Transcript procaccia08aDodgson.ppt

Approximability and
Inapproximability of Dodgson and
Young Elections
Ariel D. Procaccia, Michal Feldman
and Jeffrey S. Rosenschein
Voting: reminder?
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•
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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
a
b
b
c
a
c
b
c
Voter 1
Voter 2
Voter 3
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
a
b
c
a
b
Voter 1
a
Voter 2
b
a
c
b
c
Voter 3
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) = |{iN: 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
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
b c d e a
b c d e a
b c d e a
b c d e a
d b e a c e
d b e a c
d b e d c
b
d b e d c
d
e a c b e
e a c a e
e a c d e
a a c d e
a e a a b
a e a b b
a e a b b
e e a b b
c d b c d
c d b c d
c d b c d
c d b c d
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
b c d e a
d b e d c
c
b b d e a
d c e d c
e
b b d e a
d c e d c
e a c b e
e a c b e
e a c b b
a e a a b
a e a a b
a e a a e
c d b c d
c d b c d
c d b c d
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.
min
 jx
ij
s.t.
i, j
i,
x
ij
1
j
a  c*,
x e
a
ij ij
i, j
i, j , xij  {0,1}
 d (a)
Randomized Rounding Alg
• Solve relaxed LP to
obtain solution x.
• For k=1,..., log(m):
for all i, randomly and
independently choose
Xik = j w. prob. xij.
• For all i, Ximax = maxk
Xik.
• i pushes c* by Ximax.
min  j  xij s.t.
i, j
i,
x
ij
1
j
a  c*,
x e
a
ij ij
i, j
i, j, xij  0
 d (a)
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
1 2
5
b c d e a
b c
a
a a e d c
a a
c
d b c b e
d b
e
c e a a b
c e
b
e d b c d
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.