Axiomatic Characterization of Indirect Utility and Lexicographic Extensions

The general problem we are interested in this paper is of the following variety: We are given a finite universal set and a linear ordering on it. What is the minimal axiomatic characterization of particular extension of this linear ordering to the set of all non-empty subsets of the given set? In Kannal and Peleg[1984] we find the starting point of this literature, which basically asserts that if the cardinality of the universal set is six or more, then there is no weak order on the power set which extends the linear order and satisfies two properties: one due to Gardenfors and the other known as Weak Independence. This result was followed by a quick succession of possibility results in Barbera, Barret and Pattanaik [1984], Barbera and Pattanaik [1984], Fishburn [1984], Heiner and Packard [1984], Holzman [1984], Nitzan and Pattanaik [1984] and Pattanaik and Peleg[1984]. Somewhat later, Bossert [1989] established a possibility result by dropping the completeness axiom for the binary relation on the power set and otherwise using the same axioms as in Kannal and Peleg[1984]. In recent times Mallishevsky[1997] and Nehring and Puppe[1999] have addressed the problem of defining an “indirect utility preference’. Malishevsky [1997] addresses the integrability problem: given a weak order on the power set, under what conditions is it an indirect utility preference? A similar question is also addressed in Nehring and Puppe[1999]. In our framework, a binary relation on the power set is an indirect utility extension if given two non-empty sets, the first is as good as the second if the best element: (with respect to the linear order) of the first set is as good as the best element of the second. In this paper, we provide a minimal set of assumptions which uniquely characterizes the indirect utility extension. The indirect utility extension is easily observed to be a slight modification of the weak ordering extension due to Barbera and Pattanaik[1984]. In a final section of this paper we consider the problem of axiomatically characterizing the so called “lexicographic” extension. It is similar to the extension considered by Bossert[1989]. However unlike the extension due to Bossert our extension is complete, and though it satisfies Gardenfor’s Property if fails to satisfy Weak Independence. Given a set we consider the pair consisting of its best and worst point. Now given to wets the first is atleast as good as the second, if either the best point of the first set is better than the best point of the second or they both share the same best point, in which case the worst point of the first is required to be atleast as good as the worst point of the second. In a way, the decision maker becomes pessimistic only if he/she has not much to choose between the best points of two sets.