4 En guise de conclusion : Que Faire?
4.2 Aller au monde
Il nous faut également à présent prendre véritablement la mesure du caractère appli- qué de notre discipline. Nous parviendrons d’autant mieux à nous « vendre » que nous aurons travaillé à notre doctrine et que nous saurons rester attentifs aux transformations des organisations27. Ceci soulève des questions délicates sur ce que devrait être la poli-
tique de publication des revues de RO, voire même sur la nature souhaitable des travaux de recherche en RO. Nous ne les aborderons pas ici tout en espérant que ces quelques remarques permettront de susciter un débat. Nous nous contenterons d’indiquer que la RO, si elle veut préserver son avenir, ne saurait rester plus longtemps absente du débat public. Si l’on songe aux « experts » convoqués aujourd’hui pour réfléchir à des pro- blèmes aussi divers et importants que : l’impact de la loi sur les 35 heures, la localisation du 3ème aéroport parisien, le tracé du TGV Est, l’organisation du ferroutage, la préven- tion des risques industriels, la réforme du mode de scrutin en France, la réorganisation des services hospitaliers, la politique de lutte contre la délinquance, etc., on constatera la quasi-absence des chercheurs opérationnels. Il semble pourtant que ceux-ci, forts d’une tradition cinquantenaire au service de la rationalité dans les organisations, n’ont rien à envier à ces « experts », bien au contraire. De ce défi à relever, on peut en attendre non seulement une plus grande visibilité de notre discipline (avec son corollaire en termes de réelle reconnaissance au niveau académique) mais aussi, et surtout, la mise au service de nos concitoyens de nos savoirs et savoir-faire. Si le chemin à parcourir pour rattraper nos amis britanniques sur ce plan est certainement long et difficile, aucun autre ne nous semble en valoir réellement la peine.
27. Une étude fine de l’évolution de la « demande de RO » de la part des organisations serait ici éclairante. Il semble que la rapidité des mutations techniques (transports, NTIC) ou légales (privatisation, dérégulation) soient devenus des éléments déterminants de cette demande (que l’on songe au secteur des télécommunica- tions ou du transport aérien par exemple). On pourra se reporter ici à [27].
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Measurement
Denis Bouyssou
∗, Marc Pirlot
†Résumé
Dans le modèle classique du mesurage conjoint on s’intéresse à une représenta- tion additive de préférences transitives. Ce modèle a été étendu dans deux directions avec, d’une part, le mesurage conjoint additif et non transitif et, d’autre part, le me- surage conjoint transitif décomposable. Cet article propose divers modèles de mesu- rage conjoint combinant ces deux types d’extension. Ces modèles décomposables et non transitifs contiennent comme cas particulier de nombreuses règles d’agrégation proposées dans la littérature. Une analyse axiomatique de ces modèles est proposée.
Mots-clefs : Mesurage conjoint, Conditions de simplification, Préférences non tran-
sitives, Modèles décomposables
Abstract
Traditional models of conjoint measurement look for an additive representation of transitive preferences. They have been generalized in two directions. Nontransi- tive additive conjoint measurement models allow for nontransitive preferences while retaining the additivity feature of traditional models. Decomposable conjoint mea- surement 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 treat- ment that shows the pure consequences of several cancellation conditions used in traditional models. These nontransitve decomposable conjoint measurement mod- els encompass a large number of aggregation rules that have been introduced in the literature.
1. This text is forthcoming in the Journal of Mathematical Psychology. 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.
† CNRS – LAMSADE, Université Paris-Dauphine, 75775 Paris,bouyssou@lamsade.dauphine.fr
Key words : Conjoint measurement, Cancellations conditions, Nontransitive prefer-
ences, Decomposable models.
1
Introduction
Given a binary relation on a set X = X1× X2× ··· × Xn, the theory of conjoint
measurement consists in finding conditions under which it is possible to build a homo- morphism between this relational structure and a relational structure onR. In traditional models of conjoint measurement is supposed to be complete and transitive and the nu- merical representation is sought to be additive; the desired measurement model is such that: x y ⇔ n
∑
i=1 ui(xi) ≥ n∑
i=1 ui(yi) (1)where ui are real-valued functions on the sets Xi and it is understood that x = (x1,x2,... ,xn) and y = (y1,y2,...,yn).
