The shoot-off in archery: what one arrow decides

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

Five sets played, five-all. The board reads 5-5 and the rules do not provide a sixth set, because there were five sets and five have been shot. They provide one arrow. The two archers come back to the line, shoot one each, and whoever puts it closer to the centre wins. The match, the round, and sometimes the medal are settled in a little over forty seconds and two shots.

Everyone has an opinion about this rule, and it is always one of the same two. The first says it is a lottery, that reducing a contest to a single shot is an elegant way of tossing a coin. The second replies that an arrow is still an arrow, that the stronger archer emerges under pressure, and that the better archer makes that shot better precisely because they are the better archer. Both are respectable positions, they have been colliding for fifteen years, and the argument has almost always gone on without figures.

Of a measuring instrument, though, you can ask a question you cannot ask of an opinion: not whether it is fair, but how often it is wrong. Out of a hundred shoot-offs between two archers of different ability, how many times does that arrow really point to the more precise of the two? It is a percentage, and it follows from the geometry of the target and the rules of competition without anyone going to a field. The calculation comes from work I published in the Lab, where it is set out at length with the code to redo it.

The procedure, in full

In individual recurve a match is played over the best of five sets of three arrows. Each set is worth two points to whoever scores higher, one each if the two are level, and the match goes to whoever reaches six set points. The sum of the arrows does not count: the sets count. That produces an outcome cumulative scoring could not produce, because if after five sets the board says 5-5 there is no way to continue, and the rulebook changes register.

It then calls for one arrow each and awards the win to whichever lands closer to the centre. Formally there are two criteria in sequence, first the score of the ring and then, on an equal ring, the distance from the centre. But for two arrows striking the scoring zone on a target made of concentric circles the two criteria always coincide, since an arrow closer in cannot end up in a lower-value ring. If the distance is identical too the arrows are shot again, and if both miss the scoring zone both archers shoot another.

In international finals shooting is alternate, one archer at a time with twenty seconds per arrow, so a shoot-off is two shots and a little over forty seconds on the clock, plus the time the judges take to measure. The procedure has changed over the years: for a while two tens forced a reshoot, and that provision is no longer in place. Compound works differently, because the match is cumulative over fifteen arrows, but there too a tie is settled with a single arrow, and everything that follows applies to both bows.

Nobody arrives at a shoot-off by accident

Before measuring how often that arrow is wrong, one point needs making that changes how every subsequent number should be read: shoot-offs do not happen between any two archers. They happen between two archers who have just played five sets without either managing to pull away from the other, and that selects them.

The frequency shows it plainly. Between two identical archers at 690 points a set match ends five-all in 20.3% of cases, which is one match in five. At ten points apart it falls to 16.8%, at twenty points to 10.8%. The more alike the two are, the likelier the contest is to reach that point, and it is exactly why shoot-offs select similar pairs.

The practical result is that the mean gap between two archers who actually reach a shoot-off runs between nine and fourteen points over 72 arrows, depending on how the field is made up. They are not two strangers drawn at random: they are two archers who, on the arrows already shot, failed to distinguish themselves.

What a single arrow actually says

Before quantifying the error it helps to look at the mechanism, because it explains both why the rule works and why it works so little. An arrow landing near the centre is evidence in favour of the better archer, one landing far out is evidence against, and the strength of that evidence can be calculated point by point across the target.

Take an archer at 690 points against one at 670. An arrow planted one centimetre from the centre is 68% more likely to have come from the more precise of the two. One at twelve centimetres is two and a half times more likely to have come from the less precise, and a little past twelve and a half centimetres exactly three times more likely. Between those two extremes there is a point of indifference, around seven centimetres from the centre, where the arrow says nothing whatever about which of the two shot it.

From this follows a property of the rule worth knowing, because it is the opposite of what its critics maintain: reading that arrow by who came closest to the centre is the best possible reading. There is no different way of interpreting that single shot that would do better, because the probability that the evidence favours the better archer rises continuously as the arrow gets closer to the centre, and ordering by distance is precisely ordering by strength of evidence.

The detail of the distance, as against the ring alone, contributes something too: reading only the value of the ring and drawing lots on a tie would lower reliability by about two percentage points. The millimetre measurement the judges make, in short, is not a formality.

The limit lies elsewhere, and it is quantitative before it is conceptual. An archer at 690 puts the arrow in the ten six times in ten, one at 650 three times in ten: the two groups overlap across most of their extent, and a single draw from two distributions that similar is not enough to say which of the two it came from.

