Eliciting second-order beliefs

We study elicitation of subjective beliefs of an agent facing ambiguity (model uncertainty): the agent has a non-singleton set of (first order) priors on an event and a second-order distribution on these priors. Such a two-stage decomposition of uncertainty and non-reduction of subjective compound lotteries resulting from non-neutrality to the second-order distribution plays an important role in resolving the Ellsberg Paradox. However, a key unanswered question is whether we can actually observe and separate the sets of first and second order subjective probabilities. We answer this question and show that it is indeed possible to pin the two sets down uniquely and therefore separate them meaningfully. We introduce prize variations and ensure that the tangent plane at any point on the surface of the certainty-equivalent function is reported truthfully. Any basis for this plane consists of derivatives in two different directions, and these combine first and second order beliefs in different ways. We show that this is enough to ensure that reporting both sets of beliefs truthfully is uniquely optimal. The technique requires knowledge of the changes in certainty equivalent of a mixture of acts as a result of variations in prizes, which we also elicit.