Counting the Number of People That Are Less Clever Than You

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.