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Computational Aspects of Approval Voting and Declared-Strategy Voting Dissertation defense 17 April 2008 Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu Ron Cytron Robert Pless Steven Brams Itai Sened Jeremy Buhler Aaron Stump Themes of research • Approval voting systems • Susceptibility to insincere strategy – encouraging sincere ballots • Evaluating effectiveness of various strategies • Internalizing insincerity – separating strategy from indication of preferences • Complex voting protocols – complexity of finding most effective ballot – complexity of calculating the outcome 2 What is “manipulation”? • Broadly, effective influence on election outcome • Election officials can . . . – exclude/include alternatives [Nurmi ’99] – exclude/include voters [Bartholdi, Tovey & Trick ’92] – choose election protocol [Saari ’01] • Alternatives may be able to . . . – drop out to avoid a vote-splitting effect • Voters can . . . – find the ballot that is likeliest to optimize the outcome • This last sense is what we mean 3 Let’s vote! 45 voters 35 voters 20 voters A B C (1st) sincere preferences C C B (2nd) B A A (3rd) 4 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 5 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 6 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 7 Manipulation decision problem 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 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? • My generalization of problems from the literature: [Bartholdi, Tovey & Trick ’89] [Conitzer & Sandholm ’02] [Conitzer & Sandholm ’03] 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? • 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 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 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 11 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 (single-winner STV) [Bartholdi & Orlin ’91] – Borda [Conitzer & Sandholm ’02] 12 What can we do to make manipulation hard? • 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 13 Declared-Strategy Voting [Cranor & Cytron ’96] cardinal rational preferences strategizer ballot election outcome state 14 Declared-Strategy Voting [Cranor & Cytron ’96] sincerity strategy 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 sincere preferences 15 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 16 Can DSV be hard to manipulate? DSV can be made to be NP-hard to manipulate in the EPWCB sense. [LeGrand ’08] Proof by reduction: • Simulate Hare by using particular declared strategy in DSV • Hare is NP-hard to manipulate [Bartholdi & Orlin ’91] • If this DSV system were easy to manipulate, then Hare would be • DSV can be made NP-hard to manipulate So why use “tweaks”? (DSV is better!) 17 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 18 Approve both! 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 19 Approval voting [Ottewell ’77] [Weber ’77] [Brams & Fishburn ’78] • Allows approval of any subset of alternatives • Single alternative with most votes wins • Used historically [Poundstone ’08] – Republic of Venice 1268-1789 – Election of popes 1294-1621 • Used today [Brams ’08] – Election of UN secretary-general – Several academic societies, including: • Mathematical Society of America • American Statistical Association 20 Strands of 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 Strands of 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] 22 Strands of 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] 23 Approval ratings 24 Approval ratings • Aggregating film reviewers’ ratings – Rotten Tomatoes: approve (100%) or disapprove (0%) – Metacritic.com: ratings between 0 and 100 – Both report average for each film – Reviewers rate independently 25 Approval ratings • Online communities – Amazon: users rate products and product reviews – eBay: buyers and sellers rate each other – Hotornot.com: users rate other users’ photos – Users can see other ratings when rating • Can these “voters” benefit from rating insincerely? 