Suzumura‐consistent relations: An overview

• DOI: http://doi.org/10.1111/ijet.12139
• Author: Walter Bossert
• Date: 01 March 2018
• Published date: 01 March 2018

doi: 10.1111/ijet.12139

Suzumura-consistent relations: An overview

Walter Bossert∗

This paper provides a brief introduction to the use and usefulness of Suzumura consistency, a

coherence requirement for binary relations that weakens transitivity. The property was intro-

duced by Suzumura in the context of collective choice but, as demonstrated in some recent

contributions, its applicability reaches beyond the boundaries of social-choice theory. In addi-

tion to a summary of its mathematical underpinnings, some recent applications in individual

and collective decision-making are provided. Several examples are employed to illustrate the

property and how it distinguishes itself from alternative weakeningsof t ransitivity such as quasi-

transitivity or acyclicity.

Key wor ds Suzumura consistency,indiv idual and collectivechoice

JEL classification D01, D63, D71

Accepted 5 May2017

1 Introduction

The notion of Suzumura consistency originates in a seminal contribution by Suzumura (1976b). This

property weakens the well-established transitivity postulate that is ubiquitous in economic theory.

The main purpose of this paper is to illustrate the use and usefulness of this fundamental coherence

property. In addition to an explanation of the condition and its mathematical underpinnings, some

recent approachesthat make use of it are summarized. No proofs are presented because they can easily

be found in the original publications. Instead, I aim to provide a unified perspective to illustrate the

potential of Suzumura consistency as a general tool that is not restricted to the few specific examples

of applications discussed here.

First and foremost, Suzumura consistency can be used to obtain a substantial generalization of a

classical result by Szpilrajn (1930). Szpilrajn’s extension theorem proves that any transitive relation

can be extended to a complete and transitiverelation. While the original version of the result is stated

for strict relations, it can also be phrased in a setting where any two objects may be equally good.

Formulated in this manner, Szpilrajn’s theorem shows that transitivity is a sufficient condition for

the existence of an ordering extension—that is, an extension of a relation that is reflexive, complete,

and transitive. The generalization achieved by Suzumura (1976b) representsa major step forward: he

shows not only that Suzumura consistency is a weaker sufficient condition for an ordering extension

but also that it is necessary and, therefore, provides a clear and unambiguous dividing line between

*Department of Economics and CIREQ, University of Montreal,Montreal, Canada. Email: walter.bossert@videotron.ca

This paper is dedicated to Kotaro Suzumura in deep appreciation of his countless fundamental contributions to the

academic community in general and to the members of the economics profession in particular. I thank a referee

for thoughtful comments and suggestions. Financial support through grants from the Fonds de Recherche sur la

Soci´

et´

eetlaCultureofQu

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ebec and the Social Sciences and Humanities Research Council of Canada is gratefully

acknowledged.

International Journal of Economic Theory 14 (2018) 21–34 © IAET 21

International Journal of Economic Theory