A simple construction of complete single-peaked domains by recursive tiling

Abstract

Single-peakedness was introduced by Black (J Political Econ 56:23-34, 1948) as a sufficient condition to overcome Condorcet paradox. Since then it has been attracting interest from researchers in various fields. In this paper, we propose a simple recursive procedure of constructing complete single-peaked domains of tiling type explicitly for any finite alternative sets, by combining two results published in recent years, and some observations of known results and examples by the author. The underlying basic structure of tiling type and properties of single-peaked domains provided here give a good visualization and make further developments on single-peakedness more easy.

Constructing large peak-pit Condorcet domains

“….ðn þ 1Þ. Zhan (2019) observed this recursive property of the single-peaked domains considering their tilling representations.…”

Extended Random Assignment Mechanisms on a Family of Good Sets

“…Several other generalizations of allocation mechanisms with additional conditions or hybrid types have been developed, refer to the papers [2,3,6,9,10,28] and reference therein. Allocating resource summation, similar to the base equality of polymatroids, is treated in a recent paper by Bochet and Tumennasan [29], which is related to single-peakedness, Zhan [30]. Finally, in the case of the dichotomous domain with a matroidal family of goods, we expect the strategy-proofness property to be kept.…”

Sign representation of single-peaked preferences and Bruhat orders

“…The labels on the directed path of each snake form a linear order, or a permutation of alternatives of [n]. Also refer to [7,20] for more details of rhombus tilings and preference domains, refer to [10] for more general linear orders and permutations.…”

“…tiles, or reverse pairs, in each tiling on [n]. The right snakes have more inversion pairs than the left ones in a Bruhat poset, all snakes of a tiling also form a lattice [10,20].…”

“…The structure and characterization of single-peaked domains have been researched extensively [9,18], see also a review paper [15]. Recently, single-peakedness is studied in various topics, for applications in matching and assignment [2], counting and distribution [13], construction by tiling [20], forbidden configuration [3], and finding optimal committees under proportional approval voting [17].…”