  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.
