(Apologies for the light posting recently. Over the next week, I plan on posting a number of (relatively) recent results from decision theory, then returning to standard fare thereafter. The decision theory results tend to be technical, so I’m going to assume some working knowledge of von Neumann and Savage results.)
The classic von Neumann/Morganstern representation theorem for expected utility is powerful in that it only makes four assumptions about preferences: they are complete (every set of objects can be compared), they are transitive, they are continuous, that the domain is a Polish space, and they have the independence property (if A is preferred to B, then .8A plus .2C is preferred to .8B plus .2C). Deviations from the final property are well-studied (and will be discussed on this page shortly). The third property is a technical assumption, and does rule out things like lexicographic preferences on lotteries, but is generally not considered problematic. The Polish space (a separable metrizable space) is broad enough to encompass R^n and compact metric spaces, so assumption 4 is not worrying. The second property, transitivity, is normatively compelling (the “money pump”) and empirically justifiable depending on how the preference domain is specificied; economists tend not to worry about transitivity, though my philosopher friends generally do not find such an assumption normatively appealing. This paper discusses deviations from the first assumption, completeness.
I think even hardened economists would agree that completeness is neither true of real-world preferences nor normatively appealing – there are many pairs of objects, particularly obscure objects like two-stage AA lotteries, on which I have no preference relation. More commonly, when the agent is a firm or a household, Arrow-style results tell us that we often have no way of aggregating preferences to choose whether A or B is preferred. The authors show that if the domain of preferences is restricted to compact metric spaces, and if completeness is replaced by reflexivity (A is weakly preferred to itself), then a unique “multiutility” representation theorem still exists. That is, there is a unique (in the sense of “biggest”) set U of utility functions such that lottery a is weakly preferred to lottery b iff the expectation of u(a)>=u(b) for all continuous functions u in U. The proof involves cleverly defining a convex cone, applying an infinite-dimensional version of the separating hyperplane theorem, and noting that the polar cone of the convex cone defined earlier gives the set U; details are in the paper.
http://cowles.econ.yale.edu/P/cd/d12b/d1294.pdf (WP version – final version in JET 2004)
A Fine Theorem 