Thanks. Maybe what I'm looking for is a way to choose among the possible numbers we could select for par for those holes where the expected score isn't obvious. For those, if the definition doesn't provide an single answer, perhaps we are free to set a secondary set of selection rules. Like, perhaps, choose the one that is closest to average.
However, talking about a possible rule like that is getting a little ahead of where I'm at. I want to step back and see if there is any way to look at par on a meta basis to see if it's good or not.
Sometimes when we can't define how to make something better, it is easier to define how to make it worse and then do the opposite.
We all knew par was not as useful as it should have been 5 years ago. We didn't need to look at individual holes to know that. (Although some individual holes provided more evidence.) How did we know par was not fully useful? What was bad about it? How could it have been worse?
It seems to me that you're trying to apply some mathematical certainty and precision to some imprecise terms. In particular, "errorless". I respect the effort. But it's more than I wish to make.
For myself, with a hole without clear expectation, I tend to look at a range of results---average, most common score, median (or something a bit higher, maybe 60th percentile), and then take a stab at it. With a hope that such holes are few enough to have only a minor effect, in the grand scheme.
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I guess there are a couple of ways to judge whether par is good.
One would be, if it reflects the expected score, it's useful in all the ways previously discussed. Spectators following results, portability, etc.
The other would be, if it matches golf, at least in certain ways, then it's useful aesthetically to those who think it should.