Abstract

Fishburn’s alternating scheme domains occupy a special place in the theory of Condorcet domains as brought to light by Arkadii Slinko at The University of Auckland. Recently, Karpov (2023) generalised these domains and made an interesting observation proving that all of them are single-peaked on a circle. However, an important point that all generalised Fishburn domains are maximal Condorcet domain remained unproved. In this blog, we provide an in-depth explanation of this intriguing mathematical domain and offer a new combinatorial interpretation for generalised Fishburn's domains.

1 Preliminaries

A Condorcet domain is a set of linear orders on a given set of alternatives. This means that if all voters of a certain society have preferences over those alternatives which are represented by linear orders, the pairwise majority relation of this society is acyclic. The domain of single-peaked linear orders on a line spectrum is one of the best-known Condorcet domains. Recently, these theories have been expanded upon to generalise single-peaked domains on a circle. These intricate mathematical domains can seem nebulous, but their understanding and execution are crucial.

2 Basics of Condorcet Domains

Before diving into the complexities of Condorcet domains, let's take a look at some of the basics. For a linear order v∈ L(A) and two alternatives x, y∈A we write x≻vy if v ranks x higher than y. The set of alternatives A is often taken as [n] ={1,2, . . . , n}. Up to an isomorphism, for n= 3 there are only three maximal Condorcet domains.

Avoiding the complications that might arise from more detailed definitions, any condition of type xN{a,b,c}i with x∈ {a, b, c} and i∈ {1,2,3} is called a 'never condition', because when it is applied to a domain it requires that alternative x never takes i-th position in orders of the restriction to {a, b, c}.

3 Property and Characterization of Condorcet Domains

A granular understanding of the properties of these domains offers us an opportunity to understand how they function. A domain of linear orders D ⊆ L(A) is a Condorcet domain if and only if it satisfies a complete set of never conditions. This criterion is a well-known characterisation of Condorcet domains, as described by Sen in 1966.

4 GF-Domains and Necklaces

An arrangement of vertices on a circle, some white and some black, are numbered by integers 1,2, . . . , n and called a 'necklace'. A set of beads is said to be 'white convex' or w-convex if it meets a certain set of conditions. These include the set being an arc of the circle and not consisting of a single black bead. These theoretical principles thereafter give rise to concepts like 'flags of w-convex sets' and the 'necklaces' derived from them.

5 Conclusion

Concepts like GF-Domains, never conditions, and the Condorcet domains can be complicated to understand, but they form the crux of intricate mathematical theories. These domains, their properties and utilisation are significantly influential to the theory of mathematical domains. Therefore, while topics such as this one may appear complicated at first glance, the underlying principles are not only fascinating but form the backbone of complex theories.