clone structures in voters ’ preferences

Clone Structures in Voters’ Preferences

Edith ElkindNanyang TechnologicalUniversity, Singapore

Piotr FaliszewskiAGH Univeristy of Scienceand Technology, PolandArkadii Slinko University of AucklandNew Zealand

Example

R1: b > c > d > e >f > a > g > h > i

R2: e > f > d > c >i > g > h > b > a

R3: b > a > c > d >e > f > g > i > h

Elections and Clone Structures

8 7 6 5 4 3 2 1 0 a 10

b 17c 18d 17e 17f 14g 7h 3i 4

Def. An election is a pair (A,R) where A is the set of alternatives and R = (R1, …, Rn) is voters’ preference profile. Each Ri is a total linear order over A.

Example

R1: b > c > d > e >f > a > g > h > i

R2: e > f > d > c >i > g > h > b > a

R3: b > a > c > d >e > f > g > i > h


8 7 6 5 4 3 2 1 0 a 10

b 17c 18d 17e 17f 14g 7h 3i 4

Def. Let (A,R) be an election. A subset C of A is a clone set if members of C are ranked consecutively in all orders. C(R) is the set of all clones sets for R.

Example

R1: b > c > d > e >f > a > g > h > i

R2: e > f > d > c >i > g > h > b > a

R3: b > a > c > d >e > f > g > i > h

C(R) = { {c, d}, {e, f}, {d, e, f}, {c, d, e, f}, {g, h, i} ,{a}, {b}, {c}, {d}, {e}, {f}, {g}, {h}, {i},{a,b,c,d,e,f,g,h,i} }


8 7 6 5 4 3 2 1 0 a 10

b 17c 18d 17e 17f 14g 7h 3i 4

Example

R1: b > c > d > e >f > a > g > h > i

R2: e > f > d > c >i > g > h > b > a

R3: b > a > c > d >e > f > g > i > h

X = {c, d, e, f}Y = {g,h,i}


8 7 6 5 4 3 2 1 0 a 10

b 17c 18d 17e 17f 14g 7h 3i 4

Example

R1: b > X > a > Y

R2: X > Y > b > a

R3: b > a > X > Y

X = {c, d, e, f}Y = {g,h,i}


3 2 1 0a 3b 7X 6Y 2

Previously a member of X was winning!

Questions1. Which sets are clone structures?2. How to represent clone

structures?3. How to exploit clone structures?

An axiomatic characterization of clone structure

Compact representations of clone structures

A polynomial-time algorithm for decloning toward single-peaked elections

Preliminary results on characterizing single-peaked elections

Our Results

PQ-trees

voter representation

Part

1Pa

rt 2

A – alternative setF – a family of A subsets

F is a clone structure if and only if:

A1 {a} ∈ F for each a ∈ AA2 ∅ ∉ F, A ∈ F

Axiomatic Characterization

a b c d e



A1 {a} ∈ F for each a ∈ AA2 ∅ ∉ F, A ∈ FA3 If C1 and C2 are in F and C1 ⋂ C2

≠∅ then C1 ⋂ C2 and C1 ⋃ C2 are in F


a b c d e f





A4 If C1 and C2 are in F and C1 ⋈ C2 then C1 - C2 and C2 - C1 are in F

Axiomatic CharacterizationC1 ⋈ C2: C1 ⋂ C2 ≠∅ and

C1 - C2 ≠∅, C2 - C1≠∅

a b c d e f






A5 Each member of F has at most two minimal supersets in F.


a b

c d e f

ghi






A5 Each member of F has at most two minimal supersets in F.

A6 F is „acyclic”


a

bcd

e

fg

h

There are only two basic types of clone structures

Both satisfy our axioms, both compose induction

Proof Idea for the Characterization

(a) a string of sausages (b) a fat sausage

a b c da b c d





Our Results

PQ-treesPa

rt 1

Part

2


Clone Structure Representations

b c d e f a g h i

How to conveniently represent the above clone structure?


