 p2 p1 [0, 1, 1] (A & T) 47 Strategy comparison using the Merrill metric Current election state s [ s1 , s2 , s3 ] s1 s2 s3 Focal voter’s preferences p [ p1 , p2 , p3 ] p2 p3 p1 expected values of possible next election states: p1s1x p2 s2 1 p3 s3 1 x x V[ 0,1,1] [0, 1, 1] (A) s1 x s2 1x s3 1x p1s1x p2 s2 1 p3 s3x x V[ 0,1,0] [0, 1, 0] (T) s1x s2 1 s3x x 48 Strategy comparison using the Merrill metric Current election state s [ s1 , s2 , s3 ] s1 s2 s3 Focal voter’s preferences p [ p1 , p2 , p3 ] p2 p3 p1 so T is better than A only when: p1s1x p2 s2 1 p3 s3 1 p1s1x p2 s2 1 p3 s3 x x x x s1 s2 1 s3 1 s1x s2 1 s3 x x x x x or, equivalently: x p2 p3 s1 s 1 p3 p1 2 49 Strategy comparison using the Merrill metric Current election state s [ s1 , s2 , s3 ] s1 s2 s3 Focal voter’s preferences p [ p1 , p2 , p3 ] p2 p3 p1 so T is better than A only when: p1s1x p2 s2 1 p3 s3 1 p1s1x p2 s2 1 p3 s3 x x x x s1 s2 1 s3 1 s1x s2 1 s3 x x x x x or, equivalently: x p2 p3 s1 Intuitively, T does better than A only when: s 1 • s1 and s2 are relatively close p3 p1 2 • x is relatively small • p3 is relatively close to p1 compared to p2 50 Strategy comparison using the Merrill metric Current election state s [ s1 , s2 , s3 ] s1 s2 s3 Focal voter’s preferences p [ p1 , p2 , p3 ] p2 p3 p1 x T is better than A only when: p2 p3 s1 s 1 p3 p1 2 Corollaries: – When x is taken to infinity and s1 s2 1, strategy A dominates strategy T – When p1 p2 p3 , strategy A dominates strategy T 2 51 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. 52 Further result for strategy A More generally, it is true that if – the election state is free of ties and near-ties: s1 s2 1 s3 2 sk k 1 – and the focal voter’s cardinal preferences are tie-free: pi p j when i j – 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) 53 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 p1 p2 k p iB i 1 p2 54 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. 55 Strands of proposed research number of outcome Area of research alternatives k=1 an approval Voters approve or disapprove a rating single alternative. What is the equilibrium approval rating? k>1 m=1 Voters elect a winner by approval winner voting. What DSV-style approval strategies are most effective? k>1 m≥1 Voters elect a set of alternatives winners with approval ballots. Which set most satisfies the least satisfied voter? [Brams, Kilgour & Sanver ’04] 56 Electing a committee from approval ballots approves of k = 5 candidates 11110 00011 candidates 4 and 5 n = 6 ballots 01111 00111 10111 00001 •What’s the best committee of size m = 2? 57 Sum of Hamming distances m = 2 winners 11110 00011 2 4 4 5 01111 11000 00111 4 3 sum = 22 10111 00001 58 Fixed-size minisum m = 2 winners 11110 00011 4 0 2 1 01111 00011 00111 2 1 sum = 10 10111 00001 •Minisum elects winner set with smallest sumscore •Easy to compute (pick candidates with most approvals) 59 Maximum Hamming distance m = 2 winners 11110 00011 4 0 2 1 01111 00011 00111 2 1 sum = 10 max = 4 10111 00001 60 Fixed-size minimax [Brams, Kilgour & Sanver ’04] m = 2 winners 11110 00011 2 2 2 1 01111 00110 00111 2 3 sum = 12 max = 3 10111 00001 •Minimax elects winner set with smallest maxscore •Harder to compute? 61 Complexity Endogenous minimax Bounded-size minimax Fixed-size minimax = EM = BSM(0, k) = BSM(m1, m2) = FSM(m) = BSM(m, m) NP-hard NP-hard ? [Frances & Litman ’97] (generalization of EM) 62 Complexity Endogenous minimax Bounded-size minimax Fixed-size minimax = EM = BSM(0, k) = BSM(m1, m2) = FSM(m) = BSM(m, m) NP-hard NP-hard NP-hard [Frances & Litman ’97] (generalization of EM) (proof in written proposal) 63 Approximability Endogenous minimax Bounded-size minimax Fixed-size minimax = EM = BSM(0, k) = BSM(m1, m2) = FSM(m) = BSM(m, m) has a PTAS* no known PTAS no known PTAS [Li, Ma & Wang ’99] * Polynomial-Time Approximation Scheme: algorithm with approx. ratio 1 + ε that runs in time polynomial in the input and exponential in 1/ε 64 Approximability Endogenous minimax Bounded-size minimax Fixed-size minimax = EM = BSM(0, k) = BSM(m1, m2) = FSM(m) = BSM(m, m) has a PTAS* no known PTAS; no known PTAS; has a 3-approx. has a 3-approx. [Li, Ma & Wang ’99] (proof in written (proof in written proposal) proposal) * Polynomial-Time Approximation Scheme: algorithm with approx. ratio 1 + ε that runs in time polynomial in the input and exponential in 1/ε 65 Approximating