In the post where I interviewed G. Purdy I told the story on how we met and started collaborating. I also explained how, thanks to the precious help of L. Shieh I managed to obtain several papers that were included in Purdy's thesis bibliography as well as a digitised version of the thesis itself. The latter is a very interesting read, in particular for the detailed analysis of publications which either criticised Ulbrich's approach or tried to come to its defence.
While reading the corresponding chapter in Purdy's thesis I came across a reference to G. Holmér. Holmér, a decathlon olympic medalist (well, after the disqualification of Thorpe), had co-authored, together with A. Jörbeck the 1952 scoring tables. Following the introduction, in 1962, of the tables based on Ulbrich's erroneous theories, sponsored by Jörbeck, he was constrained to write a short note pointing out the 1962 tables' shortcomings. He remarks, quite correctly, that what should be used as the basis for establishing a fair scoring is not the velocity but the energy necessary in order to realise a performance. But what is even more interesting is the fact that Holmér suggests that combined events should have their own scoring tables, constructed from the statistics of combined events performances. Holmér was a real decathlete and it shows.
Purdy gives a set of scoring principles. Some of them state something obvious (but when one sets down principles, one must be exhaustive). I am going to present them below together with a short discussion of some of them.
Purdy principles
STP 1. Definition Principle: A point score represents a standard unit of performance.
STP 2. Universal Principle: More points are awarded to better performances than to poorer performances.
STP 3. Equivalence Principle: Every point score should represent an equivalent mark for each event.
STP 4. Uniqueness Principle: The total change (absolute and percent) between the lowest (typical) and highest (world’s best) performance marks should be different for each event in accordance with the range of achievement.
STP 5. Null Score Principle: Any non-zero performance mark should receive some non-zero point score.
STP 6. Diminishing Returns Principle: An increasing (and always positive) percentage change in the point score should result as the performance level improves.
STP 7. Progressive Change Principle: A unit change in performance must represent a larger change in points as the achievement and point level increase; conversely, a unit change in points must represent a smaller, but always non-zero, change in performance as the point level increases.
STP 8. Mathematical Consistency Principle: For each successively higher point level in an event, the first difference in the points between two adjacent performance marks must be either increasing or unchanging.
STP 9. Infinite Precision Principle: For every performance mark, the corresponding point score should be calculated to as many places as necessary to avoid ties in a multi-event competition.
STP 10. Constant Rule Principle: The rules defining an event and the manner in which it is conducted are assumed to be unchanging; and when the rules change, the scoring table may need revising.
The first four principles, as well as STP 10, are easy to understand. STP 5 on the other hand needs some discussion. Purdy stresses the fact that points should attributed to some even very low performances. Up to that point there was practically no discussion on how the performance corresponding to zero points in the IAAF tables was fixed. Purdy introduces a perfectly cogent prescription for what he calls the null-score: walking is not running, stepping is not jumping (to which we added that stretching the arm is not throwing). Thus instead of a null-score velocity of more than 6 m/s for 100 m in the 1962 IAAF tables, Purdy proposes just 2 m/s.
I do not understand what STP 6 has to do with scoring: it refers to the fact that as the athlete's performance improves he has to train more and more in order to achieve the same increment in performance. One could justify it as the rationale behind the STP 7 which, to put it in a nutshell, states that scoring tables must be progressive. STP 8 could it be understood in this spirit were it not for the fact that Purdy tackles the question of rounding, an assumption in all tables since 1934, which he opposes. Unfortunately I disagree with this lust for precision of Gerry. M. Woolf in an article in a NUTS Notes of 1978 criticised this with the obvious remark: when some performances are measured with three significant figures how can one assign to it a score with a six figure precision (Purdy proposes tables up to 1/1000 of a point). A jumper jumps 7.27 or 7.28 m, everything in between is inconsequential. So, to Gerry's dismay, I will side with Woolf and the IAAF/WA practice concerning rounding the points to the lowest integer. Simplicity trumps (illusory) precision.
In order to have progressive tables Purdy decided to introduce an exponential relation between the performance and the number of points. This was already tried in the 1934 IAAF tables by J. Ohls but, curiously, without much success. Given that Purdy's tables were setting the null-score at a very low performance it was imperative that they provide a way to attribute points to a large spectrum of performances. The solution was to add to the exponential part a linear one. This resulted to the expression
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