The single-peaked domain revisited: A simple global characterization

Abstract

for particularly helpful comments.Abstract It is proved that, among all restricted preference domains that guarantee consistency (i.e. transitivity) of pairwise majority voting, the single-peaked domain is the only minimally rich and connected domain that contains two completely reversed strict preference orders. It is argued that this result explains the predominant role of single-peakedness as a domain restriction in models of political economy and elsewhere. The main result has a number of corollaries, among them a dual characterization of the single-dipped domain; it also implies that a single-crossing ('order-restricted') domain can be minimally rich only if it is a subdomain of a single-peaked domain. The conclusions are robust as the results apply both to domains of strict and of weak preference orders, respectively. JEL Classification D71, C72

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

“…The four orders {123, 213, 231, 321} on the left path form a single-peaked domain, the four orders {123, 132, 321, 321} on the right path from a pit domain. Now we introduce a theorem related to single-peaked domains in paper (Puppe 2018) published recently. It is one of two main results which our recursive procedure is based on.…”

“…Given a single-peaked domain, a winner can be determined by a simple majority voting rule. Since then, single-peaked domains and related topics have been extensively researched, see the review paper (Monjardet 2009) and also a recently published paper by Puppe (2018).…”

Testing a mixture model of single-peaked preferences

“…Ballester and Haeringer [1] give two forbidden substructures, whose absence is a necessary and sufficient condition for the given preference profile to be consistent with single-peakedness. Puppe [26] gives a global characterization of the singlepeaked domain. Furthermore, Trick [33] provides an algorithm for recognizing single-peakedness on trees, which again runs in polynomial time.…”

On single-peaked domains and min–max rules

“…Lemma 6.1(Elkind et al (2014),Puppe (2016)). Every regular single-crossing set of preferences is single-peaked.…”

“…See, for instance, the domain restriction considered in models of voting(Tullock (1967),Arrow (1969)), taxation and redistribution(Epple and Romer (1991)), determining the levels of income redistribution(Hamada (1973),Slesnick (1988)), and measuring tax reforms in the presence of horizontal inequity(Hettich (1979)). Recently,Puppe (2016) shows that under mild conditions these domains form subsets of the maximal single-peaked domain.4 Chatterji and Sen (2011) provide a sufficient condition on a domain so that every unanimous and strategy-proof SCF on it is tops-only. However, an arbitrary single-peaked domain does not satisfy their condition.5 The top-connectedness property is well studied in the literature (seeBarberà and Peleg (1990),Aswal et al (2003),Chatterji and Sen (2011),Chatterji et al (2014),Chatterji and Zeng (2015), Puppe (2016)) andRoy and Storcken (2016).…”