Some remarks on the notion of compensation in MCDM. (English) Zbl 0598.90057

Bouyssou, Denis

Eur. J. Oper. Res. 26, 150-160 (1986).

This paper revisits the notion of compensation of independent multi- attribute preference structures. Firstly the author proposes four definitions for noncompensation in the line of those of P. C. Fishburn [Synthese 33, 393-403 (1976; Zbl 0357.90004)]:
Assume that \((X,\succcurlyeq,\succcurlyeq_ 1,\succcurlyeq_ 2,...,\succcurlyeq_ n)\) is a multi-attribute preference structure on \(X=X_ 1\times X_ 2\times...\times X_ n\). We can define \(P(x,y)=\{i\in \{1,2,...,n\}\) \(|\) \(x_ i\succ_{i}y_ i\}\), \(P(y,x)=\{i\in 1,2,...,n\}\) \(|\) \(y_ i\succ_{i}x_ i\}\), and \(I(x,y)=I(y,x)=\{i\in \{1,2,...,n\}\) \(|\) \(x_ i\succ_{i}y_ i\}\). Then the preference structure is noncompensatory if: \((P(x,y)=P(z,w)\) and \(P(y,x)=P(w,z))\Rightarrow (x\succcurlyeq y\) iff \(z\succcurlyeq w).\)
Different variations in the definition of ”noncompensatoriness” rely on the differences in the role of the set I. The ultimate purpose of the author is apparently to obtain a ”compensatoriness scale” to order the aggregation procedures. This goal remains far from being attained.
In the second part of the paper, the author is interested in the intransitive and noncompensatory models as introduced by K. Tversky with his ”additive preference model” [”Intransitivity of preferences”, Psychological Review 76, 31-48 (1969)]. He states conditions on the structure of preferences in order that there exist real-valued functions \(p_ i\) such that \[ (1)\quad x\succcurlyeq y\quad \Leftrightarrow \quad \sum p_ i(x_ i,y_ i)\geq 0\quad and\quad p_ i(x_ i,y_ i)=-p_ i(y_ i,x_ i). \] Remember that Tversky’s model requires real-valued, strictly increasing functions \(\Phi_ i\) such that \[ x\succcurlyeq y\quad \Leftrightarrow \quad \sum \Phi_ i(u_ i(x_ i)-u_ i(y_ i))\geq 0\quad and\quad \Phi_ i(\delta)=-\Phi_ i(-\delta). \] In the case of \(X=X_ 1\times X_ 2\) the author gives various conditions on the preference structure \((\succcurlyeq,\succcurlyeq_ 1,\succcurlyeq_ 2)\) (mainly completeness and cancellation) which entail the existence of \(p_ 1\) and \(p_ 2\) satisfying (1). Finally he provides a condition which amounts to relaxing the strictly increasing assumption in Tversky’s model.
In a few words, it is an important paper for people who are interested in compensation and with its extensive bibliography it should be useful for further research on this topic.

Reviewer: J.-Ch.Pomerol

MSC:

90B50	Management decision making, including multiple objectives
91B06	Decision theory
91B08	Individual preferences
91B16	Utility theory

Keywords:

decision theory; measurement; aggregation procedures; compensation of independent multi-attribute preference structures; noncompensation

Citations:

Zbl 0357.90004

Software:

ELECTRE

References:

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