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On the single-peakedness property Jimmy Devillet University of - - PowerPoint PPT Presentation
On the single-peakedness property Jimmy Devillet University of - - PowerPoint PPT Presentation
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 1 : 0 sandwich
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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
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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 ✲ ✻ a1 a2 a3 a4 a5 a6 a6 a1 a2 a5 a3 a4
- ✁
✁ ✁
- ❆
❆ ❆ ❇ ❇ ❇ ❇ ❇ r r r r r r ✲ ✻ a1 a2 a3 a4 a5 a6 a4 a3 a1 a2 a5 a6
- ❆
❆ ❆ ❅ ❅✄ ✄ ✄ ✄ ✄ ✄
- r
r r r r r
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Single-peaked orderings
- Definition. (Black, 1948)
Let ≤ and be total orderings on Xn = {a1, . . . , an}. Then is said to be single-peaked for ≤ if the following patterns are forbidden ✲ ✻ ai aj ak aj ai ak ❅ ❅✁ ✁ ✁ ✁ s s s ✲ ✻ ai aj ak aj ak ai ❆ ❆ ❆ ❆
- s
s s
Mathematically: ai < aj < ak = ⇒ aj ≺ ai
- r
aj ≺ ak
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Single-peaked orderings
ai < aj < ak = ⇒ aj ≺ ai
- r
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−1 total orderings on Xn that are single-peaked for ≤
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Weak orderings
Recall that a weak ordering (or total 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
- f Xn
C1 ≺ · · · ≺ Ck where C1, . . . , Ck 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
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Single-plateaued weak orderings
- Definition. (Black, 1948)
Let ≤ be a total ordering on Xn and let be a weak ordering on Xn. Then is said to be single-plateaued for ≤ if the following patterns are forbidden ✲ ✻ ai aj ak aj ai ak ❅ ❅✁ ✁ ✁ r r r ✲ ✻ ai aj ak aj ak ai ❆ ❆ ❆
- r
r r ✲ ✻ ai aj ak aj aj ∼ ak ❆ ❆ ❆✁ ✁ ✁ r r r ✲ ✻ ai aj ak ai ∼ aj ak ✁ ✁ ✁ r r r ✲ ✻ ai aj ak aj ∼ ak ai ❆ ❆ ❆ r r r
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Single-plateaued weak orderings
Mathematically: ai < aj < ak = ⇒ aj ≺ ai
- r
aj ≺ ak
- r
ai ∼ aj ∼ ak Examples a3 ∼ a4 ≺ a2 ≺ a1 ∼ a5 ≺ a6 a3 ∼ a4 ≺ a2 ∼ a1 ≺ a5 ≺ a6 ✲ ✻ a1 a2 a3 a4 a5 a6 a6 a1 ∼ a5 a2 a3 ∼ a4
- ❆
❆ ❆ ❅ ❅ r r r r r r ✲ ✻ a1 a2 a3 a4 a5 a6 a6 a5 a1 ∼ a2 a3 ∼ a4
- ❆
❆ ❆ ❅ ❅ r r r r r r
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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 =
1 2(1 +
√ 2)n+1 + 1
2(1 −
√ 2)n+1 =
- k≥0
n+1
2k
- 2k
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
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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 ′ defined by a1∼a2≺a3∼a4 and a1≺′a2∼′a3∼′a4 Yes! Consider ≤ defined by a3 < a1 < a2 < a4 ✲ ✻ a3 a1 a2 a4 a3 ∼ a4 a1 ∼ a2 ✡ ✡ ✡ ❏ ❏ ❏ r r r r No!
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2-quasilinear weak orderings
Definition. We say that is 2-quasilinear if 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
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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
- 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, ...
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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.