26 Approval ratings 27 Average of ratings r 0.4, 0.7, 0.8, 0.8, 0.9 v 0.4, 0.7, 0.8, 0.8, 0.9 outcome: f avg (v ) 0.72 0.72 0 1 data from Metacritic.com: Videodrome (1983) 28 Average of ratings r 0.4, 0.7, 0.8, 0.8, 0.9 v 0, 0.7, 0.8, 0.8, 0.9 outcome: f avg (v ) 0.64 0.64 0 1 Videodrome (1983) 29 Another approach: Median r 0.4, 0.7, 0.8, 0.8, 0.9 v 0.4, 0.7, 0.8, 0.8, 0.9 outcome: f med (v ) 0.8 0.8 0 1 Videodrome (1983) 30 Another approach: Median r 0.4, 0.7, 0.8, 0.8, 0.9 v 0, 0.7, 0.8, 0.8, 0.9 outcome: f med (v ) 0.8 0.8 0 1 Videodrome (1983) 31 Another approach: Median • Immune to insincerity [LeGrand ’08] – 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 32 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? • And will an equilibrium always be found? 33 Equilibrium-finding algorithm r 0.4, 0.7, 0.8, 0.8, 0.9 v 0.4, 0.7, 0.8, 0.8, 0.9 0.72 0 1 Videodrome (1983) 34 Equilibrium-finding algorithm r 0.4, 0.7, 0.8, 0.8, 0.9 v 0, 0, 0, 0, 0 0 1 35 Equilibrium-finding algorithm r 0.4, 0.7, 0.8, 0.8, 0.9 v 0, 0, 0, 0,1 0.2 0 1 36 Equilibrium-finding algorithm r 0.4, 0.7, 0.8, 0.8, 0.9 v 0, 0, 0,1,1 0.4 0 1 37 Equilibrium-finding algorithm r 0.4, 0.7, 0.8, 0.8, 0.9 v 0, 0,1,1,1 0.6 0 1 38 Equilibrium-finding algorithm r 0.4, 0.7, 0.8, 0.8, 0.9 v 0, 0.5,1,1,1 equilibrium! 0.7 0 1 • Is this algorithm is guaranteed to find an equilibrium? 39 Equilibrium-finding algorithm r 0.4, 0.7, 0.8, 0.8, 0.9 v 0, 0.5,1,1,1 equilibrium! 0.7 0 1 • Is this algorithm is guaranteed to find an equilibrium? • Yes! [LeGrand ’08] 40 Expanding range of allowed votes r 0.4, 0.7, 0.8, 0.8, 0.9 v 1, 1, 2, 2, 2 0.8 1 2 • These results generalize to any range [LeGrand ’08] 41 Multiple equilibria can exist r 0.4, 0.7, 0.7, 0.8, 0.9 v 0, 0.5,1,1,1 v 0, 0.6, 0.9,1,1 v 0, 0.75, 0.75,1,1 outcome in each case: f avg (v ) 0.7 • Will multiple equilibria will always have the same average? 42 Multiple equilibria can exist r 0.4, 0.7, 0.7, 0.8, 0.9 v 0, 0.5,1,1,1 v 0, 0.6, 0.9,1,1 v 0, 0.75, 0.75,1,1 outcome in each case: f avg (v ) 0.7 • Will multiple equilibria will always have the same average? • Yes! [LeGrand ’08] 43 Average-Approval-Rating DSV r 0.4, 0.7, 0.8, 0.8, 0.9 v 0.4, 0.7, 0.8, 0.8, 0.9 outcome: f aveq (v , 0,1) 0.7 0.7 0 1 Videodrome (1983) 44 Average-Approval-Rating DSV r 0.4, 0.7, 0.8, 0.8, 0.9 v 0, 0.7, 0.8, 0.8, 0.9 outcome: f aveq (v , 0,1) 0.7 0.7 0 1 • AAR DSV is immune to insincerity in general [LeGrand ’08] 45 Evaluating AAR DSV systems • Expanded vote range gives wide range of AAR DSV systems: a ,b (v ) 0 a 1 0 b 1 • If we could assume sincerity, we’d use Average • Find AAR DSV system that comes closest • Real film-rating data from Metacritic.com – mined Thursday 3 April 2008 – 4581 films with 3 to 44 reviewers per film – measure root mean squared error 46 Evaluating AAR DSV systems b 0.5 RMSEa ,0.5 a minimum at a 0.3240 47 Evaluating AAR DSV systems: hill-climbing b 0.4820 RMSE a , 0.4820 a minimum at a 0.3647 48 Evaluating AAR DSV systems: hill-climbing a 0.3647 RMSE 0.3647 ,b b minimum at b 0.4820 49 Evaluating AAR DSV systems 0.3647,0.4820 (v ) f avg (v ) 50 AAR DSV: Future work • Website: ratingsbyrob.com – Users can rate movies, books, each other, etc. – They can see current ratings without being tempted to rate insincerely • Find more strategy-immune rating systems • Richer outcome spaces – Hypercube: like rating several films at once – Simplex: dividing a limited resource among several uses – How assumptions about preferences are generalized is important 51 Strands of 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] 52 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 53 DSV-style approval strategies s [30, 25,15,10] p [0, 1, 0.8, 0.3] • Strategy Z: b [0, 1, 1, 0] – Approve alternatives with higher-than-average cardinal preference (zero-information strategy) [Merrill ’88] 54 DSV-style approval strategies s [30, 25,15,10] p [0, 1, 0.8, 0.3] • Strategy Z: b [0, 1, 1, 0] • Strategy T: b [0, 1, 0, 0] – Approve favorite of top two vote-getters, plus all liked more [Ossipoff ’02, Poundstone ’08] – Simplest generalization of plurality DSV strategy [Cranor & Cytron ’96] 55 DSV-style approval strategies s [30, 25,15,10] p [0, 1, 0.8, 0.3] • Strategy Z: b [0, 1, 1, 0] • Strategy T: b [0, 1, 0, 0] • Strategy J: b [0, 1, 1, 0] – Use strategy Z if it distinguishes between top two vote- getters; otherwise use strategy T [Brams & Fishburn ’83] 56 DSV-style approval strategies s [30, 25,15,10] p [0, 1, 0.8, 0.3] • Strategy Z: b [0, 1, 1, 0] • Strategy T: b [0, 1, 0, 0] • Strategy J: b [0, 1, 1, 0] • Strategy A: b [0, 1, 1, 1] – Approve all preferred to top vote-getter, plus top vote- getter if preferred to second-highest vote-getter [LeGrand ’02] . . . but how to evaluate these strategies? 