State-Dependent Utilities

Abstract

Several axiom systems for preference among acts lead to the existence of a unique probability and a state-independent utility such that acts are ranked according to their expected utilities. These axioms have been used as a foundation for Bayesian decision theory and the subjective probability calculus. In this paper, we note that the uniqueness of the probability is relative to the choice of what counts as a constant outcome. Although it is sometimes clear what should be considered constant, there are many cases in which there are several possible choices. Each choice can lead to a different “unique” probability and utility. By focusing attention on state-dependent utilities, we determine conditions under which a truly unique probability and utility can be determined from an agent’s expressed preferences among acts. Suppose that an agent’s preference can be represented in terms of a probability P and a utility U. That is, the agent prefers one act to another if and only if the expected utility of the one act is higher than that of the other. There are many other equivalent representations in terms of probabilities Q, which are mutually absolutely continuous with P, and state-dependent utilities V, which differ from U by possibly different positive affine transformations in each state of nature. An example is described in which two different but equivalent state-independent utility representations exist for the same preference structure. What differs between the two representations is which acts count as constants. The acts involve receiving different amounts of one or the other of two currencies and the states are different exchange rates between the currencies. It is easy to see how it would not be possible for constant amounts of both currencies to simultaneously have constant values across the different states. Savage (Foundations of statistics. John Wiley, New York, 1954, sec. 5.5) discovered a situation in which two seemingly equivalent preference structures are represented by different pairs of probability and utility. Savage attributed the phenomenon to the construction of a “small world”. We show that the small world problem is just another example of two different, but equivalent, representations treating different acts as constants. Finally, we prove a theorem (similar to one of Karni, Decision making under uncertainty. Harvard University Press, Cambridge, 1985) that shows how to elicit a unique state-dependent utility and does not assume that there are prizes with constant value. To do this, we define a new hypothetical kind of act in which both the prize to be awarded and the state of nature are determined by an auxiliary experiment.