Criteria for projected discovery and exclusion sensitivities of counting experiments

The projected discovery and exclusion capabilities of particle physics and astrophysics/cosmology experiments are often quantified using the median expected $p$-value or its corresponding significance. We argue that this criterion leads to flawed results, which for example can counterintuitively project lessened sensitivities if the experiment takes more data or reduces its background. We discuss the merits of several alternatives to the median expected significance, both when the background is known and when it is subject to some uncertainty. We advocate for standard use of the "exact Asimov significance" $Z^{\rm A}$ detailed in this paper.
Comment: 10 pages. v2: new information about the extreme no-background limit included, two new figures added, other minor changes