There are two scorecards on the judges' table carrying the same total. The archers shot the same number of arrows, finished with the same figure at the bottom of the column, and the rulebook still has to put one of them ahead of the other. The answer everybody knows is on the line below: count the tens, and whoever has more is placed ahead.
It is a criterion as old as scorecards, and it looks like the most reasonable thing anyone could propose. Across all matches the intuition does hold: the archer who shoots more tens is usually the better one. But the criterion is not used across all matches. It is used only after a tie, and there things change: when the two totals are identical, counting tens sends the weaker archer through more often than the better one, 55 times in 100 for the pair of archers I took as the reference.
The calculation comes from work I published in the Lab, where it is set out in full, in exact rather than simulated form, and where I also record that I had written down the opposite prediction before doing the sums.
Where the rulebook counts tens
The ground needs marking out first, because the criterion has a much wider reputation than it has actual use. In the international rulebook currently in force, Book 3 in its March 2026 version, counting tens appears in exactly one place: ordinary ties in the qualification ranking, where the Xs are looked at first and then, if those are level too, the tens.
In elimination matches it does not appear at all. At five-all the rulebook calls for a shoot-off arrow, and even the ties that decide who makes the elimination rounds are not settled by counting rings. What follows therefore concerns a rule with a single formal use and a much broader informal one, because counting tens is how anybody in the stands or in front of a scorecard decides which of two archers shot better.
One last point of geometry is needed for what follows: the target face used at 70 metres is 122 cm across and divided into ten scoring rings of equal width, 6.1 cm each. Inside the 10 there is a smaller circle of about 3 cm radius called the X ring: it scores ten like the rest of the ring, but it is counted separately, specifically so that ties can be broken.
Why it looks like the fairest criterion
The intuition behind counting tens is sound, and it can be checked. Take two archers of different ability, say one worth an average of 690 points out of 720 and one worth 680, look at every match they play against each other, and the picture is exactly what you would expect: the better archer shoots more tens and throws away fewer arrows.
| Over 15 match arrows | The 690 archer | The 680 archer |
|---|---|---|
| Tens, across all matches | 9.110 | 7.588 |
| Arrows scoring 8 or less, all matches | 0.357 | 0.894 |
| Tens, only where the totals are level | 8.343 | 8.495 |
| Arrows scoring 8 or less, totals level | 0.443 | 0.587 |
The first two rows confirm the intuition: nine tens against seven and a half, and less than half as many arrows thrown away. Anyone looking at a scorecard and counting tens is using an indicator that, on any ordinary population of matches, points to the better archer 71.1 times in 100 in the model.
The last two rows are the same measurement on the same model, with one difference: they look only at matches where the two totals came out equal. There the better archer shoots 8.343 tens and the weaker one 8.495, so the sign flips: the difference between the two goes from plus 1.52 to minus 0.15, and that swing is not simulation noise, because the calculation is exact.
What happens when you look only where it is used
The effect just described has a general cause, and it is better explained before the numbers because it applies to any tie-break rule and not only to this one. A tie is the product of a filter: out of all the scorecards, it keeps precisely those on which the scores failed to separate the two archers, so a tie is not a neutral circumstance that happens to land on some ordinary match. Measuring a rule across all matches and then applying it only to ties amounts to promising a performance that ties do not deliver.
The essay measures that shortfall for five different criteria, comparing what each gets right across all matches with what it gets right on matches that actually ended five-all.
| Criterion | Across all matches | On five-all matches only |
|---|---|---|
| Sum of squared distances from the centre | 77.8 | 68.8 |
| Mean distance from the centre | 76.9 | 67.1 |
| Weighted count of the rings | 75.1 | 62.7 |
| Total score of the match | 74.3 | 60.5 |
| Counting tens | 71.1 | 53.6 |
Predictably, every rule loses something: the ones that read distances lose about a third of their advantage over a coin, the ones that read rings lose about half, and counting tens loses five sixths. It falls from 71.1 to 53.6, which means it gets it right only 3.6 times in 100 more often than a coin would. Of every criterion examined, it is the one that suffers most from the move to the population where it is actually used.
