Archery qualification and winning: Park 2016 study

The study asks how much the qualification position weighs on the final result at major international events (Olympic Games, World Championships, World Cups), where the 72-arrow qualification round is followed by head-to-head elimination matches. For each starting position the model calculates the probability of winning every match needed: six at the Games, and at World Championships and World Cups five for the top eight seeds and seven for everyone else. The probability of winning the event falls exponentially with qualification position, with a constant between 4.56 and 5.96 places depending on the division: roughly every five or six places, the probability drops to a little over a third. Averaged across the four divisions (recurve and compound, men and women), the model gives a probability of 0.76 that the winner is one of the archers seeded first to eighth, and 0.18 that it is one of those seeded ninth to sixteenth.

The comparison with real results uses 182 events from 2000 to 2014, counting bow type and sex separately. The gold went to an archer seeded in the top eight in 76% of events (in 54% to one in the top four) and to one of the top sixteen in 90%, slightly fewer than the model predicts: when someone seeded below sixteenth wins, it is mostly the archers seeded ninth to sixteenth who pay for it. The finer counts are also close to expectation: at the London 2012 Games, in the men's recurve, nine archers seeded between 33rd and 64th won their first match, against 8.1 predicted; across the 182 events an average of 1.6 of the eight archers seeded ninth to sixteenth finished higher than ninth (2.1 expected), and the top eight seeds filled on average 4.2 of the eight best final places (4.3 expected).

Finally, the paper tracks the qualification score of the eighth-placed archer over time, which rises by 0.7 points a year in the men's recurve, 1.2 in the women's recurve, 1.8 in the men's compound and 2.3 in the women's compound. For the 2016 Games the projection was 670 points, give or take 7, for the men's recurve and 656, give or take 8, for the women's.

The model builds on an earlier paper by the same author, which links the spread of an archer's group to skill level and distance. For each event Park fits the line along which the qualification score falls with position (about half a point per place, 0.54, in the men's recurve at London 2012), leaving out the top two, who usually shoot above their own average, and the tail of archers who had problems or lack the level. The score read off that line is treated as the archer's average; from it comes the spread of the archer's group, assumed normal and centred, and the probability of hitting each scoring ring.

These probabilities give the score of every three-arrow end (1,331 combinations) and then the outcome of a match. For recurve the set system is used at 70 metres on the 122 cm face: five sets of three arrows, two set points for a won set and one for a tie, first to six wins, and at 5-5 a single-arrow shoot-off decides it, won by the arrow closer to the centre. For compound at 50 metres on the 80 cm face the 15-arrow total counts. For the whole event the author assumes that the higher seed always wins the other matches and then rescales the probabilities so that the right number of archers advance at each stage; he also assumes that everyone shoots at the same level in the matches as in qualification.

The data are World Archery's official results, which do not record the weather. Qualification rounds of 144 arrows were converted to their 72-arrow equivalent.

The weak point is the central assumption: a single day's qualification score is treated as the archer's average, because major events are few each year and there are not enough data on individual athletes to estimate it. The model therefore cannot tell an archer having a bad day from one who always shoots at that level, and it assumes that the pressure of head-to-head matches changes nobody's level. Wind and the day's conditions are also missing, since the data do not record them.

The agreement with the 182 events shows that the average picture holds, but it says nothing about an individual archer: the author himself describes his subject as archers "on average". The 182 cases are competition divisions, not people, and the same athlete can appear many times. The match format changed several times over the period, while the model only calculates matches under the rules in force when the paper was written. The 2016 projection is a straight line extended forwards, and the author notes that as scores approach the maximum of 720 a curve that flattens out will be needed.

For anyone shooting international events the study puts a size on the weight of qualification: it does not just set the bracket, it settles much of the probability of winning. In the model three wins in four go to the top eight seeds; archers seeded lower can win a few matches, and sometimes knock out a favourite, but rarely win enough in a row. The byes given to the top eight at World Championships and World Cups matter less than one might think: for the eighth seed they are worth 12% more probability of reaching the last 32, because the early opponents would have been much weaker anyway.

On matches there is one finding worth knowing. With low averages, as at many club events, a 10-point gap between two archers counts for little; at the level of major events the same gap leaves the weaker archer very little chance. A single-arrow shoot-off, by contrast, stays very open even between archers of quite different levels, and the author draws a warning from this for those who write the rules rather than for those who shoot. The eighth-place score, finally, gives an idea of the threshold to expect at a major event, bearing in mind that it is a trend over many years and that individual events scatter around it with a standard deviation between 4.8 and 7.9 points.