A characterization of single-peaked preferences via random social choice functions

Abstract

This paper proves the following result: every path‐connected domain of preferences that admits a strategy‐proof, unanimous, tops‐only random social choice function satisfying a compromise property is single‐peaked. Conversely, every single‐peaked domain admits a random social choice function satisfying these properties. Single‐peakedness is defined with respect to arbitrary trees. The paper provides a justification of the salience of single‐peaked preferences and evidence in favor of the Gul conjecture ([Barberà, 2010] ).

On mechanisms eliciting ordinal preferences

“…They also show by example that domain conditions known to imply dictatorship results in the deterministic setting are not sufficient in the randomized setting. Other work in this vein includes Ehlers et al (), Chatterji et al (), Peters et al (), Chatterji and Zeng ().Randomization has also been studied in rationing problems ( Ehlers 2002 , Ehlers and Klaus 2003 ).…”

“…In the object allocation problem of Bogomolnaia and Moulin (), the requirements of ordinal efficiency and equal treatment of equals can be expressed by an SCC that, at each profile of ordinal preferences, deems a lottery over allocations to be acceptable if and only if it is consistent with these requirements.In a voting problem, unanimity can be expressed by the SCC where, at any profile where all voters have the same preferred outcome, only the degenerate lottery on that outcome is acceptable, and at any other preferences, all lotteries are acceptable.Since our framework is specifically focused on possibility/impossibility results, it does not capture characterizations such as the random dictatorship result of Chatterji et al (). However, one could refine the SCC so as to rule out random dictatorship (for example, by imposing a “compromise” requirement as in Chatterji et al (), specifying that at some particular profiles, an outcome that is not anyone's top choice should be chosen with some minimum probability). Then the result of Chatterji et al () translates into an impossibility theorem, which fits within our framework.…”

“…Since our framework is specifically focused on possibility/impossibility results, it does not capture characterizations such as the random dictatorship result of Chatterji et al (). However, one could refine the SCC so as to rule out random dictatorship (for example, by imposing a “compromise” requirement as in Chatterji et al (), specifying that at some particular profiles, an outcome that is not anyone's top choice should be chosen with some minimum probability). Then the result of Chatterji et al () translates into an impossibility theorem, which fits within our framework.…”

The single-peaked domain revisited: A simple global characterization

“…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. It seems a worthwhile task for future research to further explore whether, and how, the present methodology can contribute to our understanding of the weaker domain restrictions that still enable consistent preference aggregation and/or non-dictatorial strategyproof social choice.…”

Necessary and sufficient conditions for pairwise majority decisions on path-connected domains

“…Characterizing domains by means of the choice functions that they admit is considered as an important problem in the literature. Chatterji et al (2016) characterize single-peaked domains on arbitrary trees by means of strategy-proof, unanimous, tops-only random social choice functions satisfying a compromise property, and Puppe (2018) shows that every minimally rich and connected Condorcet domain which contains at least one pair of completely reversed orders must be single-peaked.…”