The Law of Diminishing Returns and Archery

I have been “remote coaching” a student for a bit and he has been impressed by his rapid progress. I commented that this is normal but will taper off. The conversation got me thinking about “why?” There were two things that came to mind: what economists call the Law of Diminishing Returns and all target rounds having an upper bound (a perfect score).

The Law of Diminishing Returns is a complicated way of describing the general nature of things. If you are cooking a dish and you add the tiniest amount of salt to it, the taste of that dish doesn’t change much. If you add a bit more, it gets tastier, and a bit more it is just right (which is why chefs are trained to season their dishes lightly and then to “adjust” the seasoning just before finishing a dish). But if you add a great deal of salt, the dish is ruined as all one can taste is the salt. Similarly, there is the idea of best dose in medicine: too little of a medicine and you get no effect, too much and you can kill the patient (many medicines are quite poisonous), the trick is to get the dosage “just right.”

In any complex process, there is a spectrum of causes and effects from “too little” to “too much.” Because of this when you add a little of something (salt, drug, effort, extra production capacity, etc.) you will get a return. The closer you get to the amount that is ideal, the greater the return, and after you have reached that point, adding more of that thing gives you less and less return, possibly even a “negative” return, aka harm.

This is the basic nature of things.

Also, progress in an archer’s skill is typically gauged by scores in competitive or noncompetitive “rounds.” I am sure that this practice started out in the distant past, probably five minutes after the second bow was built, as a “hit your mark” challenge first and as the competitors learned to regularly hit their mark, the competition advanced to the best of five, the best of ten, etc. resulting in the archery “rounds” we have in use today (and a great many more which have fallen into disuse). Allow me to use the indoor 300 round as an example. But before I do that I do want to mention that perfection is not acceptable in these competitions. As an example of that, consider the change in the NFAA targets in the 1970’s. The old targets were different from the new in that the central aiming dot did not receive a special score. Any arrow hitting the dot got a “five” and any arrow hitting the rather large circle around that aiming dot also got a five. The outermost ring got a score of three. This target was fine for the first 30 years or so of the NFAA’s existence, but then equipment and more practice started resulting in perfect scores … many of them. Consequently the targets were changed so that the central dot got five, the ring around it got a four, and the outer ring a three. (This was somewhat oversimplified but correct in essence.) Any time perfect scores abound, rounds are changed. This is because the purpose of the round is to determine winners and many place ties at the end of a competition are less than satisfying.

So, having an upper bound, a perfect score, in any round, creates another impediment to progress. Back to the 300 round (either the NFAA version or USAA version, doesn’t matter). While still a beginner one is glad to break the 100 point barrier. Later, a serious archer will quickly pass a score of 200 points, a 100% improvement! But to get from 200 points to 250 points is only a 25% improvement, from 250 to 275 is a 10% improvement and to get from 295 to 300 is only about a 2% improvement. Yet, it seems like that last five points takes considerable more effort than the effort to get from 100 to 200 points. It seems that way because it is.

When one’s scores are in the 100’s, an arrow missing the target completely doesn’t affect the score very much. When trying to get from 295 to 300 (been there, done that), missing the spot severely affects your score.

If you look at this with regard to area on the target face it makes more sense. My first archery outcome goals (at least, so I remember) were to shoot rounds that were “clean,” meaning all arrows scored. Think about what that means. If I may use the NFAA, 5-4-3 targets, the rings are in the following proportions: the diameters of the three scoring rings (ignoring the X-ring) are in the following proportions: 5:3:1, so the areas of the three rings are in the proportions of 25:9:1. The first task is to get all of the arrows to score, because the penalty for nonscoring arrows is huge. If one gets all of the arrows to score, the minimum score outcome is 180 points (60 arrows x 3 points each). If one gets all of the arrows in the inner rings (no 3’s) the minimum score is 240 points (60 arrows x 4 points each). To get a perfect score, all of the arrows must be in the inner ring (60 arrows x 5 points each). But look at the challenges. To get all of the arrows to land in the four ring or higher, the area those arrows must land in is only 36% of the entire scoring area (9/25ths). You have to reduce your group size by a whopping 74% to pull that off. To get all of the arrows into the 5-ring, the area involved is only 4% of the total scoring area (1/25th). This involves reducing your group size another 89%!

And then there are the moving goalposts. Once you get the perfect round (300/300) the next thing to do is to get all X’s (never made that).

So, coaches, beginners make all kinds of progress when they are starting out because they are increasing their consistency in leaps and bounds. But things done to get one’s arrows to score at all may have to be corrected to get the arrows to score at the highest. Everything has to come together to get those near perfect scores: the archer’s physical technique, their mental game, their equipment, everything. The lower scores can be had with less than perfect equipment, quite less than perfect technique, and even no mental game at all.

The challenge is to keep your archers “on the path” and to not let them get discouraged when their rate of progress slows almost to a stop as it is almost guaranteed to do.

It is all natural.

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