49 research outputs found
Condorcet domains of tiling type
A Condorcet domain (CD) is a collection of linear orders on a set of
candidates satisfying the following property: for any choice of preferences of
voters from this collection, a simple majority rule does not yield cycles. We
propose a method of constructing "large" CDs by use of rhombus tiling diagrams
and explain that this method unifies several constructions of CDs known
earlier. Finally, we show that three conjectures on the maximal sizes of those
CDs are, in fact, equivalent and provide a counterexample to them.Comment: 16 pages. To appear in Discrete Applied Mathematic
Towards a classification of maximal peak-pit Condorcet domains
In this paper, we classify all maximal peak-pit Condorcet domains of maximal width for n ≤ 5 alternatives. To achieve this, we bring together ideas from several branches of combinatorics. The main tool used in the classification is the ideal of a domain. In contrast to the size of maximal peak-pit Condorcet domains of maximal width themselves, the size of their associated ideal is constant
Condorcet Domains, Median Graphs and the Single Crossing Property
Condorcet domains are sets of linear orders with the property that, whenever
the preferences of all voters belong to this set, the majority relation has no
cycles. We observe that, without loss of generality, such domain can be assumed
to be closed in the sense that it contains the majority relation of every
profile with an odd number of individuals whose preferences belong to this
domain.
We show that every closed Condorcet domain is naturally endowed with the
structure of a median graph and that, conversely, every median graph is
associated with a closed Condorcet domain (which may not be a unique one). The
subclass of those Condorcet domains that correspond to linear graphs (chains)
are exactly the preference domains with the classical single crossing property.
As a corollary, we obtain that the domains with the so-called `representative
voter property' (with the exception of a 4-cycle) are the single crossing
domains.
Maximality of a Condorcet domain imposes additional restrictions on the
underlying median graph. We prove that among all trees only the chains can
induce maximal Condorcet domains, and we characterize the single crossing
domains that in fact do correspond to maximal Condorcet domains.
Finally, using Nehring's and Puppe's (2007) characterization of monotone
Arrowian aggregation, our analysis yields a rich class of strategy-proof social
choice functions on any closed Condorcet domain
A family of Condorcet domains that are single-peaked on a circle
Fishburn's alternating scheme domains occupy a special place in the theory of
Condorcet domains. Karpov (2023) generalised these domains and made an
interesting observation proving that all of them are single-picked on a circle.
However, an important point that all generalised Fishburn domains are maximal
Condorcet domain remained unproved. We fill this gap and suggest a new
combinatorial interpretation of generalised Fishburn's domains which provide a
constructive proof of single-peakedness of these domains on a circle. We show
that classical single-peaked domains and single-dipped domains as well as
Fishburn's alternating scheme domains belong to this family of domains while
single-crossing domains do not.Comment: 9 page
Bipartite peak-pit domains
In this paper, we introduce the class of bipartite peak-pit domains. This is
a class of Condorcet domains which include both the classical single-peaked and
single-dipped domains. Our class of domains can be used to model situations
where some alternatives are ranked based on a most preferred location on a
societal axis, and some are ranked based on a least preferred location. This
makes it possible to model situations where agents have different rationales
for their ranking depending on which of two subclasses of the alternatives one
is considering belong to. The class of bipartite peak-pit domains includes most
peak-pit domains for alternatives, and the largest Condorcet domains
for each .
In order to study the maximum possible size of a bipartite peak-pit domain we
introduce set-alternating schemes. This is a method for constructing
well-structured peak-pit domains which are copious and connected. We show that
domains based on these schemes always have size at least and some of
them have sizes larger than the domains of Fishburn's alternating scheme. We
show that the maximum domain size for sufficiently high exceeds .
This improves the previous lower bound for peak-pit domains from
\cite{karpov2023constructing}, which was also the highest asymptotic lower
bound for the size of the largest Condorcet domains