Expected utility theory without the completeness axiom

Abstract

We study the problem of obtaining an expected utility representation for a potentially incomplete preference relation over lotteries by means of a set of von Neumann-Morgenstern utility functions. It is shown that, when the prize space is a compact metric space, a preference relation admits such a multi-utility representation provided that it satisfies the standard axioms of expected utility theory. Moreover, the representing set of utilities is unique in a welldefined sense. r

Expected utility without full transitivity

“…There are several studies that analyze expected-utility theory in the context of reflexive and transitive but not necessarily complete relations over probability distributions, such as those carried out by Aumann (1962) and by Dubra, Maccheroni and Ok (2004). However, these authors retain full transitivity as an assumption and, as a consequence, obtain results that are quite different in nature from ours.…”

“…Thus, this complexity argument can be invoked in support of our approach which allows for non-comparability as well. That completeness may be a rather strong assumption in the context of expected-utility theory has been argued in many earlier contributions-von Neumann and Morgenstern (1944;1947) themselves make this point; other authors who question the completeness axiom include Thrall (1954), Luce and Raiffa (1957), Aumann (1962) and Dubra, Maccheroni and Ok (2004). After all, there are several instances where the imposition of completeness might create artificial puzzles and even impossibilities; the earlier contributions just cited are merely examples of such problems that may be triggered by the completeness assumption.…”

“…Moreover, Aumann (1962) and Dubra, Maccheroni and Ok (2004) do not impose a property akin to the monotonicity condition alluded to earlier, which is a major reason why completeness does not follow from their axioms even in the presence of full transitivity. This is yet another feature that distinguishes these earlier approaches from ours.…”

“…[this] approach falls short of yielding a representation theorem, for it does not characterize the preference relations under consideration." Dubra, Maccheroni and Ok (2004) establish an expected multi-utility theorem that characterizes reflexive and transitive but possibly incomplete preferences on probability distributions by means of a continuity property and a version of the independence axiom. The idea underlying the multi-utility approach is that a reflexive and transitive dominance criterion can be established by means of a set of possible utility functions such that a distribution is considered at least as good as another if and only if the expectation according to the former is greater than or equal to that of the latter for all utility functions in this set.…”

Nested identification of subjective probabilities

“…But already in von Neumann and Morgenstern (1944, p.19), and next in Aumann (1962), reference is made to the possibility of replacing the completeness axiom by a partial preference ordering over lotteries. The more structured result in that vein is the "extended multi-utility theorem" in Dubra et al (2004): "there exists a uniquely defined, closed and convex set of utility functions verifying the expected utility theorem".…”