I agree that it's probably (let's say 60% confidence) impossible to satisfy all of these at once. That said, if we have to pick just one, I think it's worth as

I agree that it's probably (let's say 60% confidence) impossible to satisfy all of these at once. That said, if we have to pick just one, I think it's worth asking what information is conveyed by the test. For instance, if all that is reported is "pass/fail", we should pick a scoring rule that maximizes a test taker's chance of passing when they are well calibrated. Most interesting to me is how different these scoring rules wind up being...