Many axiom systems have been proposed in order to obtain such a representation (Krantz, Luce, Suppes, & Tversky, 1971; Wakker, 1989). Two main cases arise:
• When X is finite (but of arbitrary cardinality), it is well-known that the system of axioms necessarily involves a denumerable number of cancellation conditions guar- anteeing the existence of solutions to a system of (finitely many) linear inequalities through the use of various versions of the theorem of the alternative (see Scott (1964) or Krantz et al. (1971, Chapter 9). For recent contributions, see Fishburn (1996, 1997)).
• When X is infinite the picture changes provided that conditions are imposed in order to guarantee that the structure of X is “close” to the structure ofR and that
behaves consistently in this continuum; this is traditionally done using either
an archimedean axiom together with some solvability assumption (Krantz et al., 1971, Chapter 6) or imposing some topological structure on X and a continuity requirement on (Debreu, 1960; Wakker, 1989). Under these conditions, it is well- known that model (1) obtains using a finite—and limited—number of cancellation conditions (for recent contributions, see Gonzales (1996, 2000) and Karni and Safra (1998); for an alternative approach, extending the technique used in the finite case to the infinite one, see Jaffray (1974)). As opposed to the finite case, these structural assumptions allow us to obtain nice uniqueness results for model (1): the functions uidefine interval scales with a common unit.
In the finite case the axiom system is hardly interpretable and testable. In the infinite case, it is not always easy to separate the respective roles of the (unnecessary) structural
assumptions from the (necessary) cancellation conditions as forcefully argued by Krantz et al. (1971, Chapter 9) and Furkhen and Richter (1991). One possible way out of this difficulty is to study more general models replacing additivity by a mere decomposability requirement. Krantz et al. (1971, Chapter 7) introduced the following decomposable model:
x y ⇔ F(u1(x1),u2(x2),...,un(xn)) ≥ F(u1(y1),u2(y2),...,un(yn)) (2)
where F is increasing in all its arguments.
When X is denumerable (i.e. finite or countably infinite), necessary and sufficient conditions for (2) consist in a transitivity and completeness requirement together with a single cancellation condition requiring that the preference between objects differing on a single attribute is independent from their common level on the remaining n−1 attributes. In the general case these conditions turn out to have identical implication when supple- mented with the obviously necessary requirement that a numerical representation exists for. Though (2) may appear as exceedingly general when compared to (1), it allows us to deal with the finite and the infinite case in a unified way using a simple axiom system while imposing nontrivial restrictions on. The price to pay is that uniqueness results for model (2) are much less powerful than what they are for model (1), see Krantz et al. (1971, Chapter 7). It should be finally observed that proofs for model (2) are considerably simpler than they are for model (1). A wide variety of models “in between” (1) and (2) are studied in Luce, Krantz, Suppes, and Tversky (1990).
Both (1) and (2) imply that is complete and transitive. Among many others, May (1954) and Tversky (1969) (see however the interpretation of Tversky’s results in Iver- son and Falmagne (1985)) have argued that the transitivity hypothesis is unlikely to hold when subjects are asked to compare objects evaluated on several attributes; more recently, Fishburn (1991a) has forcefully shown the need for models encompassing intransitivities (it should be noted that intransitive preferences have even attracted the attention of some economists, see e.g. Chipman (1971) or Kim and Richter (1986)). Tversky (1969) was one of the first to propose such a model generalizing (1), known as the additive difference model, in which: x y ⇔ n
∑
i=1 Φi(ui(xi) − ui(yi)) ≥ 0 (3)whereΦiare increasing and odd functions.
It is clear that (3) allows for intransitive but implies its completeness. When at- tention is restricted to the comparison of objects that only differ on one attribute, (3), as well as (2) and (1), implies that the preference between these objects is independent from their common level on the remaining n− 1 attributes. This allows us to unambiguously define a preference relation on each attribute; it is clearly complete. Although model (3) can accommodate intransitive, a consequence of the increasingness of the Φi is that
the preference relations defined on each attribute are transitive. This, in particular, ex- cludes the possibility of any perception threshold on each attribute which would lead to an intransitive indifference relation on each attribute (on preference models with thresh- olds, allowing for intransitive indifference relations, we refer to Fishburn (1985), Pirlot and Vincke (1997), or Suppes, Krantz, Luce, and Tversky (1989)). Imposing thatΦiare
nondecreasing instead of being increasing allows for such a possibility. This gives rise to what Bouyssou (1986) called the “weak additive difference model”.
