The board says 4-6 and the scorecard says 142-136. The archer who lost put six more points on the target than their opponent, shot 30, 27, 28, 30, and 27 across the five sets against 26, 28, 29, 25, and 28, and goes home eliminated. Nothing strange happened and nobody miscounted. The system worked exactly as it is supposed to.
Understanding why means looking at how a match is scored, and since 2010 that arithmetic has not been the one almost everyone still carries in their head. That leads to the questions that actually matter: how often something like this happens, whether it happens more to strong archers or to weak ones, and whether the format does what the federation said it wanted when it adopted it.
The figures on the format come from work I published in the Lab, where the calculation is set out in full and the code to redo it can be downloaded. At the end of the article there are also two tactical questions the calculation does not touch, whether it is better to shoot first or second in a set and whether the first set is shot lower than the others, and for those the figures come from competition data.
How a set match is counted
The rule is simple to state: the match is broken into sets of three arrows; whoever scores higher in a set takes two set points, and if the set is tied they take one each; the first archer to six set points wins the match. Since set points only ever accumulate, the match ends the moment one of the two reaches six, which can happen in three sets, four, or five, meaning after nine, twelve, or fifteen arrows.
Before 2009 matches were scored another way, the method recurve has dropped but compound still uses: add up all the arrows, twelve each in recurve back then and fifteen in compound today, and the highest total wins. They are two ways of reading exactly the same arrows, and in the very same competition they can return two different verdicts. The set system entered individual recurve in 2009 and became the format of the whole senior circuit the following year, and in that first season it was not yet uniform, because the earlier elimination rounds shot sets of six arrows and three-arrow sets applied only from the quarter-finals on.
Three reasons were given. Under the old system a contest could be essentially lost after a single bad end, because one disastrous arrow was enough to make the total unrecoverable, and the set format was adopted to stop that, as well as to reward consistency and to keep the match open to the finish. Whether it got what it wanted is a question that can be answered with a number, and I come to it shortly.
Where the leftover points go
The mechanism that produced the scorecard at the top of this article sits entirely in one property of the rule: inside a set, winning by one point is worth exactly as much as winning by ten. The margin is wiped out at the boundary of the set and does not carry over, so a set won 30-25 and a set won 28-27 are worth the same thing, namely two set points.
That margin is information that cumulative scoring keeps, set after set, and that the set system discards. How much of it is thrown away can be measured: in the band where the format matters most, meaning when an archer averaging 692 points meets one averaging 675, a set that is won is won by 1.76 points on average, and in more than half of cases by two or more; between two identical archers the mean margin falls to 1.41.
For the pair at 692 and 675, the stronger archer takes the individual set 55.5% of the time, ties it 24.7% of the time, and loses it 19.8%. The fact that one set in four ends level matters more than it looks, because a tied end gives each archer one point and therefore separates nobody: it is a stretch of the match that passes without producing any information that helps decide who is the better archer.
What counting this way costs
The essay sets aside whether the set system is fair; the question it asks is how often the format points to the more precise of the two archers, and how it behaves against the old count on the same arrows. The difference can be calculated exactly, without simulating competitions, and the result is that the set format does cost some precision, but not much.
At the point where the two formats differ most, with an archer at 692 against one at 675, cumulative scoring identifies the more precise archer with probability 0.870 and the set system with 0.847. Translated into matches, out of a hundred contests the cumulative count gets it right 87 times and the set system 85: two matches in a hundred change hands, one in forty-three.
The cost is not the same at every gap, though, and the shape of the curve says something that frees a coach from a common misconception.
| Gap between the two archers | Cost of the set system | Why |
|---|---|---|
| Archers level | zero | There is no more precise archer to recognise: no format beats a coin |
| 2 points | half a percentage point | The gap starts to be legible, and the format starts to lose part of it |
| 5 points | a little over one | This is the band where the discarded margin weighs most |
| 17 points | 2.32 percentage points (the maximum) | The gap is clear on the arrows but the set count returns only part of it |
| 35 points | a little over one | The stronger archer wins nearly every set: both formats recognise it |
| 45 points | six tenths | The gap is so wide that the way of counting stops mattering |
The disadvantage, then, is not greatest between near-level archers, as you might suppose, but in the middle. The reason, once stated, is almost obvious: when two archers are nearly identical no format in the world can do better than a coin toss, because the contest really is a toss-up and settling it at random is not a measurement error; and when one is clearly stronger, every rule recognises it without effort. Only in the middle band does the choice of format have anything to lose.
