On a naïve proposal for decathlon scoring

John Barrow was a mathematics professor at Cambridge (he passed away in 2020). Among his many interests was that for mathematics of sports. He authored the book Mathletics 

where he is presenting a collection of most interesting back-of-the-enveloppe calculations that would appeal to every physicist, even those who are not particularly interested in sports. He was the director of the Millenium Mathematics Project. On the project's site one can find a slew of interesting articles. Among the latter one encounters an article entitled "Decathlon: the Art of Scoring Points". It's the one I would like to comment upon in this post.

Barrow had an "on the surface" great idea: "let's do away with the scoring tables". So he proposed to score the combined events by multiplying the lengths of the jumps and throws, dividing them by the product of the times of the races.

Let's see how this (what Barrow calls the "special total") would work for performances that correspond to 1000 and 0 points in the current men's decathlon tables. For 1000 we have the succession of performances

10.39, 7.78, 18.40, 2.21, 46.17, 13.80, 56.17, 5.29, 77.19, 3:53.79 (233.79 in seconds).

The special total in this case is 4.688.

In the case of 0 points (the tables do not list the zero point but it is straightforward to obtain it from the parameters of the scoring formulae) we have

18.00, 2.20, 1.50, 0.75, 82.00, 28.50, 4.00, 1.00, 7.00, 8 min (480 s).

The special score in this case is 0.000003.

Already there is a factor of one million between the highest and lowest score something that would appear exaggerated even to scoring-untrained eyes. But wait, there is more. Had we decided to use empire measuring units (God forbid) for the jumps and throws, we would have obtained a totally different score. This comes from the fact that multiplying arbitrarily numbers that have a dimension does not make much sense, unless one specifies the measuring unit. The special score we obtained above is a number with dimensions m^6/s^4 and its value has a meaning only in these units.

The least one can do when one tries to set up a system for scoring is to give a dimensionless score. The easiest solution to this is to divide each performance by a reference one, for instance the performance corresponding to 1000 points in the scoring table. This would amount to dividing the special score by 4.688. The advantage of this is that the result is independent of the units in which one measures the performances. (It is impossible that Barrow ignored such basic a fact. My guess is that, given that the conversion to a dimensionless unit is an overall factor, he decided not to enter into those details so as to keep his presentation as simple as possible).

Let us now do a real life comparison, using the special score recipe for the 6 best decathletes of all times (the data are available on the World Athletics site). Since the overall factor is the same for all athletes we can simply neglect it. 

We find

Athlete	decathlon	special score
K. Mayer	9126	2.61
A. Eaton	9045	2.03
R. Sebrle	9026	2.29
D. Warner	9018	2.13
T. Dvorak	8994	2.40
D. O'Brien	8891	2.13

While Mayer emerges as the best decathlete ever, A. Eaton is, surprisingly, beaten by all the other decathletes in the top-6 list. This reveals the real inadequacy of the special score recipe: it is the events where large variations are possible that have a bigger effect in the final score. Let us parse this in the case of Mayer versus Eaton. Their overall ratio is 1.28 in favour of Mayer. (In fact, another way to obtain a dimensionless number using the special score is to consider the ratio of the scores of two athletes). We compute now the ratios of throws, jumps and races. We find (Mayer/Eaton for the two first and the inverse for the third) 1.45, 1.05 and  0.83. So the throws are the ones that decide the outcome. In fact this is practically true for all the athletes of the list.

Here is the list of the 6 first where the overall special score and the one obtained from the throws alone are divided by that of Mayer 

Athlete	overall ratio	throws ratio
K. Mayer	1.00	1.00
T. Dvorak	0.92	1.01
R. Sebrle	0.88	0.89
D. O'Brien	0.82	0.87
D. Warner	0.82	0.79
A. Eaton	0.78	0.69

Unsurprisingly, Eaton has the lowest relative score, while Dvorak, who has the best ratio to Mayer's score is even slightly better in the throws. 

To be fair, Barrow was aware that his system was biased and thus, in his article, he did not pursue the issue and went on to discuss the World Athletics tables. I have two versions of his article and curiously the one in the Millenium Math project site does not contain the last part of the first where the author argued that "in each event (with the possible exception of the 1500 m) whether sprinting, throwing or jumping, it is the kinetic energy generated by the athlete that counts". I beg to differ. As I have repeatedly explained, what should count, when it comes to scoring, is the energetic cost of the effort. And while for jumps the energetic cost is closely related to the kinetic energy (things are somewhat more complicated for the throws) when it comes to races, the energetic cost is roughly proportional to the first power of the velocity and not the second one as in the kinetic energy. (To be fair, there is a small velocity-squared contribution in the case of sprints but the dominant term is the one linear in velocity).

Anyhow, scoring is a delicate business, and Barrow's idea, although exceedingly naïve and not really applicable, had the advantage of being a zero-parameter one, a kind of a black swan in the complicated world of sports modelling.