The Indirect Utility Extension: Axiomatic Characterizations

In this paper, we first provide several axiomatic characterizations of the indirect utility extension. 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 a minimal axiomatic characterization of the indirect utility extension? In keeping with the prevalent terminology in economic theory, an extension of a linear order from a set to the power set is called the indirect utility extension provided it is the case that one set is considered at least as desirable as a second if and only the largest element of the first set is no smaller that the largest element of the second. In Kannai and Peleg (1984) we find the starting point of the related literature, wherein it is asserted 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 Barbers, 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 axioms as in Kannai and Peleg (1984).