As suggested by Bouyssou (1986), Fishburn (1990b, 1990a, 1991b) and Vind (1991), the subtractivity requirement in (3) can be relaxed. This leads to nontransitive additive conjoint measurement models in which:
x y ⇔
n
∑
i=1pi(xi,yi) ≥ 0 (4)
where the piare real-valued functions on Xi2and may have several additional properties
(e.g. pi(xi,xi) = 0, for all i ∈ {1,2,...,n} and all xi∈ Xi).
This model is an obvious generalization of the (weak) additive difference model. It allows for intransitive and incomplete preference relations as well as for intransitive and incomplete “partial preferences”. An interesting specialization of (4) obtains when piare
required to be skew symmetric i.e. such that pi(xi,yi) = −pi(yi,xi). This skew symmetric
nontransitive additive conjoint measurement model implies the completeness of and that the preference between objects only differing on some attributes is independent from their common level on the remaining attributes.
Fishburn (1991a) gives an excellent overview of these nontransitive models. Several axiom systems have been proposed to characterize them. Fishburn (1990b, 1991b) gives axioms for the skew symmetric version of (4) both in the finite and the infinite case. Nec- essary and sufficient conditions for a nonstandard version of (4) are presented in Fishburn (1992b). Vind (1991) gives axioms for (4) with pi(xi,xi) = 0 when n ≥ 4. Bouyssou
(1986) gives necessary and sufficient conditions for (4) with and without skew symmetry in the denumerable case when n= 2.
The additive difference model (3) was axiomatized in Fishburn (1992a) in the infinite case when n≥ 3 and Bouyssou (1986) gives necessary and sufficient conditions for the weak additive difference model in the finite case when n= 2. Related studies of nontran- sitive models include Croon (1984), Fishburn (1980), Luce (1978) and Nakamura (1997). The implications of these models for decision-making under uncertainty were explored in Fishburn (1990c) (for a different path to nontransitive models for decision making under risk and/or uncertainty see Fishburn (1982, 1988)).
It should be noticed that even the weakest form of these models, i.e. (4) without skew symmetry, involves an addition operation. Therefore it is unsurprising that the difficulties that we mentioned concerning the axiomatic analysis of traditional models are still present
here. Except in the special case in which n= 2, this case relating more to ordinal than to conjoint measurement, the various axiom systems that have been proposed involve either:
• a denumerable set of cancellation conditions in the finite case or
• a finite number of cancellation conditions together with unnecessary structural as- sumptions in the general case (these structural assumptions generally allow us to obtain nice uniqueness results for (4): the functions piare unique up to the multi-
plication by a common positive constant).
The nontransitive decomposable models that we study in this paper may be seen both as a generalization of (2) dropping transitivity and completeness and as a generalization of (4) dropping additivity. In their most general form they are of the type:
x y ⇔ F(p1(x1,y1), p2(x2,y2),..., pn(xn,yn)) ≥ 0 (5)
where F and pi may have several additional properties, e.g. pi(xi,xi) = 0 or F nonde-
creasing or increasing in all its arguments.
This type of nontransitive decomposable conjoint models has been introduced by Goldstein (1991). They may be seen as exceedingly general. However we shall see that this model and its extensions:
• imply substantive requirements on ,
• may be axiomatized in a simple way avoiding the use of a denumerable number of conditions in the finite case and of unnecessary structural assumptions in the general case,
• permit to study the pure consequences of cancellation conditions in the absence of transitivity, completeness and structural requirements on X ,
• are sufficiently general to include as particular cases many aggregation models that have been proposed in the literature.
Our project was to investigate how far it was possible to go in terms of numerical representations using a limited number of cancellation conditions without imposing any transitivity requirement on the preference relation and any structural assumptions on the set of objects. Rather surprisingly, as we shall see, such a poor framework allows us to go rather far.
It should be clear that, in models of type (5), numerical representations are quite unlikely to possess any “nice” uniqueness properties. This is all the more true that we will refrain in this paper from using unnecessary structural assumptions on the set of
objects. Numerical representations are not studied here for their own sake; our results are not intended to provide clues on how to build them. We use them as a framework allowing us to understand the consequences of a number of requirements on.
The paper is organized as follows. We introduce our main definitions in section 2. We present the decomposable nontransitive conjoint measurement models to be studied in this paper and analyze some of their properties in section 3. We present our main results in section 4. A final section discusses our results and presents directions for future research.