On a level score, counting tens rewards the weaker archer

Part of the complete guide: How to improve your archery score

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 goes through.

It is a criterion as old as scorecards, it looks like the most reasonable thing anyone could propose, and on that point ordinary intuition is right almost always. The trouble is that almost always does not include the moment the criterion is actually used. In the one situation where a judge consults it, which is when the two totals are identical, counting tens sends the weaker archer through 55 times in 100.

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 the author also records having written down the opposite prediction before doing the sums.

Where the rulebook counts tens

The ground is worth 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 note on geometry makes the rest legible. The 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 3 cm radius called the X ring: it scores ten like the rest of the ring, but it is counted separately, and breaking ties is exactly what it exists for.

Why it looks like the fairest criterion

The intuition behind counting tens is solid, 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.

The first two rows confirm the intuition with no room for doubt: 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 works very well on any ordinary population of matches.

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, which is to say the sign flips. The difference between the two goes from plus one and a half to minus a seventh, and that swing is not simulation noise, because the calculation is exact.

What happens when you look only where it is used

What has just appeared has a general cause worth explaining before the numbers, because it applies to any tie-break rule and not only to this one. A tie is not a neutral circumstance that happens to land on some ordinary match. It is the product of a filter that selects precisely the scorecards on which the scores failed to separate the two archers. Measuring a rule across all matches and then applying it only to ties is promising a performance that will never occur in practice.

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.

Every rule loses something, and that much was predictable. 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: from 71 it falls to 53.6, which is getting it right four times in a hundred 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 repairs nothing, because the column with both criteria stacked stays below fifty across the whole grid as well.

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 close to iron. Measuring, on the reference pair, how far the difference in tens between the two archers tracks the difference in arrows thrown away, the correlation between the two quantities on scorecards with level totals comes to 0.9898, and it 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.

It should be said that this is not an absolute arithmetic necessity, because scorecards can be constructed 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. The point is that the 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, which is to say only after the very condition that flips the sign of the measurement.

There is also a check that closes off one possible misreading. You might think the phenomenon depends on the five-all, meaning on that particular form of tie produced by the set format. The essay 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 is the levelling of the totals, not 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 is in the steps. Going from an arrow lost 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 threw an arrow away and the other did not 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.

Counting tens puts all of its weight on exactly the least informative ring there is, which is why it sits at the bottom of the table. How much the choice of cut matters is shown by a comparison worth quoting: counting the arrows below 9 gets it right 62.4 times in 100 ties, counting the arrows below 10 gets it right 55.9. Two rules identical but for one ring of difference, and six and a half correct decisions per hundred between them.

Counting the bad arrows has a further merit no other criterion on this list can claim: it can be checked from the stands without instruments, because counting the arrows below 9 is within reach of anyone looking at a scorecard. On its own it gets it right 59.3 times in 100 ties, and with the shoot-off arrow appended it reaches 62.44.

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 widest of the high-level pairs.

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. 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 qualification the essay itself highlights is needed. 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 has been measured directly on the ranking is a different thing: 120,000 simulated qualification rounds, 64 archers between 690 and 665 points, 72 arrows each. Starting from a ranking in which 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 competitions in which the top eight really are the best eight more than doubles.

On one point, and I say it because the essay does not hide it, the volume is not entirely consistent with itself. One chapter states that the reversal strikes wherever tens are used to break a tie on total, and therefore in the ranking as well; another writes that in the ranking the criteria do no damage of the kind they do in a match, because the tie on total that reverses them is not present in the same way, and that they simply add no information. The two formulations do not coincide, and it does not seem right to credit the work with the sharper of the two. What holds is that in the ranking, counting tens has been measured and 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.

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 at the exact moment you consult it points at the wrong person. 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 rereading a defeat, the translation is less technical. 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. They say your errors were concentrated into two or three arrows instead of 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.

The counterargument, which the essay supplies itself

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, without circumlocution, 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 true frequency of those anomalous arrows has not been estimated, and the essay states that it cannot be estimated with existing data: 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 to use, and the discussion moves to a quantity nobody has yet measured.

A second factor works in the same direction and should be added. Wind softens the reversal, taking the reference pair from 44.66 to 47.27 with 6 cm of wind error and to 48.84 with 10 cm, but in neither case does the criterion reach the coin. What is interesting for once is the direction: almost every other factor left out of the model works against the traditional rule, and wind is one of only two exceptions.

Where this goes next

The route that would genuinely lead out of the problem does not run through a better criterion among those available, but through the kind of data that gets recorded. 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. What instrumentation like that would cost, the essay says plainly it cannot estimate.

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. The piece about ties in matches I have written about in how often a match ends five-all, and the piece about seeding in what a qualification ranking is really worth.

What is left of this chapter is one thing, and it bears on the rulebook as much as on the way anyone reads their own scorecard: two archers who finished on the same total did not shoot equally well, and the sheet says so. You only have to look at the right box, and it is not the one holding the tens.

Questions I get asked most

How is a tie broken in archery?
It depends where the tie occurs. In the qualification ranking, on a level total, the international rulebook looks first at the number of Xs and then at the number of tens. In elimination matches the count does not come into it at all: at five-all a shoot-off arrow is shot, and whoever puts it closer to the centre wins.

Does the archer with more tens shoot better?
Across any ordinary population of matches, yes, and clearly so: between an archer worth 690 and one worth 680, the first shoots an average of 9.11 tens over 15 arrows against 7.59. Looking only at the scorecards where the two totals coincide, which is the only situation where the criterion is consulted, the sign flips and counting tens points to the better archer 44.7 times in 100.

Why does counting tens point to the weaker archer on a level score?
Because with the total fixed, every extra ten has to be paid back somewhere else. The archer with more tens lost the same points concentrated into a few very poor arrows, while the other spread them across many mediocre ones. In the model the link between the two gaps has a correlation of 0.9898, so the count ends up measuring how concentrated the errors were rather than how good the shooting was.

What is the best tie-break criterion available right now?
The total over the 15 arrows, which is already written on the scorecard: it says something in two cases out of three, and when it speaks it is right between six and nine times in ten, from 60.8% on a one-point margin to 93.4% on five or more. Next comes counting the arrows below 9, which gets it right 62.4 times in 100 ties against 55.9 for counting tens.