On the single-peakedness property

On the single-peakedness property

Jimmy Devillet

University of Luxembourg Luxembourg

Single-peaked orderings

Motivating example (Romero, 1978)

Suppose you are asked to order the following six objects in decreasing preference:

a

: 0 sandwich a

: 1 sandwich a

: 2 sandwiches a

: 3 sandwiches a

: 4 sandwiches

a

: more than 4 sandwiches

We write a

≺ a

if a

is preferred to a

Single-peaked orderings

a₁: 0 sandwich a₂: 1 sandwich a₃: 2 sandwiches a4: 3 sandwiches a5: 4 sandwiches

a6: more than 4 sandwiches

after a good lunch: a1≺ a2≺ a3≺ a4≺ a5≺ a6

if you are starving: a6≺ a5≺ a4≺ a3≺ a2≺ a1

a possible intermediate situation: a₄≺ a3≺ a5≺ a2≺ a1≺ a6

a quite unlikely preference: a₆≺ a₅≺ a₂≺ a₁≺ a₃≺ a₄

Single-peaked orderings

Let us represent graphically the latter two preferences with respect to the reference ordering a1< a2< a3< a4< a5< a6

a₄≺ a3≺ a5≺ a2≺ a1≺ a6 a₆≺ a5≺ a2≺ a1≺ a3≺ a4

- 6

a₁ a₂ a₃ a₄ a₅ a₆ a₆

a1

a2

a5

a3

a4

A

A A

B B

B B

B r

r r

r

r

r

- 6

a₁ a₂ a₃ a₄ a₅ a₆ a₄

a3

a1

a2

a5

a6

A A

A

@@









 r

r

r r

r r

Single-peaked orderings

Definition. (Black, 1948)

Let ≤ and  be total orderings on Xn= {a1, . . . , an}.

Then  is said to besingle-peaked for ≤ if the following patterns are forbidden

- 6

a_i a_j a_k aj

ai

a_k

@

@

s

s s

- 6

a_i a_j a_k aj

ak

a_i A

A A

A s

s s

Mathematically:

a

< a

< a

=⇒ a

≺ a

or a

≺ a

Single-peaked orderings

ai< aj < ak =⇒ aj≺ ai or aj ≺ ak

Let us assume that Xn= {a1, . . . , an} is endowed with the ordering a1< · · · < an

For n = 4

a1≺ a2≺ a3≺ a4 a4≺ a3≺ a2≺ a1

a₂≺ a1≺ a3≺ a4 a₃≺ a2≺ a1≺ a4

a₂≺ a3≺ a1≺ a4 a₃≺ a2≺ a4≺ a1

a₂≺ a₃≺ a₄≺ a₁ a₃≺ a₄≺ a₂≺ a₁

There are 2ⁿ⁻¹total orderings  on Xn that are single-peaked for ≤

Weak orderings

Recall that aweak ordering(ortotal preordering) on Xn is a binary relation - on Xⁿ that is total and transitive.

Defining a weak ordering on Xn amounts to defining an ordered partition of Xn

C1≺ · · · ≺ Ck

where C₁, . . . , C_k are the equivalence classes defined by ∼ For n = 3, we have 13 weak orderings

a₁≺ a₂≺ a₃ a₁∼ a₂≺ a₃ a₁∼ a₂∼ a₃ a₁≺ a₃≺ a₂ a₁≺ a₂∼ a₃

a2≺ a1≺ a3 a2≺ a1∼ a3

a2≺ a3≺ a1 a3≺ a1∼ a2

a3≺ a1≺ a2 a1∼ a3≺ a2

a3≺ a2≺ a1 a2∼ a3≺ a1

Single-plateaued weak orderings

Definition. (Black, 1948)

Let ≤ be a total ordering on Xnand let - be a weak ordering on Xⁿ. Then - is said to besingle-plateaued for ≤if the following patterns are forbidden

