  In the setting where we ask participants multiple similar possibly subjective
multi-choice questions (e.g. Do you like Bulbasaur? Y/N; do you like Squirtle?
Y/N), peer prediction aims to design mechanisms that encourage honest feedback
without verification. A series of works have successfully designed multi-task
peer prediction mechanisms where reporting truthfully is better than any other
strategy (dominantly truthful), while they require an infinite number of tasks.
A recent work proposes the first multi-task peer prediction mechanism,
Determinant Mutual Information (DMI)-Mechanism, where not only is dominantly
truthful but also works for a finite number of tasks (practical). However, the
existence of other practical dominantly-truthful multi-task peer prediction
mechanisms remains to be an open question. This work answers the above question
by providing 1. a new family of information-monotone information measures:
volume mutual information (VMI), where DMI is a special case; 2. a new family
of practical dominantly-truthful multi-task peer prediction mechanisms,
VMI-Mechanisms. To illustrate the importance of VMI-Mechanisms, we also provide
a tractable effort incentive optimization goal. We show that DMI-Mechanism may
not be not optimal but we can construct a sequence of VMI-Mechanisms that are
approximately optimal. The main technical highlight in this paper is a novel
geometric information measure, Volume Mutual Information, that is based on a
simple idea: we can measure an object A's information amount by the number of
objects that is less informative than A. Different densities over the object
lead to different information measures. This also gives Determinant Mutual
Information a simple geometric interpretation.
