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Approximating Lewis Carroll’s Voting Rule. 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

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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
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
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

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some voting rules
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
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
STV: example

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marquis de condorcet
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
The Condorcet Paradox

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condorcet voting rules
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
Voting trees

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an awesome example
An awesome example

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Condorcet

Plurality

STV

Borda

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

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dodgson score example1
Dodgson score example

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def(b,a)

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dodgson s rule1
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
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
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
It’s alive!

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greed pays off
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
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 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

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lower bounds
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
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
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?
slide26

Borda

Us

Dodgson

greedy is nonmonotonic
Greedy is nonmonotonic

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greedy is nonmonotonic1
Greedy is nonmonotonic

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Greedy is nonmonotonic

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