57 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? 58 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 Wi k i s j 1 x j 59 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) 60 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 When p2 p3 p1, A is better than T if and only if: 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, A always does better than T when: s 1 • s1 is much larger than s2, p3 p1 2 • x is large, or • p3 is relatively close to p2 compared to p1 61 Strategy comparison using the Merrill metric • Also compared other strategy pairs [LeGrand ’08] • As x goes to infinity (3 alternatives): – Strategy A dominates strategy T – Strategy A dominates strategy J – Strategy A dominates strategy Z – Neither strategy T nor strategy J dominates the other • As x goes to infinity (4 alternatives): – Strategy A dominates strategy T 62 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 approval strategies according to the Merrill metric [LeGrand ’08] 63 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 64 Branching-probabilities metric: strategy A 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 number of future ballots is taken to infinity then strategy A dominates all other approval strategies according to the branching-probabilities metric [LeGrand ’08] 65 Approval DSV strategies: Future work • Consider different strategy-evaluation metrics • Study strategy-A equilibria – How “good” are the outcomes? – How often are strong Nash equilibria found? • How strategy-vulnerable is Approval DSV with strategy A? – How often will submitting insincere preferences benefit a voter? 66 Strands of 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] 67 Electing a committee from approval ballots approves of k = 5 alternatives 11110 00011 alternatives 4 and 5 n = 6 ballots 01111 00111 10111 00001 •What’s the best committee of size m = 2? 68 Sum of Hamming distances m = 2 winners 11110 00011 2 4 4 5 01111 11000 00111 4 3 sum = 22 10111 00001 •What if we elect alternatives 1 and 2? 69 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 HD sum •Easy to compute (pick alternatives with most approvals) 70 Maximum Hamming distance m = 2 winners 11110 00011 4 0 2 1 01111 00011 00111 2 1 sum = 10 max = 4 10111 00001 •One voter is quite unhappy with minisum outcome 71 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 maximum HD •Harder to compute? 72 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) 73 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) [LeGrand ’04] 74 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 75 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] [LeGrand, Markakis & [LeGrand, Markakis & Mehta ’06] Mehta ’06] * Polynomial-Time Approximation Scheme: algorithm with approx. ratio 1 + ε that runs in time polynomial in the input and exponential in 1/ε 76 Susceptibility to insincerity Endogenous minimax Bounded-size minimax Fixed-size minimax = EM = BSM(0, k) = BSM(m1, m2) = FSM(m) = BSM(m, m) insincere voters insincere voters insincere voters can benefit can benefit can benefit [LeGrand, Markakis & [LeGrand, Markakis & [LeGrand, Markakis & Mehta ’06] Mehta ’06] Mehta ’06] But our 3-approximation for FSM is immune to insincere strategy! 77 Fin Thanks to – my advisor, Ron Cytron – Steven Brams – members of my committee – co-authors Vangelis Markakis and Aranyak Mehta – Morgan Deters and the rest of the DOC Group Questions? 78 Rational [m,M]-Average strategy • Allow votes between m 0 and M 1 • 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 , m), M ) • After voter i uses this strategy, one of these is true: – f avg (v ) ri and vi M – f avg (v ) ri – f avg (v ) ri and vi m 79 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 80 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 81 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 82 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 83 An equilibrium always exists? • At equilibrium, v must satisfy (i) vi min(max(ri n j i v j , m), M ) I proposed to prove that, given a vector r , at least one equilibrium exists. A particular algorithm will always find an equilibrium for any r . . . 