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 a 2 : 1 sandwich a 3 : 2 sandwiches a 4 : 3 sandwiches a 5 : 4 sandwiches a 6 : more than 4 sandwiches We write a i ≺ a j if a i is preferred to a j

Single-peaked orderings a 1 : 0 sandwich a 2 : 1 sandwich a 3 : 2 sandwiches a 4 : 3 sandwiches a 5 : 4 sandwiches a 6 : more than 4 sandwiches after a good lunch: a 1 ≺ a 2 ≺ a 3 ≺ a 4 ≺ a 5 ≺ a 6 if you are starving: a 6 ≺ a 5 ≺ a 4 ≺ a 3 ≺ a 2 ≺ a 1 a possible intermediate situation: a 4 ≺ a 3 ≺ a 5 ≺ a 2 ≺ a 1 ≺ a 6 a quite unlikely preference: a 6 ≺ a 5 ≺ a 2 ≺ a 1 ≺ a 3 ≺ a 4

Single-peaked orderings Let us represent graphically the latter two preferences with respect to the reference ordering a 1 < a 2 < a 3 < a 4 < a 5 < a 6 a 4 ≺ a 3 ≺ a 5 ≺ a 2 ≺ a 1 ≺ a 6 a 6 ≺ a 5 ≺ a 2 ≺ a 1 ≺ a 3 ≺ a 4 ✻ ✻ a 4 a 6 r r �❆ � � � a 3 a 5 ❆ r r ✁ ✄ ❆ a 5 ✁ a 2 ✄ r r ❇ �❆ ✁ � ✄ a 2 ❇ a 1 ❆ r r � ✄ ❇ � ❆ a 1 a 3 ✄ ❇ r r ❅ ❇ ❅✄ a 6 a 4 r r ✲ ✲ a 1 a 2 a 3 a 4 a 5 a 6 a 1 a 2 a 3 a 4 a 5 a 6

Single-peaked orderings Definition . (Black, 1948) Let ≤ and � be total orderings on X n = { a 1 , . . . , a n } . Then � is said to be single-peaked for ≤ if the following patterns are forbidden ✻ ✻ a k a i s s ✁ ❆ ✁ ❆ a i a k s s ✁ ❆ ❅ � ❅✁ ❆� a j a j s s ✲ ✲ a j a j a i a k a i a k Mathematically: = ⇒ a j ≺ a i or a j ≺ a k a i < a j < a k

Single-peaked orderings a i < a j < a k = ⇒ a j ≺ a i or a j ≺ a k Let us assume that X n = { a 1 , . . . , a n } is endowed with the ordering a 1 < · · · < a n For n = 4 a 1 ≺ a 2 ≺ a 3 ≺ a 4 a 4 ≺ a 3 ≺ a 2 ≺ a 1 a 2 ≺ a 1 ≺ a 3 ≺ a 4 a 3 ≺ a 2 ≺ a 1 ≺ a 4 a 2 ≺ a 3 ≺ a 1 ≺ a 4 a 3 ≺ a 2 ≺ a 4 ≺ a 1 a 2 ≺ a 3 ≺ a 4 ≺ a 1 a 3 ≺ a 4 ≺ a 2 ≺ a 1 There are 2 n − 1 total orderings � on X n that are single-peaked for ≤

Weak orderings Recall that a weak ordering (or total preordering ) on X n is a binary relation � on X n that is total and transitive. Defining a weak ordering on X n amounts to defining an ordered partition of X n C 1 ≺ · · · ≺ C k where C 1 , . . . , C k are the equivalence classes defined by ∼ For n = 3, we have 13 weak orderings a 1 ≺ a 2 ≺ a 3 a 1 ∼ a 2 ≺ a 3 a 1 ∼ a 2 ∼ a 3 a 1 ≺ a 3 ≺ a 2 a 1 ≺ a 2 ∼ a 3 a 2 ≺ a 1 ≺ a 3 a 2 ≺ a 1 ∼ a 3 a 2 ≺ a 3 ≺ a 1 a 3 ≺ a 1 ∼ a 2 a 3 ≺ a 1 ≺ a 2 a 1 ∼ a 3 ≺ a 2 a 3 ≺ a 2 ≺ a 1 a 2 ∼ a 3 ≺ a 1

