Noncompensatory and generalized noncompensatory preference structures
Suppose \(X=X_ 1\times X_ 2\times X_ 3\times...\times X_ n\) is a nonempty set of alternatives and P is an asymmetric binary relation on X. We interpret P as a strict preference relation on a set of multiattributed alternatives. We define a binary relation \(P_ i\) on \(X_ i\) by \(x_ iP_ iy_ i\) if and only if (iff) \(((x_ j)_{j\neq i}),x_ i)P((x_ j)_{j\neq i},y_ i)\) for all \((x_ j)_{j\neq i}\in \prod_{j\neq i}X_ i\). P on X is said to be noncompensatory following \textit{P. C. Fishburn} [Synthese 33, 393-403 (1976; Zbl 0357.90004)] iff \([x_ iP_ iy_ i\) iff \(z_ iP_ iw_ i\) and \(y_ iP_ ix_ i\) iff \(w_ iP_ iz_ i]\) imply [xPy iff zPw] i.e. when there is no tradeoff between the various attributes. We provide a representation theorem stating necessary and sufficient conditions for P to be built additively using ''weights'' assigned to each attribute. We then generalize the idea of noncompensatory preferences and say that (X;P) is a generalized noncompensatory preference structure iff \([x_ iP_ iy_ i\) iff \(z_ iP_ iw_ i\) and \(y_ iP_ ix_ i\) iff \(w_ iP_ iz_ i]\) imply [xPy implies not (wPz)] and \([x_ iP_ iy_ i\) for some \(i\in \{1,2,...,n\}\) and \(Not(z_ jP_ jw_ j)\) for all \(j\in \{1,2,...,n\}]\) imply xPy. We show that most of the interesting properties of noncompensatory preferences are preserved using this generalized definition and study the links between these two definitions. We provide conditions under which a generalized noncompensatory preference system can be numerically represented additively using weights on each attribute and ''veto'' thresholds. As a conclusion we show how such preference structures can be used for decision-aid.
scientific article; zbMATH DE number 3978736
scientific article