Approximating Lewis Carroll’s Voting Rule

Approximating Lewis Carroll’s Voting Rule Speaker: Ariel Procaccia1 Joint work with: Ioannis Caragiannis2, Jason Covey3, Michal Feldman1, Chris Homan3, Christos Kaklamanis2, Nikos Karanikolas2, and Jeff Rosenschein1 1 Hebrew University of Jerusalem, Israel 2 University of Patras, Greece 3 Rochester Institute of Technology, USA

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

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

Some voting rules • 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

Single Transferable Vote • 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

STV: example c a b b d b a b b a b c d d a c d

Marquis de Condorcet • 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

The Condorcet Paradox c c c b b b a a a c a b c a a b b b c a c

Condorcet voting rules • 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

Voting trees c c a a b b b b c c a a ? a a b b c c ? ? c a c ? b a

An awesome example a b c b d d c c b e d a Condorcet Plurality STV Borda e a e c d e e e c b c b d b d a a a

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

Dodgson’s rule • Score of x = minimum # of exchanges between adjacent candidates needed to make x a Condorcet winner

Dodgson score example b b d d c e e a c e a a c d b e a 3 3 2 2 3 3 3 3 d 3 3 3 2 e 2 3 2 2 P(a,b) P(a,c) P(a,d) P(a,e) b c a b c d

Dodgson score example b b d d c e e a c e a a c d b a e d 0 0 0 0 e 0 0 0 1 def(b,a) def(b,c) def(b,d) def(b,e) b c a b c d

Dodgson’s rule • 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

Complexity of Dodgson • Dodgson-Score: given candidate x, a preference profile, and a threshold k, is the Dodgson score of xat most k ? • [BTT 89] Dodgson-Score is NP-complete, Dodgson-Winner is NP-hard • [HHR 97] Dodgson-Winner is complete for Parallel access to NP

Greedy algorithm • 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 aliveiff 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

It’s alive! d d d d d a c b c c d e4 c b a x c c e9 e13 b b e5 x e16 b b e10 e17 a a e6 x e14 a a x e1 e11 e15 e7 e2 e12 e8 e3 x x

Greed pays off • Theorem: The greedy alg has an approx ratio of Hm-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 Hm-1 and show that shrunk assignment is feasible

ILP for Dodgson • Variables yij: boolean, 1 iffi pushes xj positions • Constants ijc: boolean, 1 iff pushing xj positions by i gives x additional vote against c

Randomized rounding algorithm • Randomized Rounding alg: • Solve relaxed LP to obtain solution y • For k = 1,...,2log(m): for all i, randomly and independently choose Yik = j w. prob. yij • For all i, Yi* = kYik • Theorem: The randomized rounding alg gives a valid solution that is an 8log(m) approx with prob.  1/2 3 5 3 2 1 3 k = 1 0 3 0 k = 2 1 1 0 k = 3

Lower bounds • Theorem: [essentially Mc 06] It is NP-hard to approximate Dodgson-Score 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

The coolest result • 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

Approximation algs as voting rules • 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?

Borda Us Dodgson

Greedy is nonmonotonic d d c d d d b a a c b c d e4 c c e9 e13 b c e5 e14 e16 b b e10 e17 e6 e11 e15 x a a b a a e1 e7 e12 x e2 e8 x e3 x x x

Greedy is nonmonotonic d d d a c b e4 c c e9 e13 e5 e14 e16 b b e10 e17 e6 e11 e15 x a a e1 e7 e12 x e2 e8 x e3 x x x

Greedy is nonmonotonic d d d d d a c b c c d e4 c b a x c c e9 e13 b b e5 x e16 b b e10 e17 a a e6 x e14 a a x e1 e11 e15 e7 e2 e12 e8 e3 x x

A new agenda • 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

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

Thank You!

Approximating Lewis Carroll’s Voting Rule