Preference, Indifference, Incomparability: Binary Relations and Basic Structures

  • Bernard Roy
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 12)


In the first section we present concepts that describe an actor Z’s stated preference judgments when comparing two actions of A. In Section we use examples to introduce the basic preference situations with which Z can be faced. We then state the axiom of limited comparability (Axiom 7.1.1), which serves as a point of departure from classic decision theory.

Based on the four binary relations of indifference (I), strict preference (P), weak preference (Q), and incomparability (R), we show in Section that all of Z’s preferences on A can be modeled by a basic system of preference relations, denoted BSPR, or, if necessary, by a consolidated system of preference relations, denoted CSPR. We discuss the two nontraditional relations Q and R in Section and discuss transitivity of the relations I, P, Q, and R in Section

We then investigate the situations and consolidated relations defined in Table 7.1.5 and the most noteworthy CSPR’s that they generate. The one on which classical decision theory is based consists of two consolidated relations: ~ (nonpreference) and ≻ (preference). In Section we discuss how the axiom of limited comparability is replaced by the axiom of complete transitive comparability in this theory. After discussing the three consolidated relations J, K, and S, in Section, we introduce a final CSPR, one in which the outranking relation S plays a fundamental role. The section ends by examining relationships among the various situations and motivating certain choices of SPR models.

We use the first part of the second section to introduce graphical conventions used in the rest of the book. We also present a new example concerning a mayor’s preferences. In the two subsections that follow, we present the principal structures associated with the most interesting SPR’s and their functional representations. In Section 7.2.2 we look at those that exclude incomparability, and in Section 7.2.3 at those that acknowledge it. The following table synthesizes the principal structures studied in these two subsections in terms of the relations that constitute the SPR.

We touch upon the fundamental problem of comparing preference differences or exchanges (elements of A × A) in Section and examine the relationships between SPR’s on A and on A × A in Section

Readers put off by the terse nature of Section 2 should only skim its first two subsections.


Preference Relation Binary Relation Strict Preference Weak Preference Loan Application 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer Science+Business Media Dordrecht 1996

Authors and Affiliations

  • Bernard Roy
    • 1
  1. 1.LAMSADEUniversité Paris-DauphineFrance

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