Lexicographic Tradeoff Structures

In selecting a house from a limited domain of houses, price dominates one's considerations when the difference in price between two houses is sufficiently large. But when the two prices are sufficiently close, other factors begin to enter into consideration. We do not ignore price it is hardly possible to pay no attention to the difference between $49,500 and $55,000 but we begin to acknowledge that such a difference can be compensated for by factors such as location, style, lot size, spaciousness, etc. Examples such as this exhibit three main features, two of which do not seem to be jointly captured in any of our existing decision models. First, the choice situation can be described as a conjoint structure with at least two factors affecting the decision attribute. Second, one factor, call it the first, is most important in the sense that for sufficiently large differences on that factor it is totally controlling. Third, when, however, the difference on the first factor is not too large, the second factor, and others if they exist, come into play and tradeoffs exist between the two factors, as in the usual theories of conjoint measurement. Tversky (1969) discussed this problem. He pointed out that, to a degree, the behavior is captured by a lexicographic semiorder: a structure that is lexicographic in the sense that control shifts to the second component when indifference exists on the first; however indifference is assumed to be nontransitive and, in fact, to arise from a semiorder (see below). As Tversky notes on p. 40, " ... despite its intuitive appeal, it is based on a noncompensatory principle that is likely to be too restrictive in many contexts." He then goes on to suggest an additive difference representation where there are functions ~b i assigned to the n components X i and increasing weighting functions f / such that x is prefererred or indifferent to y if and only if n i ~1 ~ [r r % ) ] > 0, where j~ ( 6) = -~j~ (6). This, however, does not capture very well the lexicographic character of some choices.