X

X = {a, b, c, d, e, f, g, h, i}

b c d e f a g h i

X


b Y a Z

X = {a, b, c, d, e, f, g, h, i}Y = {c,d, e, f}, Z = {g, h, i}

b c d e f a g h i

X

b Y a Z


b Y a g h i

X = {a, b, c, d, e, f, g, h, i}Y = {c,d, e, f}, Z = {g, h, i}

b c d e f a g h i

X

b Y a Z

g h i


b c d U a g h i

X = {a, b, c, d, e, f, g, h, i}Y = {c,d, e, f}, Z = {g, h, i}U = {e, f}

b c d e f a g h i

X

b Y a Z

g h ic d U


b c d e f a g h i


b c d e f a g h i

X

b Y a Z

g h ic d U

e f


b c d e f a g h i


b c d e f a g h i

X

b Y a Z

g h ic d U

e f

P-node – fat sausageQ-node – string of sausage

a b c d

How Many Voters Needed to Represent a Clone Structure?

Strings of sausages

a > b > c > d

A single voter suffices

a b c d

Fat sausages

a > b > c > dc > a > d > b

Two voters suffice …

a b c

a > b > ca > c > bb > a > c

The only fat sausage that needs three voters!


a b c 1 2 3 4

X

a 1 2 3 4 c

a > b > cb > a > c

1 > 2 > 3 > 44 > 2 > 3 > 1

a > 1 > 2 > 3 > 4 > c4 > 2 > 3 > 1 > a > c

Y X with Y in place of b


a b c 1 2 3 4

X Y X with Y in place of b

a 1 2 3 4 c

a > b > cb > a > c

1 > 2 > 3 > 44 > 2 > 3 > 1

a > 1 > 2 > 3 > 4 > c4 > 2 > 3 > 1 > a > c1 > 3 > 2 > 4 > a > c


a b c 1 2 3 4

X Y X with Y in place of b

a 1 2 3 4 c

a > b > cb > a > c

1 > 2 > 3 > 44 > 2 > 3 > 1

a > 1 > 2 > 3 > 4 > c1 > 3 > 2 > 4 > a > c

Theorem. For every clone structure F over alternative set A, there are three orders R1, R2, R3 that jointly generate F.





Our Results

PQ-trees

Part

1Pa

rt 2


Single-peakedness models votes in natural elections

Part 2: Clones in Single-Peaked Elections

a b c d

b > c > d > aa > b > c > d

Def. An election (A,R) is single-peaked with respect to an order > if for all c, d, e in A such that c > d > e (or e > d > c) and all Ri it holds that:

c Ri d ⇒ c Ri e

c > b > a > d

Single-peakedness models votes in natural elections

Part 2: Clones in Single-Peaked Elections

a b c d1 d2

b > c > d1 > d2 > aa > b > c > d1 > d2

Def. An election (A,R) is single-peaked with respect to an order > if for all c, d, e in A such that c > d > e (or e > d > c) and all Ri it holds that:

c Ri d ⇒ c Ri e

c > b > a > d2 > d1

Profile losessingle-peakednessdue to cloning

Decloning a clone set in (A,R)◦ Operation of contracting a clone-set into a single candidate

We have a polynomial-time algorithm that finds a decloning of a preference profile such that:◦ The profile becomes single-peaked◦ Maximum number of candidates remain in the election

Decloning Toward Single-Peakedness

b a

c d g h i

e f

Decloning◦ Operation of contracting a clone-set into a single candidate



b a

c d g h i

e f


b a

c d g h i

e f

Decloning◦ Operation of contracting a clone-set into a single candidate


It would be interesting to know what clones structures can be implemented by single-peaked profiles

◦ Not all clone structures can be!

◦ However, all clone structures whose tree representation contains P-nodes only can be implemented

◦ Work in progress!

Characterizing Single-Peaked Clone Structures

Clone structures form an interesting mathematical object

Clones can be used in various ways to manipulate elections; understanding clone structures helps in this respect.

Clones can spoil single-peakedness of an election; decloning toward single-peakedness can be a useful preprocessing step when holding an election.

Conclusions

Thank You!

Intermediate Preferences

a > b > c > d > eb > a > c > d > eb > c > a > d > ec > b > a > e > dc > b > e > a > d

Breaking News!

Intermediate Preferences

a > b > c > d > eb > a > c > d > eb > c > a > d > ec > b > a > e > dc > b > e > a > d

Breaking News!

Every clone structure can be implemented

Decloning toward intermediate preferences is NP-complete

COMSOC-2012 in Kraków, Poland

clone structures in voters ’ preferences

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