- 6

ai aj ak

a_j a_i a_k

@@

r

r r

- 6

ai aj ak

a_j a_k a_i

A A

A r

r r

- 6

ai aj ak

a_j a_j ∼ ak

A A

A

r

r r

- 6

a_i a_j a_k a_i ∼ a_j

a_k

r r

r

- 6

a_i a_j a_k a_j ∼ a_k

a_i A

A A r

r r

Single-plateaued weak orderings

Mathematically:

a_i< a_j < a_k =⇒ a_j ≺ ai or a_j ≺ ak or a_i ∼ aj ∼ ak

Examples

a₃∼ a4≺ a2≺ a1∼ a5≺ a6 a₃∼a₄≺ a2∼ a1≺ a5≺ a6

- 6

a1 a2 a3 a4 a5 a6

a6

a1∼ a5

a2

a3∼ a4

A A

A

@@ r

r r r

r r

- 6

a1 a2 a3 a4 a5 a6

a6

a5

a1∼ a2

a3∼ a4

A A

A

@@ r r

r r

r r

Single-plateaued weak orderings

n ∈ N

u(n) : number of weak orderings on Xn that are single-plateaued for ≤ (OEIS: A048739)

Proposition (Couceiro,D.,Marichal, 2019) We have the closed-form expression

2 u(n) + 1 = ¹₂(1 +√

2)ⁿ⁺¹+¹₂(1 −√

2)ⁿ⁺¹ = P

k≥0 n+1

2k
2^k

u(0) = 0, u(1) = 1, u(2) = 3, u(3) = 8, u(4) = 20, ...

Example. u(3) = 8

a1≺ a2≺ a3 a1∼ a2≺ a3 a1∼ a2∼ a3

a2≺ a1≺ a3 a2≺ a3≺ a1 a2≺ a1∼ a3

a3≺ a2≺ a1 a3∼ a2≺ a1

Single-plateaued weak orderings

Q: Given - is it possible to find ≤ for which - is single-plateaued?

Example: On X4= {a1, a2, a3, a4} consider-and-⁰ defined by a1∼a2≺a3∼a4 and a1≺⁰a2∼⁰a3∼⁰a4

Yes! Consider ≤ defined by a₃< a₁< a₂< a₄

- 6

a₃ a₁ a₂ a₄ a3∼ a4

a1∼ a2

J

J r J

r r r

No!

2-quasilinear weak orderings

Definition.

We say that - is2-quasilinearif

a ≺ b ∼ c ∼ d =⇒ a, b, c, d are not pairwise distinct

Proposition (D., Marichal, Teheux) We have

- is 2-quasilinear ⇐⇒ ∃ ≤ for which - is single-plateaued

2-quasilinear weak orderings

v (n) : number of weak orderings on Xn that are 2-quasilinear (OEIS:

A307005)

Proposition (D., Marichal, Teheux) We have the closed-form expression

v (n) =

n

X

k=0

n!

(n + 1 − k)!Gk, n ≥ 1, where Gn=

√ 3 3 (¹⁺

√ 3 2 )ⁿ−

√ 3 3 (¹⁻

√ 3 2 )ⁿ.

v (0) = 0, v (1) = 1, v (2) = 3, v (3) = 13, v (4) = 71, ...

Some references

S. Berg and T. Perlinger.

Single-peaked compatible preference profiles: some combinatorial results.

Social Choice and Welfare, 27(1):89–102, 2006.

D. Black.

On the rationale of group decision-making.

J Polit Economy, 56(1):23–34, 1948.

M. Couceiro, J. Devillet, and J.-L. Marichal.

Quasitrivial semigroups: characterizations and enumerations.

Semigroup Forum, 98(3):472–498, 2019.

J. Devillet, J.-L. Marichal, and B. Teheux.

Classifications of quasitrivial semigroups.

arXiv:1811.11113.

Z. Fitzsimmons.

Single-peaked consistency for weak orders is easy.

In Proc. of the 15th Conf. on Theoretical Aspects of Rationality and Knowledge (TARK 2015), pages 127–140, June 2015. arXiv:1406.4829.