# Separating Probability Elicitation from Utilities

Authors:
Joseph B. Kadane, Robert L. Winkler
Carnegie-Mellon University, Duke Univeristy

Jul 31 1987

This paper concerns the separation of probability elicitation from utilities. We show that elicited probabilities can be related to utilities not just through the explicit or implicit payoffs related to the elicitation process, but also through other stakes the expert may have in the events of interest. We study three elicitation procedures -- lotteries, scoring rules, and promissory notes -- and show how the expert's utility function and stakes in the events can influence the resulting probabilities. Particularly extreme results are obtained in an example involving a market at equilibrium. The applicability of a no-stakes condition and some implications for probability elicitation are discussed.
Let π represent an expert's probability for an event A and let p denote the elicited probability from some elicitation procedure. We determine the value of p that maximizes the expert's expected utility. When utility is linear in money, p=π for all of the procedures studies here. Under nonlinear utility, the lottery procedure still yields p=π in the presence of nonlinear utility. If the no-stakes condition holds and the elicitation-related payoffs approach zero, then p=π in the limit. For all three procedures, the combination of nonlinear utility and other stakes can lead to values of p other than π. Furthermore, an analysis of the promissory-note procedure in a market setting gives a very extreme result: in a complete market at equilibrium for such promissory notes, the elicited probability on the market prince, not on π.
Is the no-stakes condition reasonable? We suggest that it often is not, since experts are likely to already have significant states, particularly in important situations. Moreover, it may be difficult to determine exactly what those stakes are (and perhaps to obtain accurate information about the expert's utility function). This creates somewhat of a dilemma for probability elicitation, implying that, at least in theory, it is difficult to separate probability elicitation from utilities.

##### Keywords:
subjective probability, no-stakes condition, lotteries, scoring rules, promissory notes