  To extend a partial order to a total order, we can map each element s to the
number of elements that are less than s. Due to the transitivity of a partial
order, the obtained total order preserves the original partial order. In a
continuous measurable space, we can map each element s to the volume of the
space that consists of elements that are less than s.
  This simple idea generates a new family of mutual information measures,
volume mutual information (VMI). This new measure family has an application in
peer prediction. In the setting where participants are asked 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 subjective feedback without verification. We use VMI to design
a family of mechanisms where truth-telling is better than any other strategy
and all participants only need to answer a small constant number of tasks.
Previously, Determinant Mutual Information (DMI)-Mechanism is the only
mechanism that satisfies the two properties. We also give DMI a geometric
intuition by proving that DMI is a special case of VMI. Finally, we provide a
visualization of multiple commonly used information measures as well as the new
VMI in the binary case.
