Computational social choice Lirong Xia Todays schedule - - PowerPoint PPT Presentation

Lirong Xia

Computational social choice

ØComputational social choice: the easy-to- compute axiom

• voting rules that can be computed in P
• satisfies the axiom
• Kemeny: a full proof of NP-hardness
• IP formulation of Kemeny

2

Today’s schedule

ØKendall tau distance

• K(R,W)= # {different pairwise comparisons}

ØKemeny(P)=argminW K(P,W) =argminW ΣV∈P K(V,W) ØFor single winner, choose the top-ranked alternative in Kemeny(P)

3

Kemeny’s rule

K( b ≻ c ≻ a , a ≻ b ≻ c ) =

2

Profile P WMG K(P,a ≻ b ≻ c ≻ d)=0+1+2*3+2*5=17

4

Example

c d a b 2 2 2 2 2

a ≻ b ≻ c ≻ d b ≻ a ≻ c ≻ d d ≻ a ≻ b ≻ c c ≻ d ≻ b ≻ a

1 1 2 2

ØFor each linear order W (m! iter)

• for each vote R in D (n iter)
• compute K(R,W)

ØFind W* with the smallest total distance

• W*= argminW K(D,W)=argminW ΣR∈DK(R,W)
• top-ranked alternative at W* is the winner

ØTakes exponential O(m!n) time!

5

Computing the Kemeny winner

Ø Ranking W → direct acyclic complete graph G(W) Ø Given the WMG G(P) of the input profile P Ø K(P,W) = Σa→b∈G(W)#{V∈P: b ≻ a in V} =Σa→b∈G(W)(n+w(b→a))/2 = nm(m-1)/4 + Σa→b∈G(W) w(b→a)/2 Ø argminW K(P,W)=argminWΣa→b∈G(W) w(b→a) =argminWTotal weight on inconsistent edges in WMG

6

Kemeny

a ≻ b ≻ c ≻ d b a c d

ØTotal weight on inconsistent edges between W and P is: 20

7

Example

W= a ≻ b ≻ c ≻ d

b a c d

Profile P:

a b c d

20 16 14 12 8 6

ØReduction from feedback arc set (FAC)

• Given a directed graph G and a number k
• does there exist a way to eliminate no more

than k edges to obtain an acyclic graph?

8

Kemeny is NP-hard to compute

d a b c

• J. Bartholdi III, C. Tovey, M. Trick, Voting schemes for which it can be difficult to tell who

won the election, Social Choice Welfare 6 (1989) 157–165.

ØThe KendallDistance problem:

• Given a profile P and a number k,
• Does there exist a ranking W whose total Kendall

distance is at most k?

9

Proof

Instance of FAC Instance of KendallDistance Yes No Yes No P-time d a b c

k

k’=2k+nm(m-1)/4-5

c d a b 2 2 2 2 2 WMG(P): G

ØFor any edge a→b∈G, define ØPa→b= {a ≻ b ≻ others, Reverse(others) ≻ a ≻ b} ØP = ∪ a→b∈G Pa→b

10

Constructing the profile

d a b c WMG(Pa→b)= 2

ØA voting rule satisfies the easy-to- compute axiom if computing the winner can be done in polynomial time

• P: easy to compute
• NP-hard: hard to compute
• assuming P≠NP

11

The easy-to-compute axiom

ØGiven: a voting rule r ØInput: a preference profile P and an alternative c

• input size: nmlog m

ØOutput: is c the winner of r under P?

12

The winner determination problem

ØIf following the description of r the winner can be computed in p-time, then r satisfies the easy-to-compute axiom ØPositional scoring rule

• For each alternative (m iter)
• for each vote in D (n iter)

– find the position of m, find the score of this position

• Find the alternative with the largest score (m iter)
• Total time O(mn+m)=O(mn)

13

Computing positional scoring rules

ØFor each pair of alternatives c,d (m(m-1) iter)

• let k = 0
• for each vote V∈P
• if c>d add 1 to the counter k
• if d>c subtract 1 from k
• the weight on the edge c→d is k

14

Computing the weighted majority graph

15

Satisfiability of easy-to-compute

Rule Complexity

Positional scoring P Plurality w/ runoff STV Copeland Maximin Ranked pairs Kemeny NP-hard Slater Dodgson

ØFor each pair of alternatives a, b there is a binary variable xab

Øxab = 1 if a>b in W Øxab = 0 if b>a in W

Ømax Σa,bw(a→b)xab

s.t. for all a, b, xab+xba=1 for all a, b, c, xab+xbc+xca≤2

ØDo we need to worry about cycles of >3 vertices? Homework

16

Solving Kemeny in practice

No edges in both directions No cycle of 3 vertices

17

Manpulation

Manipulation under plurality rule

(lexicographic tie-breaking)

> > > > > >

> >

Plurality rule Alice Bob Carol

Strategic behavior (of the agents)

ØManipulation: an agent (manipulator) casts a vote that does not represent her true preferences, to make herself better

• ff

ØA voting rule is strategy-proof if there is never a (beneficial) manipulation under this rule

ØVoting time! ØN>M>O à O>M>N

20

Using STV?

×4

> > > >

×2

> >

×2

Any strategy-proof voting rule?

ØNo reasonable voting rule is strategyproof

• Gibbard-Satterthwaite Theorem [Gibbard Econometrica-73,

Satterthwaite JET-75]: When there are at least three

alternatives, no voting rules except dictatorships satisfy

• non-imposition: every alternative wins for some profile
• unrestricted domain: voters can use any linear order

as their votes

• strategy-proofness

ØRandomized voting rules [Gibbard Econometrica-77] ØMultiple winners [Duggan &Schwartz SCW 00]

Ø Using Arrow’s impossibility theorem

• Arrow’s theorem: no ranking rule satisfies non-dictatorship,

universal domain, unanimity, and IIA

ØSuppose G-S does not hold for r (strategy-proof + non-dictatorial), construct a contraduction f to Arrow’s theorem

• 1. for any profile P and any pair of alternatives (a,b), raise

them to the top-2 to obtain Pab. Let a>b in f(P) iff r(Pab)=a

• 2. prove that f(P) is a linear order
• 3. prove that f satisfies non-dictatorship, universal domain,

unanimity, and IIA è contradiction

22

Proof idea

ØRelax non-dictatorship: use a dictatorship ØRestrict the number of alternatives to 2 ØRelax unrestricted domain: mainly pursued by economists

• Single-peaked preferences

23

A few ways out

10% Participation 40% 70%

ØThere exists a social axis S

• linear order over the alternatives

ØEach voter’s preferences V are compatible with the social axis S

• there exists a “peak” a such that
• [b≺c≺a in S] implies [c≻b in V]
• [a≻c≻b in S] implies [c≻b in V]
• alternatives closer to the peak are more preferred
• different voters may have different peaks, but

must share the same social axis

24

Single-peaked preferences

25

Examples

rank Axis

Ø The median rule

• given a profile of “peaks”
• choose the median in the social axis

Ø Theorem. The Median rule is strategy-proof. Ø The median rule with phantom voters [Moulin PC 80]

• parameterized by a fixed set of “peaks” of phantom voters
• chooses the median of the peaks of the regular voters and

the phantom voters

Ø Theorem. Any strategy-proof rule for single-peaked preferences are median rules with phantom voters

26

Strategy-proof rules for single- peaked preferences