approximability and inapproximability of dodgson and young elections
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Approximability and Inapproximability of Dodgson and
Young ElectionsAriel D. Procaccia, Michal Feldman
and Jeffrey S. Rosenschein
Voting: reminder?
• Set of voters V={1,...,n}.• Set of Candidates C={a,b,c...}; |C|=m. • Voters (strictly) rank the candidates.• Preference profile: a vector of rankings.
a
b
c
a
c
b
b
a
c
2
Condorcet winner
• 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.
3
The Condorcet Paradox
c
b
a
a
c
b
b
a
c
4
Condorcet voting rules
• Copeland: a’s score is num of other canidates a beats in a pairwise election.– If a is a Condorcet winner, score = m-1, and for
any b≠a, score < m-1.• P(a,b) = |{iN: a >i b}|• Maximin: a’s score is minbP(a,b)– If a is a condorcet winner, minbP(a,b) > n/2, any
for any b≠a, P(b,a) < n/2.• Voting trees.
5
Dodgson’s voting rule • (Dodgson)• Find candidate closest to a
Condorcet winner.• distance/score of c = number of
exchanges between pairwise candidates needed for c to become a Condorcet winner.
• Alternatively: number of places each voter has to push c.
• Elect candidate with minimal distance/score.
Example
1 2 3 4 5
b c d e a
d b e d c
e a c b e
a e a a b
c d b c d
1 2 3 4 5
b c d e a
d b e d c
e a c b e
a e a a b
c d b c d
1 2 3 4 5
b c d e a
d b e d c
e a c a e
a e a b b
c d b c d
1 2 3 4 5
b c d e a
d b e a c
e a c d e
a e a b b
c d b c d
1 2 3 4 5
b c d e a
d b e a c
a a c d e
e e a b b
c d b c d
1 2 3 4 5
b b d e a
d c e d c
e a c b e
a e a a b
c d b c d
1 2 3 4 5
b b d e a
d c e d c
e a c b b
a e a a e
c d b c d
b d e
c e
Hardness and Approximation• Bartholdi, Tovey and Trick 89: NP-hard to
compute Dodgson score.• Hemaspaandra et al. 97: Even harder to compute
Dodgson winner. (Why not in NP?)• Poly time if either n or m is constant.• We want to approximate the Dodgson score.• Discussion: essentially gives us a new voting rule
(can satisfy desiderata).• Existing lower bound: log(m). Also works for
random algs, unless NP = RP.
Trivial alg• Given: profile, c*. • Alg:– Let C’ be the candidates not beaten by c* in a
pairwise election.– While C’ is not empty:
• Choose some a in C’.• Perform minimal number of exchanges needed to make c*
beat a.• Recalculate C’.
• Step 2 in while: d(a) is deficit w.r.t. a; sufficient to choose d(a) voters which require smallest number of exchanges.
Trivial claim about trivial alg
• Claim: alg gives m-approx.• Proof: – Let a be the candidate which requires the max
number t of exchanges to get c* to beat a.– Score of c* >= t.– Each iteration of the while loop performs <= t
flips. There are at most m iterations.• Trivial alg which gives n-approx: at every
stage, each voter pushes c* one place up.
LP for Dodgson
• Notations:– Variables xij: binary,
1 iff i pushed c* j positions.
– d(a) – deficit of c* w.r.t. a.
– constants eija:
binary, 1 iff pushing c* j positions by i gives c* additional vote against a.• (Example)
• ILP is NP-hard.
}1,0{,,
)(*,
1,
..min
,
,
ij
aij
jiij
jij
jiij
xji
adexca
xi
tsxj
Randomized Rounding Alg
• Solve relaxed LP to obtain solution x.
• For k=1,..., log(m): for all i, randomly and independently choose Xi
k = j w. prob. xij.
• For all i, Ximax = maxk Xi
k.
• i pushes c* by Ximax. 0,,
)(*,
1,
..min
,
,
ij
aij
jiij
jij
jiij
xji
adexca
xi
tsxj
Young’s rule
• Also chooses candidate “closest” to Condorcet winner.
• Score of c*: maximum subset of voters for which c* is a Condorcet winner. – 0 is no nonempty subset.
• Alternatively: minimum number of voters one has to remove.
Example
1 2 3 4 5
b c d e a
a a e d c
d b c b e
c e a a b
e d b c d
1 2 5
b c a
a a c
d b e
c e b
e d d
About Young• Same hardness results.• Can also formulate as LP. • Young is nonmonotonic: if it is possible to make
c* a winner on k voters, it doesn’t mean that it’s possible on 0< r < k voters.
• Theorem: NP-hard to approximate the Young score to any factor.
• Specifically: It is NP-hard to determine whether there is a nonempty subset of voters on which c* is a Condorcet winner.– Discussion.
Related work
• Not...• Ask me if you’re interested.
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