Separating Probability Elicitation From Utilities

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Abstract
This article deals with 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 studied here. Under nonlinear utility, the lottery procedure still yields p = π as long as the expert has no other stakes involving the occurrence or nonoccurrence of A (the no-stakes condition). With the scoring-rule and promissory-note procedures, the no-stakes condition is no longer sufficient for p = π in the presence of nonlinear utility. If the no-stakes condition holds and the elicitation-related payoffs approach 0, 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 depends on the market price, not on π. Is the no-stakes condition reasonable? We suggest that it often is not, since experts are likely to have significant stakes already, 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.

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