The absolute level of the two contenders matters as well, and it matters in the convenient direction. The peak cost is 1.99 percentage points between archers at 700, rises to 2.32 between archers at 692, to 2.63 between those at 680, and to 2.97 between those at 660: between a world-class match and a good national-level match the cost of the format goes from just under two points to almost three. Higher up, the model's arrows cluster around the centre, sets resemble each other and the two arithmetics converge, which means the format costs least exactly where it is used most.
Those levels are expected scores over the 72 arrows of the qualification round, and in the model an archer at 692 puts about six arrows in ten into the ten. Arrows in real matches score less: in the head-to-head matches World Archery recorded between 2013 and 2019, recurve and compound together, the average in sets is 9.15 points per arrow, in line with the nine or so points the registry gives for a top archer at 70 metres, and even at the highest level most arrows in competition are not tens. So when I apply these numbers to a real match, I also look at the lower rows of the scale, where the format costs a little more.
The match ending early is not the problem
An explanation you often hear puts the cost down to brevity: a set match can close in nine arrows instead of fifteen, and fewer arrows mean less information. In the reference pair the average duration is 4.268 sets, meaning 12.80 arrows rather than 15, so a few arrows genuinely do go unshot.
The calculation dismantles that explanation decisively. Force the two archers to play all five sets even when one has already reached six, and the probability of winning does not change by a thousandth: the two versions return the same value to the last digit the computer can display. The reason is arithmetic: five sets hand out ten set points in total, and the loser never gets beyond four set points against the six or more of the winner, so the arrows the set system leaves unshot are exactly the ones that could not have changed the verdict.
There is a second piece of evidence, and it runs against intuition. If the match is made longer rather than shorter, the gap between the two formats grows: with a single set it is exactly zero, because there is no internal boundary to cross and the two arithmetics coincide; with three sets it is 1.84 percentage points, with five it is 2.32, and it keeps rising to around six sets before coming back down, because at twenty-one sets both formats are close to infallible and there is no room left for a large difference. The cost, in other words, arises at the boundaries between one set and the next: every extra boundary throws more margin away, and only beyond about six sets do the extra arrows start to outweigh the boundaries.
How often the lower total wins
A result like the one on the scorecard at the top of this article is called a reversal, and its frequency can be calculated: between two essentially level top-flight archers it happens in 7.1% of matches counting the arrows actually shot, and in 7.6% if you imagine all fifteen being shot, about one match in fourteen.
The frequency changes with the gap, and it changes in the way you would expect.
| Situation | The lower total wins |
|---|---|
| Two level top-flight archers | one match in 14 |
| 17 points apart | one match in 21 |
| 25 points apart | one match in 33 |
| 35 points apart | one match in 69 |
| Two level archers at 670 points | 9.2% of matches |
| Two level archers at 650 points | above 10%, one match in ten |
The point to take away runs exactly counter to the indignation the scorecard produces: the reversal is most frequent precisely where the format betrays nobody, because it grows the closer the two archers are and its maximum falls near equality, where the cost in precision is zero. Between two identical archers reversals are very common indeed, one in fourteen, and not one of them is an error, because there is no more precise archer to betray and whoever wins, the verdict is legitimate. As the gap widens the reversals thin out, down to one in sixty-nine.
The reason is geometric. For the winner to have posted the lower total, the two totals have to be close enough to swap places, and that happens when the two archers resemble each other. When one is much stronger their total sits a long way from the other's, and chance is no longer enough to make them cross.
A reversal should not be confused with an upset, meaning a win for the weaker archer, which is a different thing and is looked at on competition data. That in head-to-head matches upsets happen when the two archers are of similar level is more an expected consequence of the format than a measured result, and in the registry the claim is contested: at the World Cup stage studied the archer ranked higher in qualification usually wins, and upsets cluster where the bracket pairs archers close together in the ranking, but at the World Championships the same year, shot in very strong gusts, many low-ranked compound archers finished near the top, so the upsets came from archers far apart in the ranking.