How far apart they have to be for that arrow to notice

The question can be turned around the way you turn the question about a scale: what is the smallest difference the instrument can perceive? Answering it takes a definition of "better", and the essay adopts a deliberately spare one, namely whoever groups tighter and nothing else, declared before any result is looked at. Defining the better archer as whoever won would make every rule infallible by construction.

With that definition the calculation closes in a line: the probability that the shoot-off arrow points to the more precise archer depends only on the ratio between the widths of the two groups. And the scale on which every answer is read has two fixed endpoints, 50%, which you get by tossing a coin without even looking at the target, and 100%.

The ten millimetres on the second row deserve a comment, because that is the distance separating a top-level archer from a good one, not two archers who resemble each other. To reach sixty times in a hundred, which is a reliability nobody would call satisfactory, that arrow needs almost fifteen points of gap over 72 arrows.

The reverse is clearer still. Between two archers separated by five points over a full round, which is worth three and a half millimetres of group, the shoot-off points to the more precise archer about fifty-four times in a hundred. That is four percentage points away from a coin toss.

And the picture gets worse as the level drops, for a geometric reason: wide groups overlap far more than tight ones, so the same relative difference becomes harder to read. Reaching 60% reliability takes 12.5 points of gap between archers at 700, 14.8 between archers at 690, 19.3 between archers at 670, and 23 between archers at 650. Anyone running a regional competition is therefore working with a blunter instrument than anyone running a world final.

The number that answers the question

Put the two things together, the resolution of the instrument and the fact that shoot-offs select similar pairs, and you get the answer to the original question. In the situations where it is actually called upon, the shoot-off points to the more precise of the two between 55% and 57% of the time.

The comparison that makes the number interesting is with the scenario in which shoot-offs happened between archers drawn at random from the field, without the filter of five sets played level: in that case the same rule would point to the more precise archer between 57% and 61% of the time. The difference between the two figures is produced entirely by the selection.

The diagnosis that comes out is precise and runs against what you hear on shooting fields. The shoot-off is close to a coin toss not because it reads the arrow badly, but because it gets called upon exactly when the two archers resemble each other too closely for one arrow to tell them apart. The problem is not the reading, it is the amount of information available: 72 arrows against one.

Why shooting more is not the shortcut it looks like

The proposal that occurs to everybody is to add arrows. It works, and it can be quantified: for a pair twelve points apart, one arrow points to the more precise archer 58% of the time, two arrows 62%, three arrows 65%. The first additions are also the most profitable, because the second and third arrows are each worth more than any that follow and on their own recover two fifths of the entire gain you would get by going to nine.

The cost in time is modest: two extra arrows each in alternate shooting is four shots, a little over a minute.

The problem arrives afterwards, and it has the shape of a familiar law. The precision of a repeated measurement grows as the square root of the number of trials, so going from one arrow to four multiplies the advantage by 2.13, going from four to sixteen multiplies it by 1.85, and going from sixteen to sixty-four only by 1.45. The return falls while the time grows linearly, which means there is no reasonable number of arrows capable of turning a shoot-off into a reliable instrument.

Pressure, put into numbers

The commonest objection to all of this is that the model does not account for pressure, and that a shoot-off arrow is not an arrow like the others. The objection is legitimate, and the essay meets it by quantifying it rather than dismissing it.

Pressure can be expressed as a temporary worsening of the group, measured in equivalent score points, and you can see how far it moves the result. For the typical pair twelve points apart, with no pressure the shoot-off points to the more precise archer 58.3% of the time; with a drop equivalent to ten points it falls to 56.5%, with twenty to 55%. Genuinely demolishing the instrument would take a great deal more: a drop of twenty-two points to get down to 55%, of seventy-eight to reach 52%, of a hundred and forty to touch 51%.

That last number gives the measure of the thing, because a drop of a hundred and forty points would mean an archer at 690 shooting, in that moment, like one at 550. As long as pressure strikes the two contenders alike, then, it moves the result by a few percentage points and does not change the conclusion. If it acted differently on the two of them, the calculation does not cover it, and the essay says so openly among its own limitations.

On the bracket, where it matters most

The consequences show up best across a whole tournament. In a bracket of 64 archers, with 63 matches, the set format sends between eight and ten matches to a shoot-off, a little over twice what cumulative scoring would send, which would stop at around five.

