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Speaker: Ariel Procaccia 1 Joint work with: Ioannis Caragiannis 2, Jason Covey 3, Michal Feldman 1, Chris Homan 3, Christos Kaklamanis 2, Nikos Karanikolas 2, and Jeff Rosenschein 1 1 Hebrew University of Jerusalem, Israel 2 University of Patras, Greece 3 Rochester Institute of Technology, USA

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Background on Voting Approximability of Carroll’s rule Greedy alg Randomized rounding alg Inapproximability Epilogue: on the desirability of approx algs as voting rules 2

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Set of voters {1,...,n} Set of m candidates {a,b,c...} Voters (strictly) rank the candidates Preference profile: a vector of rankings Voter 1Voter 2Voter 3 a b c a c b b a c 3

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Voting rule: a mapping from preference profiles to candidates; designated winner Examples (Positional Scoring): Plurality: each voter awards one point to candidate ranked first Borda: each voter awards m-k points to candidate ranked k’th 4

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Election proceeds in rounds In each round, each voter awards one point to candidate ranked highest out of surviving candidates. Candidate with least points is eliminated Used for national elections in Ireland, Australia and Malta; for local elections in New Zealand and Scotland 5

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2 voters 1 voter a a b b a d 6 b b c c d d b b a a d d c c c c b d d b b a a

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French mathematician and philosopher. a beats b in a pairwise election if the majority of voters prefers a to b a is a Condorcet winner if a beats any other candidate in a pairwise election 7

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Voter 1Voter 2 Voter 3 c b a a c b b a c 8 a a a b b b c c c a a c c b b

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Condorcet-consistency: if a Condorcet winner exists, it must be elected Copeland: a’s score is # of other candidates a beats in a pairwise election If a is a Condorcet winner, score = m-1, and for any b≠a, score < m-1 9

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10 a a c c ? ? ? ? ? ? ? ? b b a a c c c c b b a a a a c c b b b b a a c c c c c c c c b b b b b b a a a a a a

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33 voters16 voters3 voters 8 voters18 voters22 voters a b c d e b d c e a c d b a e c e b d a d e c b a e c b d a 11

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English author and mathematician, better known as Lewis Carroll Suggested to choose a candidate “as close as possible” to a Condorcet winner 12

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Score of x = minimum # of exchanges between adjacent candidates needed to make x a Condorcet winner 13

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Voter 1Voter 2Voter 3 Voter 4 Voter 5 b d e a c b c a e d d e c a b e d b a c a e c b d P(a,b) P(a,c) P(a,d) P(a,e)

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Voter 1Voter 2Voter 3 Voter 4 Voter 5 b d e a c b c a e d d e c a b e d b a c a e c b d def(b,a) def(b,c) def(b,d) def(b,e)

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Score of x = minimum # of exchanges between adjacent candidates needed to make x a Condorcet winner Alternatively: total number of positions that the voters push x Elect candidate with minimum score 16

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D ODGSON -S CORE : given candidate x, a preference profile, and a threshold k, is the Dodgson score of x at most k ? [BTT 89] D ODGSON -S CORE is NP-complete, D ODGSON -W INNER is NP-hard [HHR 97] D ODGSON -W INNER is complete for Parallel access to NP 17

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Given x C and pref profile def(x,c) = def(c) = # additional voters that must rank x above c in order for x to beat c in a pairwise election c is alive iff def(c) > 0, otherwise dead Cost-effectiveness of push = ratio between # of live candidates overtaken and # of positions pushed Greedy Algorithm: while live candidates, perform the most cost-effective push 18

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d d c c b b a a x x d d c c b b a a x x e2e2 e2e2 e1e1 e1e1 e3e3 e3e3 d d e4e4 e4e4 e5e5 e5e5 e6e6 e6e6 e7e7 e7e7 e8e8 e8e8 c c e9e9 e9e9 e 10 e 11 e 12 b b e 13 e 14 e 15 a a x x e 16 e 17 x x x x x x a a a a a a b b b b b b c c c c c c d d d d d d 19

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Theorem: The greedy alg has an approx ratio of H m-1 Proof relies on the dual fitting technique [Vaz 01] Primal solution found by algorithm upper- bounded by infeasible dual assignment Divide dual assignment by H m-1 and show that shrunk assignment is feasible 20

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Variables y ij : boolean, 1 iff i pushes x j positions Constants ij c : boolean, 1 iff pushing x j positions by i gives x additional vote against c 21

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Randomized Rounding alg: Solve relaxed LP to obtain solution y For k = 1,...,2log(m): for all i, randomly and independently choose Y i k = j w. prob. y ij For all i, Y i * = k Y i k 22 Theorem: The randomized rounding alg gives a valid solution that is an 8log(m) approx with prob. 1/ k = 1 k = 2 k =

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Theorem: [essentially Mc 06] It is NP-hard to approximate D ODGSON -S CORE to (logm) Theorem: There is no poly-time alg that approximates Dodgson to (1/2- )lnm unless NP has quasi poly-time algs Implies that greedy alg is optimal up to a factor of 2 23

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Work in social choice shows sharp discrepancies between Dodgson ranking and other rules E.g., Dodgson ranking can be opposite of Copeland ranking [Klam 03] and Borda ranking [Klam 04] Theorem: It is NP-hard to decide if a given candidate is a Dodgson winner or in last 6 m positions Wide scope, captures many previous results 24

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Does it make sense to approximate a voting rule?? Approximation algorithm is a new voting rule How good are our approximation algorithms as voting rules? 25

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26 Condorcet Tractability Monotonicity Dodgson Condorcet Tractability Monotonicity Condorcet Tractability Monotonicity

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d d c c b b a a x x d d c c b b a a e2e2 e2e2 e1e1 e1e1 e3e3 e3e3 d d e4e4 e4e4 e5e5 e5e5 e6e6 e6e6 e7e7 e7e7 e8e8 e8e8 c c e9e9 e9e9 e 10 e 11 e 12 b b e 13 e 14 e 15 a a x x e 16 e 17 x x x x x x c c a a a a b b b b b b c c a a c c d d d d d d x x 27

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d d c c b b a a x x d d c c b b a a e2e2 e2e2 e1e1 e1e1 e3e3 e3e3 d d e4e4 e4e4 e5e5 e5e5 e6e6 e6e6 e7e7 e7e7 e8e8 e8e8 c c e9e9 e9e9 e 10 e 11 e 12 b b e 13 e 14 e 15 a a e 16 e 17 x x x x x x x x x x 28

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d d c c b b a a x x d d c c b b a a x x e2e2 e2e2 e1e1 e1e1 e3e3 e3e3 d d e4e4 e4e4 e5e5 e5e5 e6e6 e6e6 e7e7 e7e7 e8e8 e8e8 c c e9e9 e9e9 e 10 e 11 e 12 b b e 13 e 14 e 15 a a x x e 16 e 17 x x x x x x a a a a a a b b b b b b c c c c c c d d d d d d 29

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RR alg is monotonic; advantage over greedy alg as a voting rule Voting rule is strongly monotonic if pushing a winning candidate can’t make it lose Dodgson itself is not strongly monotonic Is there an approx alg that is strongly monotonic? What about other properties? Truthfulness, as in algorithmic mechanism design? Homogeneity Same goes for other hard-to-compute voting rules 30

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Our paper “On the Approximability of Dodgson and Young Elections” also contains results about Young’s rule Available from Google: “Ariel Procaccia” 31

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