In practice, scoring rules elicit good probability estimates from individuals, while betting markets elicit good consensus estimates from groups. Market scoring rules combine these features, eliciting estimates from individuals or groups, with groups costing no more than individuals.

Since utilities and probabilities jointly determine choices, event-dependent utilities complicate the elicitation of subjective event probabilities. However, for the usual purpose of obtaining the information embodied in agent beliefs, it is su#cient to elicit objective probabilities, i.e., probabilities obtained by updating a known common prior with that agent's further information. Bayesians who play a Nash equilibrium of a certain insurance game before they obtain relevant information will afterward act regarding lottery ticket payments as if they had event-independent risk-neutral utility and a known common prior. Proper scoring rules paid in lottery tickets can then elicit objective probabilities.

We investigate the problem of truthfully eliciting an expert’s assessment of a property of a probability distribution, where a property is any real-valued function of the distribution such as mean or variance. We show that not all properties are elicitable; for example, the mean is elicitable and the variance is not. For those that are elicitable, we provide a representation theorem characterizing all payment (or “score”) functions that induce truthful revelation. We also consider the elicitation of sets of properties. We then observe that properties can always be inferred from sets of elicitable properties. This naturally suggests the concept of elicitation complexity; the elicitation complexity of property is the minimal size of such a set implying the property. Finally we discuss applications to prediction markets.