The reversal
The 53.6 in that table covers every match that ended five-all, including those where the totals over 15 arrows differ and the criterion therefore has something to grip. Narrow it further, to scorecards where the totals are identical as well, and the number drops below the coin: on the reference pair, counting tens sends the better archer through 44.7 times in 100, which means it sends the weaker archer through 55.
The first sensible reaction is to ask whether it depends on that particular pair of archers. The essay computes the same quantity across eight pairs spread along the whole scale, from two archers worth 700 and 695 points down to two worth 640 and 600, and the answer is that it sits below fifty everywhere.
On the closest pair, two archers worth 700 and 695, the criterion reaches 48.70 and so stays just below a coin toss. It gets worse as the level drops, and on the widest row, between an archer worth 640 and one worth 600, it settles at 36.57, which means sending the better archer home 63 times in 100. Adding Xs as a second criterion after the tens does not repair the fault: with both criteria applied in sequence the figure stays below fifty for seven pairs out of eight, and only on the closest, 700 and 695, does it edge above, to 52.19.
The explanation is arithmetic
The reason for the reversal is easier to grasp with a scorecard in hand than with a formula. Two archers have shot 15 arrows each and both finished on 140 points. The first shot eleven tens, which is 110 points, leaving 30 points to spread over four arrows, an average of seven and a half. The second shot nine tens, which is 90 points, leaving 50 points over six arrows, an average of eight and a third.
The first archer has two more tens and the rulebook sends him through. Look at the other arrows, though, and the first archer has something the second does not, namely a couple of decidedly bad shots, and it is precisely those shots that left him room for the extra tens without going over the total. With the total fixed, every additional ten has to be paid back somewhere else, and it is paid back on the low arrows.
In the model that link is all but ironclad: on the reference pair, and on scorecards with level totals only, the difference in tens between the two archers and the difference in arrows thrown away have a correlation of 0.9898, which falls only to 0.9488 on a much wider pair, between an archer worth 680 and one worth 660. Stated precisely, the result is this: with the total fixed, counting tens measures which archer concentrated their deficit into fewer arrows, and not which archer shot better.
The link is not an absolute arithmetic necessity, though, because scorecards can be constructed on paper in which more tens go together with fewer arrows thrown away. The essay gives the counterexample explicitly and says so: the link is a regularity in the way archers distribute their errors, not a theorem. Even so, that regularity holds across every pair on the grid that was tested.
Why this matters
A criterion that works in general and reverses in one particular case would already be a defect. Here the situation is worse, because the particular case where it reverses coincides exactly with the only case in which the criterion is consulted. Nobody counts the tens of two archers who finished on different totals: tens are counted only after a tie, and so only once the very condition that flips the sign of the measurement has occurred.
There is also a possible misreading to rule out: one might think the phenomenon depends on the five-all, meaning on that particular form of tie produced by the set system. The essay therefore redoes the calculation asking only that the two totals over 15 arrows coincide, with no condition on sets at all, and the reversal holds across the whole grid, with differences of between two and seven tenths against the earlier values. The cause, then, lies in the levelling of the totals and not in the route taken to get there.
What a ten is actually worth
What remains is to understand why, of all the ways of reading a scorecard, the traditional one is the most fragile. The essay builds the answer by asking how much each ring ought to weigh in order to tell two archers apart as well as possible, and then comparing those weights against the points the rulebook assigns. Rescaled onto the same scale from zero to ten, the weights the rings deserve turn out to be strongly concave: ring 1 deserves 1.9 rather than 1, ring 2 deserves 3.6 rather than 2, ring 5 deserves 7.4, and ring 9 deserves 9.9.
The consequence shows in the steps from one ring to the next: going from an arrow off the target to an arrow in ring 1 is worth 1.9 of information; going from a 9 to a 10 is worth 0.15. Knowing that one archer missed the target where the other scored a 1 says twelve and a half times more about the difference between them than knowing that one shot a ten where the other shot a nine.
Competition counts give this reasoning its base rate. In the registry, the claim that even at the highest level most arrows are not tens is corroborated by three independent samples. In the qualification rounds of Chinese professional competitions, 20.7% of arrows land in the ten. Among eight high-level French archers, 549 arrows at 70 metres, the share rises to 30.2%, and arrows scoring 8 or less are almost as common, at 29.7%. In head-to-head matches at the Games, the World Championships and the European Grand Prix, across 6,374 sets, the first two arrows of the set are both tens in only 13% of cases, while in almost half neither of them is. Counting tens therefore looks at a single line on the target, the one between the 9 and the 10, and on the other side of that line, even at the top, sit most of the arrows: a 9 and a 6 count the same, simply as arrows that are not tens, yet by the essay's weights a difference like that says far more than the step from a 9 to a 10.
