A characterization of consistent collective choice rules

doi:10.1016/j.jet.2007.05.004

Journal of Economic Theory

Volume 138, Issue 1, January 2008, Pages 311-320

https://doi.org/10.1016/j.jet.2007.05.004 Get rights and content

Abstract

We characterize a class of collective choice rules such that collective preference relations are consistent. Consistency is a weakening of transitivity and a strengthening of acyclicity requiring that there be no cycles with at least one strict preference, which excludes the possibility of a “money pump.” The properties of collective choice rules used in our characterization are unrestricted domain, strong Pareto, anonymity and neutrality. If there are at most as many individuals as there are alternatives, the axioms provide an alternative characterization of the Pareto rule. If there are more individuals than alternatives, however, further rules become available.

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Cited by (21)

Collective choice rules with social maximality
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We consider social choice on a non-empty, finite set $X$ where all elements of $X$ are available; society will not be required to make a choice from a strict subset of $X$ . Arrow assumes that the social preference relation is an ordering of the alternatives in $X$ . However, under the assumption that the collective choice rule satisfies unrestricted domain, independence of irrelevant alternatives and the weak Pareto principle, this can be weakened to the requirement that (i) the social preference relation generates a non-empty set of maximal alternatives, and (ii) every maximal alternative is strictly socially preferred to every non-maximal alternative. Requirement (ii) can be weakened further when there are four or more alternatives. When only property (i) is satisfied, we characterize all collective choice rules that output complete social preferences and satisfy unrestricted domain, Pareto, neutrality and anonymity. These collective choice rules are related to the $S$ rules of Bossert and Suzumura (2008). Finally, we prove some equivalence theorems relating the various binary relations considered in the paper to existing coherence conditions on social preferences.
Superset-robust collective choice rules
2021, Mathematical Social Sciences
Citation Excerpt :
An alternative characterization of the Pareto rule is obtained by replacing weak Pareto with strong Pareto instead of adding finite anonymity. Recall that Bossert and Suzumura (2008) provide a characterization of the Pareto rule by employing strong Pareto, finite anonymity, and neutrality if the number of alternatives is at least as large as the population size. Corollary 8 implies that weak Pareto is enough for the characterization in the presence of non-negative responsiveness and superset robustness, while Corollary 9 implies that we can drop finite anonymity under non-negative responsiveness and superset robustness.
A new property of collective choice rules that we refer to as superset robustness is introduced, and we employ it in several characterization results. The axiom requires that if all individual preference orderings expand weakly (in the sense of set inclusion), then the corresponding social preference relation must also expand weakly. In other words, if a given profile is changed by adding instances of weak preference to some individual relations, then the social weak preference relation for the expanded profile must contain the social weak preference relation for the original—that is, the social relation cannot contract in response to the addition of pairs to the individual relations. We begin by examining social welfare functions (that is, collective choice rules such that the resulting social preferences are orderings) and then move on to rules that generate transitive (but not necessarily complete) social rankings. The remaining results of the paper focus on Suzumura-consistent collective choice rules. In all of these cases, it turns out that the property of superset robustness is closely related to classes of agreement-based collective choice rules. These are rules such that the social relation is determined by collecting the pairs on whose relative rankings the members of the decisive sets agree.
Acyclicity, anonymity, and prefilters
2020, Journal of Mathematical Economics
Citation Excerpt :
Suzumura consistency, introduced by Suzumura (1976), is a coherence condition that is intermediate in strength between transitivity and acyclicity. It has several appealing properties that are discussed in Suzumura (1976), Bossert et al. (2005), and Bossert and Suzumura (2008), for instance. Finally, we deal with the case of an infinite (to be precise, countably infinite) population.
We analyze the decisiveness structures associated with acyclical collective choice rules. In particular, we examine the consequences of adding anonymity to weak Pareto, thereby complementing earlier results on acyclical social choice. Both finite and countably infinite populations are considered. As established in contributions by Donald Brown and by Jeffrey Banks, acyclical social choice is closely linked to prefilters in the presence of the weak Pareto principle. We introduce the notion of a conditional prefilter and use it to generalize their results. In the finite-population case, adding anonymity implies that the collection of decisive sets is a special case of a conditional prefilter. We then identify the decisiveness structure that results from adding anonymity to the weak Pareto principle. Moving to infinite populations, we obtain the decisive set that consists of the entire population as a possibility, along with a new class of prefilters that we refer to as symmetric free Fréchet prefilters. The choice of the term Fréchet prefilter is motivated by the observation that they share a defining property with the well-known Fréchet filter—namely, that all sets in the requisite collection are such that their complement is finite.
Incomplete decision-making and Arrow's impossibility theorem
2018, Mathematical Social Sciences
Citation Excerpt :
Some recent works conduct social choice with S-consistency (Bossert and Suzumura, 2008, 2012a, b; Cato, 2013a). Bossert and Suzumura (2008) provide a characterization of a class of S-consistent collective choice rules. Bossert and Suzumura (2012a) and Cato (2013a, c) clarify the decisive structure underlying an S-consistent collective choice rule.
This paper is concerned with social choice without completeness of social preference. Completeness requires that pairs of alternatives are perfectly comparable. We introduce the concept of minimal comparability, which requires that for any profile, there is some comparable pair of distinct alternatives. Complete silence should be avoided according to this condition. We show that there exists no normatively desirable aggregation rule satisfying minimal comparability.
Remarks on Suzumura consistent collective choice rules
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Citation Excerpt :
Hence, Suzumura consistency serves as the logical foundation of Bergsonian welfare economics. However, in spite of its importance, Suzumura consistency has not received much attention in the context of social choice, and studies on this subject have recently become prominent (Bossert and Suzumura, 2008, 2009, 2010, 2012; Cato, forthcoming-a, b). The purpose of this paper is to provide several remarks on Suzumura consistent collective choice rules.
Suzumura consistency is a necessary and sufficient condition for the existence of a weak-order extension. This paper provides some remarks on collective choice rules that generate Suzumura consistent social preferences. We examine the properties of such collective choice rules by introducing a procedural condition on collective choice rules. As applications of the procedural condition, we first investigate the decisive structure of a Paretian collective choice rule, and then consider the assignment of individual rights. In our analysis, the concept of semi-decisiveness works effectively.
Product filters, acyclicity and Suzumura consistency
2012, Mathematical Social Sciences
In a seminal contribution, Hansson (1976) demonstrates that the collection of decisive coalitions associated with an Arrovian social welfare function forms an ultrafilter. He goes on to show that if transitivity is weakened to quasi-transitivity as the coherence property imposed on a social relation, the set of decisive coalitions is a filter. We examine the notion of decisiveness with acyclical or Suzumura consistent social preferences and without the assumption that the social relation is reflexive and complete. This leads to a new set-theoretic concept applied to product spaces.

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Notes, Comments, and Letters to the EditorA characterization of consistent collective choice rules

Abstract

Social Choice and Individual Values

Suzumura consistency

Consistent rationalizability

Economica

Decision Analysis

Quasi-transitivity, rational choice and collective decisions

Rev. Econ. Stud.

Notes, Comments, and Letters to the Editor
A characterization of consistent collective choice rules