Condorcet domains, median graphs and the single-crossing property

Clemens Puppe () and Arkadii Slinko

No 92, Working Paper Series in Economics from Karlsruhe Institute of Technology (KIT), Department of Economics and Management

Abstract: Condorcet domains are sets of linear orders with the property that, whenever the preferences of all voters of a society belong to this set, their majority relation has no cycles. We observe that, without loss of generality, every 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 voters whose preferences belong to this domain. We show that every closed Condorcet domain can be endowed with the structure of a median graph and that, conversely, every median graph is associated with a closed Condorcet domain (in general, not uniquely). Condorcet domains that have linear graphs (chains) associated with them are exactly the preference domains with the classical single-crossing property. As a corollary, we obtain that a domain with the so-called 'representative voter property' is either a single-crossing domain or a very special domain containing exactly four different preference orders whose associated median graph is a 4-cycle. Maximality of a Condorcet domain imposes additional restrictions on the associated median graph. We prove that among all trees only (some) chains can be associated graphs of maximal Condorcet domains, and we characterize those single-crossing domains which are maximal Condorcet domains. Finally, using the characterization of Nehring and Puppe [2007] of monotone Arrovian aggregation, our analysis yields a rich class of strategy-proof social choice functions on any closed Condorcet domain.

Keywords: social choice; Condorcet domains; acyclic sets of linear orders; median graphs; single-crossing property; distributive lattice; Arrovian aggregation; strategy-proofness; intermediate preferences (search for similar items in EconPapers)
JEL-codes: C72 D71 (search for similar items in EconPapers)
Date: 2016
New Economics Papers: this item is included in nep-gth, nep-mic and nep-pol
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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Persistent link: https://EconPapers.repec.org/RePEc:zbw:kitwps:92

DOI: 10.5445/IR/1000061997

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