FSM 11110 m = 2 winners 00011 00111 00111 00001 choose a ballot 10111 arbitrarily 01111 66 Approximating FSM 11110 m = 2 winners 00011 00111 coerce to 00111 00101 00001 size m choose a ballot 10111 arbitrarily 01111 outcome = m-completed ballot 67 Approximation ratio ≤ 3 optimal 11110 2 FSM set 00011 2 00111 1 00110 3 00001 2 10111 2 01111 ≤ OPT OPT = optimal maxscore 68 Approximation ratio ≤ 3 optimal chosen 11110 2 FSM set ballot 00011 2 00111 1 1 00110 00111 3 00001 2 10111 2 01111 ≤ OPT ≤ OPT OPT = optimal maxscore 69 Approximation ratio ≤ 3 optimal chosen m-completed 11110 2 FSM set ballot ballot 00011 2 00111 1 1 1 00110 00111 00011 3 00001 2 10111 2 01111 ≤ OPT ≤ OPT ≤ OPT (by triangle inequality) OPT = optimal maxscore ≤ 3·OPT 70 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 71 Local search approach for FSM 1. Start with some c {0,1}k of weight m 01001 4 72 Local search approach for FSM 1. Start with some c {0,1}k of weight m 11000 10001 2. In c, swap up to r 0-bits 5 4 with 1-bits in such a way 01100 01001 00101 that minimizes the 4 4 4 maxscore of the result 01010 00011 4 4 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 01010 4 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 01010 that minimizes the 4 maxscore of the result 75 Local search approach for FSM 1. Start with some c {0,1}k of weight m 11000 10010 2. In c, swap up to r 0-bits 5 4 with 1-bits in such a way 01100 01010 00110 that minimizes the 4 4 3 maxscore of the result 01001 00011 3. Repeat step 2 until 4 4 maxscore(c) is unchanged k times 4. Take c as the solution 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 00110 that minimizes the 3 maxscore of the result 3. Repeat step 2 until maxscore(c) is unchanged k times 4. Take c as the solution 77 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 (maxscore of solution found) – highest average ratio < 1.04 (maxscore of exact solution) • 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 78 Manipulating FSM 00110 00011 m = 2 winners 2 0 2 1 01111 00011 00111 2 1 max = 2 10111 00001 •Voters are sincere •Another optimal solution: 00101 79 Manipulating FSM 00110 11110 00011 m = 2 winners 0 2 2 2 1 01111 00110 00111 2 3 max = 3 10111 00001 •A voter manipulates and realizes ideal outcome •But our 3-approximation for FSM is nonmanipulable 80 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 81 Progress so far Area of research State 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 Completed: comparison of A and T in 3-alt. case, approval domination of A as x strategies To do: comparisons of other pairs, analysis using branching-probabilities metric Fixed-size Completed: NP-hardness proof, 3-approximation, minimax heuristic evaluation, manipulability analysis 82 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? 83 What happens at equilibrium? • The optimal strategy recommends that no voter change • So (i ) v ri vi 1 • And (i ) v ri vi 0 – equivalently, (i ) vi 0 v ri • Therefore any average at equilibrium must satisfy two equations: – (A) v n i : v ri – (B) i : v ri vn 84 Proof: Only one equilibrium average A( ) n i : ri B( ) i : ri n • Theorem: A(1 ) B(1 ) A(2 ) B(2 ) 1 2 • Proof considers two symmetric cases: – assume 1 2 – assume 2 1 • Each leads to a contradiction 85 Proof: Only one equilibrium average case 1: 1 2 (i ) 2 ri 1 ri i : 2 ri i : 1 ri i : 2 ri i : 1 ri 2n i : 2 ri A(2 ) i : 1 ri 1n B (1 ) 2n i : 2 ri i : 1 ri 1n 2 n 1n 2 1 , contradicting 1 2 86 Proof: Only one equilibrium average Case 1 shows that 1 2 Case 2 is symmetrical and shows that 2 1 Therefore 1 2 Therefore, given r , the average at equilibrium is unique 87 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 88 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) 89 Average approx. ratios found radius = 1 radius = 2 fixed-size 1.0012 1.0000 minimax 3-approx. 1.0017 1.0000 random 1.0057 1.0000 set highest- 1.0059 1.0000 maxscore performance on GTS ’03 election data k = 24 candidates, m = 12 winners, n = 161 ballots 90 Largest approx. ratios found radius = 1 radius = 2 fixed-size 1.0714 1.0000 minimax 3-approx. 1.0714 1.0000 random 1.0714 1.0000 set highest- 1.0714 1.0000 maxscore performance on GTS ’03 election data k = 24 candidates, m = 12 winners, n = 161 ballots 91 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) 92

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