On the reference pair the model is more generous with tens, more than nine in fifteen for the 690 archer, because it describes archers of international standard shooting with no wind. Lower down the scale, among archers who shoot fewer tens, the essay's grid shows a reversal that is generally sharper, not weaker.
Counting tens puts all of its weight on the least informative step of all, the one between the 9 and the 10, which is why it sits at the bottom of the table. A direct comparison shows how much the choice of cut-off matters: with the shoot-off arrow appended, counting the arrows below 9 gets it right 62.4 times in 100 ties, counting those below 10 gets it right 55.9. The two rules differ by a single ring, yet the first gets six and a half more decisions right per hundred than the second.
Counting the bad arrows also has a practical merit: among the best rules it is the easiest to check from the stands, because all it takes is counting the arrows below 9 on the scorecard. On its own, without the shoot-off arrow appended, it gets it right 59.3 times in 100 ties.
The X ring is a different case
The next question is about the other criterion in use, the count of Xs, which the rulebook consults before the tens. The X escapes the arithmetic constraint for a structural reason: it is a subdivision inside the 10, so moving from an ordinary ten to an X does not change the total and requires no repayment anywhere else. On scorecards with level totals it does indeed stay above the coin for pairs at the top: 53.9 for the two archers worth 700 and 695, 53.1 for the reference pair, and 54.3 for the 690 and 670 pair.
The advantage, though, depends on how large the X ring is compared with the archer's group, and as the level drops the group widens, as comparisons between archers of different levels also report. The essay locates the point where the ratio between the two turns over at around 670 points out of 720: above that level the X carries information, below it the X reverses just as counting tens does. The identical rule, applied at a national competition rather than a world final, changes sign without a word of the rulebook changing.
What happens in the qualification ranking
Now to the place where the criterion is formally used, ordinary ties in the qualification ranking, and here a caveat is needed, one that the essay itself stresses. All the figures for the reversal are computed on scorecards from matches of 15 arrows each, not on rounds of 72, and they must not be carried wholesale from one population to the other.
What was measured directly on the ranking is something else, across 120,000 simulated qualification rounds with 64 archers between 690 and 665 points shooting 72 arrows each. In a ranking where ties are drawn by lot, the top qualifier really is the best archer in the field in 14.0% of cases; add the count of Xs and it rises to 14.2%; add the count of tens on top, which is to say apply today's rule in full, and it stays at 14.2%. With a standard error of a tenth of a point, the Xs are worth two tenths and the tens are worth nothing.
The same holds for the other columns of the same measurement: the mean positional error goes from 8.62 places to 8.60 with the Xs and back up to 8.61 with the tens, which is to say it does not move. Order the qualification round by the sum of squared distances from the centre instead, using the information the rings discard every time they round an impact, and the mean error falls to 7.70 places while the share of qualification rounds in which the top eight are exactly the best eight goes from 0.04% to 0.09%: more than double, but still an extremely rare event.
This needs stating precisely, because in the essay the two points sit in different chapters. In the chapter on the reversal I write that it would strike counting tens wherever it is used to break a tie on total, and so in the ranking as well; the measurement made on the ranking says something less sharp, namely that there the Xs and the tens do no damage of the kind they do in a match and simply add no information. Of the two formulations I keep the measured one: in the ranking, counting tens adds nothing.
How to read a tied scorecard
For anyone looking at a scorecard and wanting to know who shot better, the hierarchy that comes out of all this is clear and uses only figures already written on the sheet. The first place to look is the total over the 15 arrows, which after a five-all stays identical in about a third of cases and says something in the other two thirds.
| Margin on the 15-arrow total | How often it happens | How often it points to the better archer |
|---|---|---|
| No margin, totals level | 34.4% | 50.48, which decides nothing |
| 1 point | 40.8% | 60.8 |
| 2 points | 17.4% | 71.3 |
| 3 points | 5.6% | 80.7 |
| 4 points | 1.5% | 87.8 |
| 5 points or more | 0.4% | 93.4 |
A single point of margin, which is the commonest case of all, already takes it to 60.8%; two points take it to 71.3; three take it to 80.7. That number is already written at the foot of the column on the scorecard and needs no extra measurement, which makes it hard to explain why the match never consults it.
