When Nobody Won: ties, competitions and seasons in archery

In Olympic recurve archery a set-system match can end five-all, and at that point the rules call for a single arrow to settle it. This volume measures how well that arrow, and every other admissible tie-break rule, identifies the better archer, and then widens the question to the tournament and to the season. The model is an archer who shoots a Gaussian group of known spread; from there matches are computed by exact convolution, tournaments by paired simulation, and the season ranking by a joint estimation of abilities and difficulties. The main results: the rule currently in force sits in the bottom third of the alternatives examined, and nine of the fourteen considered beat it on the reference pair, because they also read the arrows already shot in the match and not just the shoot-off arrow; reading the match total before the arrow is worth 0.3 points, less than recording distances is worth, and it retains that advantage even when the shoot-off is shot worse than the rest of the match; seeding matters more than bracket structure, and using the season ranking as the seeding is the most effective intervention among those that require nobody to shoot an extra arrow; and the questions the debate concentrates on, the length of the sets and the width of the repechage, move the results by fractions of a match out of sixty-three.