More interesting is how the two quantities move along the bracket, because they run in opposite directions. In the first round about 11.5% of matches finish in a shoot-off, and the share grows round by round to almost 19% in the final, because as the competition proceeds the archers left in resemble each other more and more. Over the same span the reliability of the arrow falls: in the first round it points to the more precise archer between 57% and 61% of the time, in the final between 51.5% and 55.5%. The mean gap between the two contenders goes from fifteen points to five.

Put in one sentence: about one final in five is decided by a single arrow, and that arrow, right there, points to the more precise archer a little over fifty-three times in a hundred. The instrument becomes least reliable exactly where the stakes are highest.

What the calculation does not look at

These numbers come out of a model rather than a collection of real results, and it is right to say precisely what is left out. No data from real people enters the work: the profiles at 690 or 678 are nobody's scores, but synthetic archers defined by the width of their group. Everything comes from three public and impersonal sources, namely the geometry of the target as fixed by the rules, the rules of the competition formats, and a statistical model of shot dispersion.

The definition of ability is spare by choice, and it leaves out nerve, reading the wind, and the capacity to produce the right shot when it is needed, which are exactly the qualities that weigh most at the moment that arrow has to be shot. The choice has one virtue, in that a group can be measured while the other qualities have no shared ruler, but it remains a choice.

The most serious limitation concerns the independence of the arrows. The calculation assumes the shoot-off arrow comes out of the same group as the others, and researchers who have compared shoot-off arrows with match arrows have found signs that it does not. The essay declares this the most fragile point in the whole structure.

On the shape of the group, by contrast, the result holds: testing nine different shapes, from circular to displaced or elongated, the band of results stays narrow and none of the shapes brings the shoot-off close to being reliable.

What to do with it, as an archer and as a coach

For the archer, the consequence is that a lost shoot-off is not a report on your technique. If you got there, it means that across the previous five sets you and your opponent did not separate, and that arrow, in those conditions, is close to a coin. Analysing for weeks the single shot that decided a match means looking for a signal inside a piece of data that does not contain it.

The reverse holds too, and it is the less welcome half: a shoot-off won says little about the relative merit of the two. The correct way to read either case is to look at the five sets that came before, where there are fifteen arrows and a great deal more information.

For anyone coaching, there is a consequence for preparation. If one match in five between equals finishes five-all, then the shoot-off arrow is a recurring situation rather than an exception, and it is worth training as part of the routine, with the same sequence used for every other shot. Not because that arrow is more important than the others, but because it is the only one shot in conditions that never come up in practice.

Where this goes next

One question this article does not take on is the one the essay develops in full: whether better readings of that situation exist, and what they cost. Some require no extra arrow at all, because they use information already written on the scorecard, namely the fifteen arrows of the match just finished.

The wider picture this sits in concerns the way points are counted in a match, which I have described in the article on how the set system works, and the precision of the ranking that decides who meets whom, which I have written about in how many points it takes to say one archer is better than another.

Questions I get asked most

How does a shoot-off work in archery?
When a set match ends five-all, each archer shoots one arrow and whoever puts it closer to the centre wins. Formally the ring is looked at first and then, on an equal ring, the distance from the centre, but on concentric circles the two criteria always coincide. In finals shooting is alternate with twenty seconds per arrow. In compound the match is cumulative, but a tie is settled the same way.

Is a shoot-off really a lottery?
Almost. In the situations where it is actually used it points to the more precise of the two between 55% and 57% of the time, and in a final it drops to around 53%. The reason is not that the rule reads the arrow badly, though. It is the best possible reading. The reason is that the two archers who get there are two the previous fifteen shots failed to separate.

How far apart do two archers have to be for a shoot-off to work?
Reaching sixty times in a hundred takes almost fifteen points of gap over 72 arrows at the 690 level, which is worth ten millimetres of group size. Between archers at 650 it takes twenty-three. Between two archers separated by five points the shoot-off points to the more precise one about fifty-four times in a hundred.

Would shooting more shoot-off arrows not fix it?
It helps, but less than it looks. For a pair twelve points apart you go from 58% with one arrow to 62% with two and 65% with three, and then the return falls away: precision grows as the square root of the number of arrows, so going from sixteen arrows to sixty-four increases the advantage by only 45%. No reasonable number of arrows makes a shoot-off reliable.