1. 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

2.  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

3.  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

4.  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

5.  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

6. 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

7.  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

8. 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

9.  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

10. 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

11. 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

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

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

14. 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) 23222322 33223322 33323332 33333333 14

15. 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) 00010001 00000000 15

16.  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

17.  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

18.  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

19. 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

20.  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

21.  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

22.  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.  1/2 2 2 0 0 1 1 1 1 3 3 3 3 1 1 0 0 0 0 k = 1 k = 2 k = 3 3 3 5 5 3 3

23.  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

24.  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

25.  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

26. 26 Condorcet Tractability Monotonicity Dodgson Condorcet Tractability Monotonicity Condorcet Tractability Monotonicity

27. 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

28. 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

29. 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

30.  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

31.  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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