84 An equilibrium always exists! Equilibrium-finding algorithm: • sort r so that (i j ) ri r j • for i = 1 up to n do vi min(max(ri n k i vk (n i)m, m), M ) (full proof and more efficient algorithm in dissertation) • Since an equilibrium always exists, average at equilibrium is a function, f aveq ( r , m, M ) . • Applying f aveq to v instead of r gives a new system, Average-Approval-Rating DSV. 85 Average-Approval-Rating DSV • What if, under AAR DSV, voter i could gain an outcome closer to ideal by voting insincerely ( vi ri )? I proposed to prove that Average-Approval-Rating DSV is immune to strategy by insincere voters. • Intuitively, if f aveq (v , m, M ) vi, increasing vi will not change f aveq (v , m, M ) . 86 AAR DSV is immune to strategy • If f aveq (v , m, M ) vi ri, – increasing vi will not change f aveq (v , m, M ). – decreasing vi will not increase f aveq (v , m, M ) . • If f aveq (v , m, M ) vi ri, – increasing vi will not decrease f aveq (v , m, M ) . – decreasing vi will not change f aveq (v , m, M ) . (complete proof in dissertation) • So voting sincerely ( vi ri ) is guaranteed to optimize the outcome from voter i’s point of view 87 Parameterizing AAR DSV • [m,M]-AAR DSV can be parameterized nicely using a and b, where 0 a 1 and 0 b 1: 1 m a b M m 1 M m b 1 b m b M b a a b 1 b a ,b (v ) lim f aveq v , b , b xa x x 88 Parameterizing AAR DSV • For example: 1,b (v ) f aveq (v , 0,1) 1 1 (v ) f aveq v , 1, 2 , 3 2 1 1 (v ) f aveq v , 10 ,11 , 21 2 1 (v ) f med v 0, 2 0, 0 (v ) max v 0 ,1 (v ) min v 89 Evaluating AAR DSV systems • Real film-rating data from Metacritic.com – mined Thursday 3 April 2008 – 4581 films with 3 to 44 reviewers per film 0 a 1 0 b 1 SEa,b v a,b v f avg v 2 v SEa,b v RMSEa ,b V vV v vV 90 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 91 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] 92 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 93 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 94 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 95 Further result for strategy A V[ 0, 0,...0 ] p1 , p2 , pk s1 , s2 , sk s1 s2 sk • just a weighted average of p i values • assume p1 p2 • as x , V[ 0 , 0,...0 ] p1 from below • so maximized when weights of those pi p1 are maximized, which is done by approving only alternatives i where pi p1 • p2 p1 case is similar: approve i where pi p1 • only strategy A always does this 96 Approximating FSM 11110 m = 2 winners 00011 00111 00111 00001 choose a ballot 10111 arbitrarily 01111 97 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 98 Approximation ratio ≤ 3 optimal 11110 2 FSM set 00011 2 00111 1 00110 3 00001 2 10111 2 01111 ≤ OPT OPT = optimal maxscore 99 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 100 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 101 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 102 Local search approach for FSM 1. Start with some c {0,1}k of weight m 01001 4 103 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 4 104 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 4 105 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 106 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 107 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 108 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 109 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 dissertation) 110 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 2 111 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 112 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 113 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) 114 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 115 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 116 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 117 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, strategy-immunity of AAR DSV, evaluation of AAR DSV systems DSV-style Completed: Merrill-metric comparison of A and T in 3- approval alt. case, 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 118

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