Single-plateaued weak orderings Definition . (Black, 1948) Let ≤ be a total ordering on X n and let � be a weak ordering on X n . Then � is said to be single-plateaued for ≤ if the following patterns are forbidden ✻ ✻ ✻ a j ∼ a k a k a i r r r r ✁ ❆ ❆ ✁ a i ✁ a k ❆ ❆ ✁ r r ❅ � ❅✁ ❆� ❆✁ a j a j a j r r r ✲ ✲ ✲ a i a j a k a i a j a k a i a j a k ✻ ✻ a k a i r r ✁ ❆ ✁ ❆ ✁ ❆ a i ∼ a j a j ∼ a k r r r r ✲ ✲ a i a j a k a i a j a k

Single-plateaued weak orderings Mathematically: a i < a j < a k = ⇒ a j ≺ a i or a j ≺ a k or a i ∼ a j ∼ a k Examples a 3 ∼ a 4 ≺ a 2 ≺ a 1 ∼ a 5 ≺ a 6 a 3 ∼ a 4 ≺ a 2 ∼ a 1 ≺ a 5 ≺ a 6 ✻ ✻ a 3 ∼ a 4 a 3 ∼ a 4 r r r r � ❆ � ❆ � � a 2 a 1 ∼ a 2 ❆ ❆ r r r � � ❆ ❆ a 1 ∼ a 5 a 5 r r r ❅ ❅ ❅ ❅ a 6 a 6 r r ✲ ✲ a 1 a 2 a 3 a 4 a 5 a 6 a 1 a 2 a 3 a 4 a 5 a 6

Single-plateaued weak orderings n ∈ N u ( n ) : number of weak orderings on X n that are single-plateaued for ≤ (OEIS: A048739) Proposition (Couceiro,D.,Marichal, 2019) We have the closed-form expression √ √ 2) n +1 + 1 � n +1 1 2) n +1 � 2 k 2 u ( n ) + 1 = 2 (1 + 2 (1 − = � k ≥ 0 2 k u (0) = 0, u (1) = 1, u (2) = 3, u (3) = 8, u (4) = 20, ... Example. u (3) = 8 a 1 ≺ a 2 ≺ a 3 a 1 ∼ a 2 ≺ a 3 a 1 ∼ a 2 ∼ a 3 a 2 ≺ a 1 ≺ a 3 a 2 ≺ a 3 ≺ a 1 a 2 ≺ a 1 ∼ a 3 a 3 ≺ a 2 ≺ a 1 a 3 ∼ a 2 ≺ a 1

Single-plateaued weak orderings Q: Given � is it possible to find ≤ for which � is single-plateaued? Example: On X 4 = { a 1 , a 2 , a 3 , a 4 } consider � and � ′ defined by a 1 ≺ ′ a 2 ∼ ′ a 3 ∼ ′ a 4 a 1 ∼ a 2 ≺ a 3 ∼ a 4 and Yes! Consider ≤ defined by a 3 < a 1 < a 2 < a 4 ✻ a 1 ∼ a 2 r r ✡ ❏ ✡ ❏ ✡ ❏ a 3 ∼ a 4 r r ✲ a 3 a 1 a 2 a 4 No!

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

2-quasilinear weak orderings v ( n ) : number of weak orderings on X n that are 2-quasilinear (OEIS: A307005) Proposition (D., Marichal, Teheux) We have the closed-form expression n n ! � v ( n ) = ( n + 1 − k )! G k , n ≥ 1 , k =0 √ √ √ √ ) n − 3 ( 1+ 3 3 3 ( 1 − 3 3 ) n . where G n = 2 2 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.