Six matches in a row: what happens to a whole tournament
A single match is one thing, but a 64-archer bracket is made of six rounds of single elimination, and winning takes surviving all of them. So the question can be asked again at tournament scale: how often does the favourite reach gold under one format and how often under the other, and where does the probability lost along the way end up?
Across a six-round bracket the direction never changes, in any of the fields of competitors built for the test: cumulative scoring takes the favourite to gold more often than the set system does, by a difference running from 3.3 to 4.3 percentage points. In a field whose level is spread out regularly the favourite wins gold 46.7% of the time counting by points and 42.5% counting by sets.
The most useful thing, though, is to compare that difference with another factor: if you leave the format alone and change the composition of the field, keeping the number of competitors the same but varying how closely matched they are, the favourite's probability of gold moves from 31% to 58%, twenty-seven percentage points against the four that come from the arithmetic. Who else is in the competition therefore weighs more than six times as much as the way the points are counted, a ratio that cuts down to size an argument that has been running since 2010.
The last question, the one nobody usually asks, is where the four points the favourite loses by moving to the set system end up. A tournament awards a gold medal either way, so that probability has to go to someone else: in the regularly spread field the top seed loses 4.2 of it, the group from the second to the eighth seed gains 1.9, and the group from the ninth to the thirty-second gains 2.3. The set format transfers probability from the favourite to the outsiders, and transfers more of it to the distant group than to the one immediately below.
Consistency, which the format was meant to reward
That leaves the original promise to deal with, starting with the first two things the federation wanted: to stop one unlucky arrow deciding a match, and to reward consistency. It got the first, and the calculations confirm it, because the format confines the damage of an arrow inside the set where it lands. On the second the calculations say the opposite.
The measurement is made by setting two archers with the same expected score against each other, one of whom shoots steadily while the other occasionally places a ruinous arrow. Under cumulative scoring the steady archer wins 48.5% of matches; under the set system that falls to 45.4%, and the direction does not change at the top or further down. About three percentage points of advantage therefore pass from the archer who shoots consistently to the archer who occasionally throws one away, and it is the direct consequence of the protection the format offers: isolating the catastrophic arrow inside the set where it lands means protecting whoever shoots it.
These are calculations on modelled archers, not on real competitions. The comparison between the set verdict and the cumulative verdict on the same real matches has not been put to the test, which is why in the registry the claim that the set system rewards the more consistent archer remains not studied: the calculation says that in the model the format rewards consistency less than cumulative scoring does, but it is not a measurement on athletes.
That same property is the one spectators like, because a format that protects the archer who errs produces more open matches, less predictable outcomes, and more frequent upsets, which is the third thing the federation said it wanted. On that front the set system works, and works for exactly the reason it measures worse: they are the same property under two different names, and which of the two to use is not something a calculation can decide.
What the calculation does not look at
All these numbers come out of a model in which an archer is the width of their group and nothing else, arrows are independent of one another, and form on the day does not exist. No real archer's score enters the calculation, and the archers are synthetic profiles defined by their expected score, so the conclusion is about the competition format rather than about a population of athletes.
On one point the model turned out to be fragile in an interesting way, and the essay says so. Allow that 5% of arrows are anomalous, meaning drawn from a dispersion three times wider than normal, and the cost of the set system falls from 2.32 to 0.68 percentage points, less than a third. The reason is that a thrown-away arrow does more damage to a cumulative total than to a set format, so the advantage of the cumulative count shrinks precisely when anomalous arrows really exist. That even top archers put the occasional arrow away from the centre is found by nearly every study that reports individual arrows, but in one study of seven elite Indian archers, indoors at 18 metres, no arrow scored below eight, and in the registry the claim is contested; how often it happens over a whole round nobody has counted, so which of the two numbers, 2.32 or 0.68, is closer to real competition remains open.
On other fronts the result holds up well. Changing the shape of the group, moving from a circular dispersion to a vertical ellipse or displacing the centre by three centimetres, moves the cost by a few hundredths of a percentage point, from 2.32 to 2.34 and 2.36: the verdict does not depend on the geometry of the group. Making the centre denser does matter, because taking the share of Xs from 22% to 30% with the score unchanged raises the cost from 2.32 to 3.13.
