Recent advances in multi-task peer prediction have greatly expanded our knowledge about the power of multi-task peer prediction mechanisms. Various mechanisms have been proposed in different settings to elicit different types of information. But we still lack understanding about when desirable mechanisms will exist for a multi-task peer prediction problem. In this work, we study the elicitability of multi-task peer prediction problems. We consider a designer who has certain knowledge about the underlying information structure and wants to elicit certain information from a group of participants. Our goal is to infer the possibility of having a desirable mechanism based on the primitives of the problem. Our contribution is twofold. First, we provide a characterization of the elicitable multi-task peer prediction problems, assuming that the designer only uses scoring mechanisms. Scoring mechanisms are the mechanisms that reward participants' reports for different tasks separately. The characterization uses a geometric approach based on the power diagram characterization in the single-task setting ([Lambert and Shoham, 2009, Frongillo and Witkowski, 2017]). For general mechanisms, we also give a necessary condition for a multi-task problem to be elicitable. Second, we consider the case when the designer aims to elicit some properties that are linear in the participant's posterior about the state of the world. We first show that in some cases, the designer basically can only elicit the posterior itself. We then look into the case when the designer aims to elicit the participants' posteriors. We give a necessary condition for the posterior to be elicitable. This condition implies that the mechanisms proposed by Kong and Schoenebeck are already the best we can hope for in their setting, in the sense that their mechanisms can solve any problem instance that can possibly be elicitable.

中文翻译：

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多任务对等预测的局限性

多任务对等预测的最新进展极大地扩展了我们对多任务对等预测机制的力量的认识。已经在不同的环境中提出了各种机制来引出不同类型的信息。但是我们仍然缺乏对多任务对等预测问题何时存在理想机制的了解。在这项工作中，我们研究了多任务对等预测问题的可引性。我们考虑一个设计师，他对底层信息结构有一定的了解，并希望从一组参与者那里得到某些信息。我们的目标是根据问题的原语推断拥有理想机制的可能性。我们的贡献是双重的。首先，我们提供了可引出的多任务对等预测问题的特征，假设设计者只使用评分机制。评分机制是分别奖励参与者对不同任务的报告的机制。表征使用基于单任务设置中的功率图表征的几何方法（[Lambert 和 Shoham，2009，Frongillo 和 Witkowski，2017]）。对于一般机制，我们还给出了可引出多任务问题的必要条件。其次，我们考虑设计者旨在引出一些在参与者关于世界状态的后验中线性的属性的情况。我们首先表明，在某些情况下，设计者基本上只能引出后验本身。然后我们研究设计师旨在引出参与者的后验的情况。我们给出了后验可引出的必要条件。