Dynamic Elicitation of Unobservable Information

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Most aspects of real-world decision making involve two complicating factors: uncertainty and time. The proposed research applies very broadly, but we can think of investment decisions. For example, the behavior of the stock market is largely unpredictable throughout time, and depends on many factors, many of which are not understood to lay investors. All the same, real-world decision makers are forced to make decisions in such environments. Individuals are required to choose an investment portfolio for their retirement without knowing how capital markets will fare. There are experts who are in a much better position to understand the uncertainty. These individuals are likewise more likely to recognize the importance of events occurring throughout time. For example, a professional investor may recognize when a certain stock becomes under or overvalued; and thus may revise her assessment as to the likely future performance of the stock. The proposed research seeks to construct a device which will allow the lay investor to uncover all information which the professional deems relevant throughout time. The device works by providing monetary incentives to the professional. It can be understood as a compound option. For example, an option written on Tuesday gives the individual the right to choose any of a given collection of securities on Wednesday. The value of each security depends on the uncertainty in question. A compound option written on Monday gives the individual the right to choose any of a given collection of options on Tuesday. By utilizing such a simple device, we propose to establish that a completely uninformed investor can induce the professional investor to reveal everything she knows about the behavior of the market, including how she perceives it will behave in the future at all future dates. No payments need be given to the professional until all uncertainty is resolved. Further, the payments can be chosen to be very small.Formally, we propose to adapt the notion of a proper scoring rule to a dynamic situation. An expert holds subjective probabilistic beliefs about a future state of the world, and also receives a sequence of signals as to the true state. These signals are observed throughout time. It is without loss to imagine that the expert holds a probability over probabilities, say, when there is one intermediate signal realization. The goal is to elicit this probability tree throughout time, as it is realized. All payoffs only occur after the state of the world obtains. We show how to construct a function, which at each stage, strictly induces the expert to announce their one-step-ahead belief. Results are obtained in both discrete and continuous time, and moreover, we propose to show that the class of mechanisms so constructed exhausts the class of all incentive compatible mechanisms.