The shoot-off in archery: how much is one arrow worth?

A shoot-off is not a moral judgement: it is a measuring instrument, and of a measuring instrument you can ask how often it is wrong. These pages ask one question, and it has a numerical answer: out of a hundred shoot-offs between two archers of different ability, how many times does that arrow point to the better of the two. "Better" is defined in advance and deliberately narrowly, as whoever groups tighter, because defining it as whoever won would make every rule infallible by construction. As a reading of that single arrow, the existing rule is already the best possible: only rules that also look at the arrows already shot in the match can do better, and the volume When Nobody Won compares them. And the instrument is used precisely where it knows the least. Weighting every pair by its probability of actually reaching a shoot-off, the average 72-arrow ability gap among archers who genuinely shoot one lies between 9.1 and 14.0 points, and the shoot-off identifies the more precise archer only 55.6% to 57.2% of the time.