Let us start with a simple question. In fact this is a question I asked myself when teaching modelling classes. What is the physical equivalent of money? The answer is simple once you start thinking along these lines: energy. Despite the marxist overtones, we must admit that this is how, in principle, one makes money. Through work. (And, yes, I know that I am cheating using in the same time the term work in its everyday meaning and its physical one. Moreover by "energy" I mean the "free energy" of thermodynamics which designates the energy that can be transformed to work but I should stop here lest I become too technical).
Still, a parenthesis is unavoidable (for the physicist, not for the scoring specialist). I believe that in modern societies there is another quantity that could play the role of money: information. If one pursues this reasoning one reaches the conclusion that whatever allows one to locally diminish entropy is a good physical equivalent of money. Still, since we are talking here about athletic performances we will not delve further into this matter.
With these premises we can move now to the first statement concerning the scoring of athletic performance. If we analyse an athlete's effort we can easily convince ourselves that, from a physical standpoint, energy is what conditions the result. The performance is directly related to the work done. (We are talking here about individual, quantitative sports. For sports which do not fall within this class something can still be done, thanks to Harder's approach). So our starting point for scoring will be the requirement that the score be related to the energetic cost of the performance.
Once he principle energy=>scoring is accepted the next difficulty is how do we relate the two. That's where the ideas of Dale Harder come into play. (Dale Harder is the author of an excellent treatise "Sports Comparisons". I urge you to go and buy his book. You can contact him at daleharder at comcast.net). His principle can be summarised in one sentence: different athletic achievements in different events can be compared, by comparing the number of athletes who reach any given level proportionate to the total number of athletes competing in that event. The advantage of the Harder approach is that one can compare (i.e. score) events which are not quantitative to begin with. He attributes marks for mountain climbing, which for me was the typical example of sport you could not score. And he is right too! (Climbing the Everest gives you 725 points is his system, a rather low mark but since more than 1000 climbers have reached the summit, this makes sense. For a higher mark, you have to do it solo and without oxygen, being preferably a woman at an advanced age).
Harder's scale is a 1000 point one where 100 points correspond to what an average male, in his prime, should achieve. Zero points correspond to a performance that roughly 95% of the population would realise, while 1000 correspond to the perfect score under current rules and conditions (something that 1 individual out of the total male population could ideally realise). An interpolation attributes points to performances realised by larger percentages of athletes.
At this point one can ask how can these considerations be made more quantitative. The answer is sigmoids (or S-shaped curves). From a study of several human characteristics from height to intelligence one knows that the percentage of individuals the performance of which does not exceed some level follows a curve which is vaguely S-shaped: it starts at 0, grows rapidly and then saturates towards 1. Various functional representations do exist but one who is quite convenient and allows mostly analytical treatments is one involving the hyperbolic tangent. If x is a variable (related somehow to performance) taking the values from -infinity to +infinity (an unrealistic assumption, but a more realistic domain of variation can be introduced) we can define the function
S=(1+ae-bx)-1
Clearly, S goes to 0 when x goes to -infinity, it goes to 1 when x goes to +infinity, and takes the value 1/(1+a) for x=0. The Harder scale can be perfectly simulated with this sigmoid, with the appropriate choice of a and b by assuming that x is linear in the number of points. Solving for x we find that the points are a linear function of the logarithm of S/(1-S).
From this point onwards two possibilities exist. The first corresponds to a non-quantitative or non-absolute sport: no measurable performance can be obtained or such a performance depends on the adversaries. In this case a scoring is still possible. That's where the monumental work of Harder comes into play: through a statistical analysis of results one can establish emprirically the sigmoid S and compute a point-equivalent. The second case is the one which will interest us here. It corresponds to sports where a measurable, objective performance does exist. (Classical athletics, swimming or weight-lifting are the typical representatives of this class). In this case the variable x appearing in the sigmoid is simply related to the energetic cost of the event. Even a linear relation is not unreasonable. Following the reasoning of Harder we assume that the exponent is also linear in the number of points. As a result, with these assumptions (which should be tested on a realistic data basis) we arrive at the conclusion that the points attributed can be a linear function of the energetic cost of the performance.
The expression Points=A*Energy+B can become a most convenient scoring system, provided we have a precise (empirical or theoretical) knowledge of the energetic cost of the performance.
The example I would like to present is one based on a discipline I am familiar with: (fin)swimming. The energetic cost of swimming was obtained by di Prampero as a function of velocity. He obtained an (empirical) formula E=kvc where v is the (mean) velocity and c an exponent with a value around 1.2. (This is a most interesting finding. From data obtained from the resistance to passive towing, one would have predicted a value of c close to 2. The fact that the exponent is significantly smaller means that the swimmer is adapting his style to his velocity so as to minimise resistance). Since c is very close to 1, I decided as a first approximation to work with an energetic cost which is simply E=kv. Thus the scoring tables for finswimming are based on the expression p=mv+n. (I must point out here that I am perfoming some hocus-pocus taking the results for swimming and applying them without hesitation to finswimming. The reason is that the energetic cost of finswimming is not known (and I am not familiar enough with swimming so as to be able to develop tables for its various disciplines)). The parameters m,n appearing in the formula for scoring necessitate some further assumption and some statistical analysis of performances for their values to be fixed for each event.