Dynamic Elicitation of Unobservable Information

现实世界决策的大多数方面涉及两个复杂的因素：不确定性和时间。拟议的研究适用范围非常广泛，但我们可以考虑投资决策。例如，股票市场的行为在很大程度上是不可预测的，而且取决于许多因素，其中许多因素对普通投资者来说是不被理解的。尽管如此，现实世界的决策者被迫在这样的环境中做出决定。个人被要求在不知道资本市场表现如何的情况下为自己的退休选择投资组合。有专家能更好地理解不确定性。同样，这些人也更有可能认识到时时刻刻发生的事件的重要性。例如，专业投资者可能会认识到某一股票何时被低估或高估，从而可能修改她对该股票未来可能表现的评估。这项拟议的研究旨在构建一种装置，允许非专业投资者在任何时候都能发现专业人士认为相关的所有信息。该设备通过向专业人士提供金钱激励来发挥作用。它可以被理解为一种复合期权。例如，周二制定的一项期权赋予个人在周三从给定证券集合中选择任何一种证券的权利。每种证券的价值取决于所讨论的不确定性。周一写的复合期权让个人有权在周二从给定的期权集合中选择任何一个。通过使用这样一个简单的工具，我们建议建立一个完全不知情的投资者可以诱使专业投资者透露她所知道的关于市场行为的一切，包括她对未来所有日期的市场行为的看法。在所有不确定性得到解决之前，不需要向专业人士支付任何款项。此外，我们可以选择支付非常少的款项。通常情况下，我们建议将适当的评分规则的概念调整到动态情况。专家持有关于世界未来状态的主观概率信念，也接收关于真实状态的一系列信号。这些信号会一直被观察到。想象专家持有的概率高于概率是没有损失的，比方说，当有一个中间信号实现时。我们的目标是随着时间的推移得出这个概率树，因为它被实现了。所有的收益只发生在世界的状态获得之后。我们展示了如何构造一个函数，在每个阶段，该函数严格诱导专家宣布他们领先一步的信念。得到了离散时间和连续时间的结果，并且证明了这样构造的机制类耗尽了所有激励相容机制类。

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.