What to do with it, on the shooting line
The most immediate practical consequence concerns the Monday after a defeat. If you lost a set match having outscored your opponent, and your opponent was at a level similar to yours, that scorecard is worth little as a technical diagnosis, because it describes the situation in which the format reverses most often and in which no format could do better. Rebuilding your technique on the strength of that result means correcting something that may have been working fine.
The scorecard becomes informative when the gap is wide, because reversals are rare there: at a gap of thirty-five points they happen in one match in sixty-nine, so if you lost on sets to a much weaker archer after outscoring them, it makes sense to work out what happened in the individual sets. The same logic applies in reverse to the winner: a win on sets against an equal says far less than it seems about which of the two is the better archer.
Finally, there is a consequence for preparation, which follows from the set probabilities above. If a quarter of ends finish level and a set is won by under two points on average, many ends are decided by a single point: one point saved in a close end turns a loss into a tie, or a tie into a win, and moves a set point, while three or four points recovered in a set lost by five change nothing. That is why consistency end by end counts in a set match, even though the format, as we have seen, rewards the consistent archer less over the match as a whole.
Shooting first or second, and the first set
Two tactical questions come up often when people talk about set matches, and the essay's calculation does not touch them, because its archers feel no pressure and do not know what their opponent has just shot. For both there is a single study, but one made on real competition data.
The first is the shooting order within a set. On the shooting line you hear two opposite views: the archer shooting second has the advantage because they already know the opponent's score, or the archer shooting first puts pressure on the one who follows. A study of World Archery's arrow-by-arrow data took 6,374 three-arrow sets from elite recurve individual matches, at three Olympic Games, the 2021 World Championships and a European Grand Prix, with the order perfectly balanced, 3,187 sets shot first and 3,187 shot second. On the score of the last arrow of the set the order has no measurable effect (p = 0.986), even after accounting for the two previous arrows and the state of the match. In the registry the claim that shooting first or second in a set does not change the score is reported, with two limits: the study looks only at the third arrow, and it does not separate the order from the information that comes with it, namely the opponent's score already on the target. In the same sample nearly two sets out of three, 64.5%, do not decide the match, so the sets in which order might matter most are a minority. On these data, there is no support for preparing a match differently depending on who shoots first.
The second is the first set, which a piece of shooting-line wisdom says is where you settle into the match, with the body still cold and the tension of the start. A study of World Archery matches from 2013 to 2019 that went to a shoot-off compared each archer with themselves, across 3,645 performances by adult archers, three quarters of them recurve and the rest compound, where the match is not shot in sets and the five three-arrow ends are added up. Compared with the first set, scores rise by 0.3% in the second and by the same amount in the third, where the difference is only borderline, while in the fourth and fifth they return to where they were. Widening the sample to all matches that ended in five sets, 19,838 performances, the shape stays the same: 9.138 points per arrow in the first set, 9.151 and 9.155 in the second and third, 9.145 and 9.122 in the last two.
In the registry the claim that in a match the first set is shot slightly lower than the second is reported, and the size is worth keeping in mind: the gap is measured in hundredths of a point per arrow, less than a tenth of a point over the whole set, a small gap next to the margins by which sets are won and lost. Why it happens the study does not say, and warm-up, early tension or simply settling in are possible readings that nobody has separated. For preparation this tells you little: it does not justify writing off the first set, and it does not say whether a different warm-up would change anything.
Where this goes next
The calculation about the format is only half the story, and not the bigger half. The instrument that decides who meets whom, namely the qualification round, carries a far larger uncertainty than the effect discussed here. In international qualification on the 72-arrow round, in the middle stretch of the ranking, one place is worth less than a point, from 0.39 to 0.84 points depending on the division on the trend lines projected to the 2019 World Championships, figures that all come from the same author; in the model of a compact top-level field two neighbours on the ranking are separated by four tenths of a point on average, and over a gap like that 72 arrows point to the better archer 52.9 times in a hundred, little better than a coin. I have written about it at length in the articles on how often the top qualifier really is the strongest and on how many points it takes to say one archer is better than another.
The essay carries the full calculation, the cost curve gap by gap and level by level, the robustness tests on the shape of the group, the chapter on 64-archer tournaments, and the code package with which anyone can redo the sums and check whether the published numbers are the ones the calculation really produces.