Edith Elkind Nanyang Technological University, Singapore Piotr Faliszewski AGH Univeristy of Science and Technology, Poland Arkadii Slinko University of Auckland New Zealand

Example R 1 :b>c> d>e >f>a>g>h > i R 2 :e>f> d>c >i>g>h>b > a R 3 :b>a> c>d >e>f>g>i > h a  10 b  17 c  18 d  17 e  17 f  14 g  7 h  3 i  4 Def. An election is a pair (A,R) where A is the set of alternatives and R = (R 1, …, R n ) is voters’ preference profile. Each R i is a total linear order over A.

Example R 1 :b>c> d>e >f>a>g>h > i R 2 :e>f> d>c >i>g>h>b > a R 3 :b>a> c>d >e>f>g>i > h a  10 b  17 c  18 d  17 e  17 f  14 g  7 h  3 i  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 R 1 :b>c> d>e >f>a>g>h > i R 2 :e>f> d>c >i>g>h>b > a R 3 :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} } a  10 b  17 c  18 d  17 e  17 f  14 g  7 h  3 i  4

Example R 1 :b>c> d>e >f>a>g>h > i R 2 :e>f> d>c >i>g>h>b > a R 3 :b>a> c>d >e>f>g>i > h X = {c, d, e, f} Y = {g,h,i} a  10 b  17 c  18 d  17 e  17 f  14 g  7 h  3 i  4

Example R 1 :b>X>a>Y R 2 :X >Y>b > a R 3 :b>a> X>Y X = {c, d, e, f} Y = {g,h,i} a  3 b  7 X  6 Y  2 Previously a member of X was winning! Questions 1.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 PQ-trees voter representation Part 1 Part 2

A – alternative set F – a family of A subsets F is a clone structure if and only if: A1 {a} ∈ F for each a ∈ A A2 ∅ ∉ F, A ∈ F a b c d e

A – alternative set F – a family of A subsets F is a clone structure if and only if: A1 {a} ∈ F for each a ∈ A A2 ∅ ∉ F, A ∈ F A3 If C 1 and C 2 are in F and C 1 ⋂ C 2 ≠∅ then C 1 ⋂ C 2 and C 1 ⋃ C 2 are in F a b c d e f

A – alternative set F – a family of A subsets F is a clone structure if and only if: A1 {a} ∈ F for each a ∈ A A2 ∅ ∉ F, A ∈ F A3 If C 1 and C 2 are in F and C 1 ⋂ C 2 ≠∅ then C 1 ⋂ C 2 and C 1 ⋃ C 2 are in F A4 If C 1 and C 2 are in F and C 1 ⋈ C 2 then C 1 - C 2 and C 2 - C 1 are in F C 1 ⋈ C 2 : C 1 ⋂ C 2 ≠∅ and C 1 - C 2 ≠∅, C 2 - C 1 ≠∅ a b c d e f

A – alternative set F – a family of A subsets F is a clone structure if and only if: A1 {a} ∈ F for each a ∈ A A2 ∅ ∉ F, A ∈ F A3 If C 1 and C 2 are in F and C 1 ⋂ C 2 ≠∅ then C 1 ⋂ C 2 and C 1 ⋃ C 2 are in F A4 If C 1 and C 2 are in F and C 1 ⋈ C 2 then C 1 - C 2 and C 2 - C 1 are in F A5 Each member of F has at most two minimal supersets in F. a b c d e f ghighi

A – alternative set F – a family of A subsets F is a clone structure if and only if: A1 {a} ∈ F for each a ∈ A A2 ∅ ∉ F, A ∈ F A3 If C 1 and C 2 are in F and C 1 ⋂ C 2 ≠∅ then C 1 ⋂ C 2 and C 1 ⋃ C 2 are in F A4 If C 1 and C 2 are in F and C 1 ⋈ C 2 then C 1 - C 2 and C 2 - C 1 are in F A5 Each member of F has at most two minimal supersets in F. A6 F is „acyclic” a b c d e f g h

 There are only two basic types of clone structures  Both satisfy our axioms, both compose  induction (a) a string of sausages(b) a fat sausage a b c d

 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 PQ-trees Part 1 Part 2 voter representation

bc de fag h i  How to conveniently represent the above clone structure?

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

b c d e f 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} bc de fag h i X b Y a Z g h i c d U e f

b c d e f 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} bc de fag h i X b Y a Z g h i c d U e f P-node – fat sausage Q-node – string of sausage

a b c d Strings of sausages a > b > c > d A single voter suffices a b c d Fat sausages a > b > c > d c > a > d > b Two voters suffice … a b c a > b > c a > c > b b > a > c The only fat sausage that needs three voters!

a b c X a c a > b > c b > a > c 1 > 2 > 3 > 4 4 > 2 > 3 > 1 a > 1 > 2 > 3 > 4 > c 4 > 2 > 3 > 1 > a > c YX with Y in place of b

a b c X Y X with Y in place of b a c a > b > c b > a > c 1 > 2 > 3 > 4 4 > 2 > 3 > 1 a > 1 > 2 > 3 > 4 > c 4 > 2 > 3 > 1 > a > c 1 > 3 > 2 > 4 > a > c

a b c X Y X with Y in place of b a c a > b > c b > a > c 1 > 2 > 3 > 4 4 > 2 > 3 > 1 a > 1 > 2 > 3 > 4 > c 1 > 3 > 2 > 4 > a > c Theorem. For every clone structure F over alternative set A, there are three orders R 1, R 2, R 3 that jointly generate F.

 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 PQ-trees Part 1 Part 2 voter representation

Single-peakedness models votes in natural elections a b c d b > c > d > a a > 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 R i it holds that: c R i d ⇒ c R i e c > b > a > d

Single-peakedness models votes in natural elections a b c d 1 d 2 b > c > d 1 > d 2 > a a > b > c > d 1 > d 2 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 R i it holds that: c R i d ⇒ c R i e c > b > a > d 2 > d 1 Profile loses single-peakedness due 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 b a cd g h i ef

 Decloning ◦ 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 b a cd g h i ef

b a cd ef  Decloning ◦ 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

b a cd g h i ef  Decloning ◦ 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

 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!

 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. Thank You!

Intermediate Preferences a > b > c > d > e b > a > c > d > e b > c > a > d > e c > b > a > e > d c > b > e > a > d

Intermediate Preferences a > b > c > d > e b > a > c > d > e b > c > a > d > e c > b > a > e > d c > b > e > a > d Every clone structure can be implemented Decloning toward intermediate preferences is NP-complete