Nontransitive Decomposable Conjoint Measurement

Abstract

1 We wish to thank M. Abdellaoui, J.-P. Doignon, Ch. Gonzales, Th. Marchant, P.P. Wakker and an anonymous referee for their very helpful suggestions and comments on earlier drafts of this text. The usual caveat applies. Denis Bouyssou gratefully acknowledges the support of the Centre de Recherche de l'ESSEC and the Brussels-Capital Region through a "Research in Brussels" action grant. AbstractTraditional models of conjoint measurement look for an additive representation of transitive preferences. They have been generalized in two directions. Nontransitive additive conjoint measurement models allow for nontransitive preferences while retaining the additivity feature of traditional models. Decomposable conjoint measurement models are transitive but replace additivity by a mere decomposability requirement. This paper presents generalizations of conjoint measurement models combining these two aspects. This allows us to propose a simple axiomatic treatment that shows the pure consequences of several cancellation conditions used in traditional models. These nontransitve decomposable conjoint measurement models encompass a large number of aggregation rules that have been introduced in the literature.

An axiomatic analysis of concordance–discordance relations

“…We consider binary relations on X that can be represented in the following model introduced in Bouyssou and Pirlot (2002b):…”

“…The relations defined below were introduced in Bouyssou and Pirlot (2002b). They will play a fundamental rôle in the sequel.…”

“…In doing so, we build on previous results characterizing the relations that can be obtained only using the first part of the concordance / non-discordance principle, i.e., the concordance condition Pirlot 2005a, 2007b). As first presented in Bouyssou, Pirlot, and Vincke (1997), the general strategy followed in this paper is to view outranking relations as a particular case of relations having a representation in the nontransitive decomposable models introduced in Bouyssou and Pirlot (1999, 2002b, 2004a; this was indeed our initial motivation for developing them. This particular case obtains when only a few distinct levels of preference differences are distinguished.…”

Computer science and decision theory

“…See for example [417] and the work of Pennock, Horvitz, and Giles [340] described in Section 2.1. Other relevant work can be found in [51,52,55,56,57,139,182,284,285,361,362,414]. We return to this point in our discussion of algorithmic decision theory, specifically in Section 4.1.…”

Further results on concordance relations

“…Our starting point is a general model of conjoint measurement tolerating intransitive and/or incomplete relations that was introduced in Bouyssou and Pirlot (2002) (henceforth BP02). This model investigates conditions allowing to build a numerical representation of a binary relation on a product set X = n i=1 X i such as:…”