On Strategy‐proofness and the Salience of Single‐peakedness

Abstract

We consider strategy‐proof rules operating on a rich domain of preference profiles. We show that if the rule satisfies in addition tops‐onlyness, anonymity, and unanimity, then the preferences in the domain have to satisfy a variant of single‐peakedness (referred to as semilattice single‐peakedness). We do so by deriving from the rule an endogenous partial order (a semilattice) from which the concept of a semilattice single‐peaked preference can be defined. We also provide a converse of this main finding. Finally, we show how well‐known restricted domains under which nontrivial strategy‐proof rules are admissible are semilattice single‐peaked domains.

On strategy-proofness and semilattice single-peakedness

“…The domain D is rich on (A, )if, for all x, y, z ∈ A with z / ∈ [x, y] = ∅, there exist R x i , R y i ∈ D such that yP x i z and xP y i z.To illustrate richness return to Example 1 and consider, for instance, alternativesx 3 , x 5 , x 8 ∈ A for which x 8 / ∈ [x 3 , x 5 ] = ∅. In this case, P x 3 i , P x SSP( ) are such that x 5 P x 3 i x 8 and x 3 P x 5 i x 8 .Well-known domains of preferences satisfying generalized notions of single-peakedness studied in the literature are rich (seeChatterji and Massó (2018)). However, subsets of single-peaked domains may not be rich, if they are substantially restricted; for example, the Euclidean preference domain is not rich.19 Nevertheless, the set of all single-peaked preferences SP( ) is rich on (A, ).…”

The single-peaked domain revisited: A simple global characterization

“…For instance, the existence of non-dictatorial Arrovian aggregators and/or strategy-proof social choice functions can be demonstrated under much weaker domain restrictions (Kalai and Muller [1977]). Also in this context, richness and/or connectedness assumptions have frequently been imposed, and variants of the single-peakedness condition have been found to play an important role in the derivation of possibility results (Nehring and Puppe [2007], Chatterji et al [2013], Chatterji and Massó [2015]). In a recent paper, Chatterji et al [2016] have characterized a weaker notion of single-peakedness ('single-peakedness with respect to a tree') using strategy-proofness and other conditions imposed on random social choice functions.…”

A Taxonomy of Non-dictatorial Unidimensional Domains

“…Earlier literature Barberà et al (1993) showed that if a minimally rich domain admits the median voter rule as a strategy-proof rule, the domain must be singlepeaked. Instead of considering a specific rule, Chatterji et al (2013) established that on a path-connected domain, semi-single-peakedness, rather than single-peakedness, is necessary for the existence of an anonymous, tops-only and strategy-proof rule, and Chatterji and Massó (2018) showed that semilattice single-peakedness, a generalization of semi-single-peakedness, arises as a consequence of the existence of an anonymous, tops-only and strategy-proof rule on a rich domain (where the richness condition is formulated relative to the particular rule that is assumed to exist). Recently, Barberà et al (2020) provide an insightful survey that covers these and other related issues.…”

“…In particular, our analysis highlights the role of "critical spots" embedded in the gap between semi-single-peakedness and single-peakedness (respectively, between semi-hybridness and hybridness) that display a curious and seemingly paradoxical phenomenon, namely, that adding preferences to a single-peaked domain to create critical spots, may allow non-tops-only rules to emerge in a strategy-proof way and simultaneously shrink the scope for tops-only rules. 3 To put this analysis in perspective, we note that earlier work has shown that while the Gibbard-Satterthwaite Theorem (Gibbard, 1973;Satterthwaite, 1975) is robust and survives on restricted domains with enough connectedness (see for instance Aswal et al, 2003;Sato, 2010;Pramanik, 2015), semi-single-peaked domains are implied by the existence of a tops-only and anonymous strategy-proof rule on a "rich" domain (see Chatterji et al, 2013;Chatterji and Massó, 2018). In order to obtain a more complete picture of non-dictatorial domains, we dispense with the axiom of anonymity, and show (by strengthening mildly the richness condition) that allowing for non-dictatorial, tops-only and strategy-proof rules adds exactly one class of domains, semi-hybrid domains.…”