The second place to look is the bad arrows, counting how many each archer shot below 9. The third, counting tens, is the only one of the three that in the model, at the exact moment you consult it, points more often at the weaker archer. What the essay draws from all this is a proposal to remove it from the tie-break criteria rather than reorder it, because no reordering repairs an inverted sign.
For the archer going back over a defeat, the same point can be put less technically: if you finished the match on the same total as your opponent and shot more tens than they did, those tens do not say you shot better; what they say is that your errors were concentrated into two or three arrows rather than spread across many. The work to be done in that case is on the worst arrows and not on the good ones, which is exactly the opposite of what the count suggests.
Caveats and the counterargument
Everything above holds under the dispersion model the volume adopts, and the dependence on that model is stronger here than anywhere else in the work. It is a modelling study rather than an empirical one, no competition data was used, and the starting scores are a declared assumption.
The check that counts is the one the essay puts in an appendix and calls, quite plainly, the argument available to anyone defending the traditional rule. If a small proportion of arrows falls outside the regular behaviour of the group, meaning that now and then an archer shoots an anomalous arrow for reasons the model does not describe, counting tens becomes informative again. With three anomalous arrows in a hundred it climbs back to 52.44, which is above the coin; with five in a hundred it reaches 55.54; with ten in a hundred it reaches 63.85 and becomes a good criterion.
The baseline model, for its part, is very harsh on arrows like these: for an archer worth 690 points it places an arrow below the 7 once in three million. In the data gathered in the registry, by contrast, almost every study that reports the individual arrows of high-level archers finds the occasional arrow away from the centre. Among four elite Australian archers, at 30 m on the 60 cm face, 31 arrows out of 240 scored between 5 and 7 and were classed as poor on the archer's and coach's ratings as well; among seven juniors who had shot at the Junior World Championships, indoors at 30 m, the weaker ones went down to a 5 and a 6; and in head-to-head matches recorded by World Archery between 2013 and 2019 the lowest score on a shoot-off arrow is 1. In the registry the claim is nonetheless contested, because seven elite Indian archers never dropped below the 8 over 210 indoor arrows at 18 m; and nobody has counted how often it happens over full rounds.
Those counts, moreover, are made on rings, at different distances and on different faces, and on archers who are not all worth 690 points. They show arrows far from the centre in high-level archers, but not how many of those arrows fall outside the regular behaviour of the group and how many are simply the wide part of a regular group. That is exactly the quantity that would settle the matter, and the essay states that it cannot estimate it: it would take the coordinates of the impacts, because from the rings alone frequency and severity are confounded with each other. Anyone defending the historical criterion therefore has a real argument, partly backed by data on archers, and the discussion moves to a number that ring scores do not contain.
Wind works in the same direction as the anomalous arrows, but with less force: it takes the reference pair from 44.66 to 47.27 with 6 cm of wind error and to 48.84 with 10 cm, and in neither case does the criterion reach the coin. Other departures from the model push the opposite way: an off-centre group, for instance, takes counting tens on the 700 and 690 pair from 46.80 to 41.76, deepening the reversal rather than withdrawing it.
Where this goes next
The route that would genuinely lead out of the problem runs through the kind of data that gets recorded, not through a better criterion among those available. Order by the sum of squared distances from the centre, meaning record where the arrow struck rather than only which ring it struck, and you get it right 68.8 times in 100 ties, far more than any rule based on rings, and the precision needed to do it is 3 mm. The essay states plainly that it cannot estimate what instrumentation like that would cost.
The volume this chapter comes from then widens out, measuring bracket structures, match formats and season rankings, and reconstructing what each choice of rule costs and what it returns. I have written about the part on ties in matches in how often a match ends five-all, and about the part on seeding in what a qualification ranking is really worth.
The one thing to take from this chapter bears on the rulebook as much as on the way anyone reads their own scorecard: two archers who finished on the same total may have shot in very different ways, and the sheet shows it to anyone who looks in the right box, which is not the one holding the tens.