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

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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.

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The Condorcet Paradox

c

b

a

a

c

b

b

a

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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.

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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.