Condorcet Domains, Median Graphs and the Single Crossing Property

TL;DR

This paper explores the mathematical structure of Condorcet domains, showing their connection to median graphs and the single crossing property, and characterizes maximal domains and strategy-proof social choice functions.

Contribution

It establishes a novel correspondence between closed Condorcet domains and median graphs, and characterizes single crossing and maximal domains within this framework.

Findings

Closed Condorcet domains are naturally median graphs.

Single crossing domains correspond to linear graphs (chains).

Maximal Condorcet domains are induced only by chains among trees.

Abstract

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.