Noncompensatory preferences

This paper provides a general characterization of noncompensatory preference structures in multi-attribute preference theory, then examines a number of conditions that might hold for such structures. It will be assumed throughout that >('is preferred to') is an asymmetric binary relation on a product set X = X1 x X2 x . . . x Xn, with each X~ nonempty and n I> 2. The indifference relation ~ on X is defined from >by x y iff neither x :~ y nor y > x. We shall refer both to i and to Xi as an attribute. Loosely speaking, (X, > ) will be said to be a noncompensatory preference structure if it satisfies a simple independence condition pertaining to conditional preferences on the Xi and if > between any two n-tuples x = (xa . . . . . xn) and y = (Yl, • • •, yn) depends solely on the coordinates i for which xi is conditionally preferred to yl and for which y~ is conditionally preferred to xi. Since this prohibits compensating trade-offs among different attributes, noncompensatory structures are probably much less common than compensatory ones. However, as shown by Chipman (1960), Coombs (1964), Green and Wind (1973), MacCrimmon (1973), Fishburn (1974), and others who are cited in these studies, there are many situations in which a noncompensatory preference or choice model may be a useful guide for decision making or a realistic descriptor of a decision agent's preferences. The definition of noncompensatory preference structure given below does not presume that the preference relation > is transitive or acyclic. This is in keeping with examples and arguments based on sensory or judgmental thresholds, individual feelings about what constitutes a significant difference between levels of an attribute, and other aspects (e.g., Davidson et al., 1955; Coombs, 1964; Weinstein, 1968; Tversky, 1969; Schwartz, 1972; Fishburn, 1974) that can give rise to cyclic preferences in mulfiattribute contexts. The effects of ordering assumptions for > on noncompensatory preferences will be considered later in the paper where