Theories of scoring: the energetic cost approach

In my modelling courses, I always ask the question: "what is the physical equivalent of money?" Eschewing all Marxist overtones, one can safely answer that the physical quantity which is the best candidate is free energy, i.e. the amount of energy that can be converted into work. Extrapolating the situation in a sport setting, we can decide that the reward of a performance, in terms of points attributed, must be closely related to the work necessary in order to produce the performance. It is thus of capital importance to know the energetic cost of the various disciplines.

In athletics the energetic cost of running is essentially proportional to the velocity, except for the sprints where a small contribution, proportional to the square of the velocity, does exist. For jumps and throws the energetic cost is proportional to the length of the jump (horizontal or vertical) or of the throw. The current WA scoring tables are of the form

for track events, where p are the points and v the velocity. The meaning of v₀ is clear: it is the velocity corresponding to precisely 0 points. The interesting quantity is the exponent c. For the decathlon track events the value of c is 1.81-1.92. When one plots the points as a function of the velocity one obtains a quasi-linear dependence (the dashed line materialising the best linear fit). 

For field events the corresponding expression is

where L₀ is the performance corresponding to 0 points. The exponent c for jumps is 1.35-1.42 and for throws 1.05-1.10. Plotting the points versus the length for long jump one obtains the figure below. 

While there is a definite positive curvature (due to the requirement of the tables to be "progressive") the departure from a straight line fit is not enormous. The linear dependence is much more pronounced for throws as can be assessed  from the figure corresponding to the discus throw.

Summarising the situation we can conclude that the number of points for scoring in athletics are linearly related to the energetic cost of the performance. However there is always a cut-off, i.e. zero points do not correspond to zero performance. In my previous post I showed that it is possible to introduce a scoring system which remedies this while staying very close to the official scoring for all but the smallest performances. It can be done in this case as well but then the proportionality between the performance and the points is somewhat skewed: low quality work gets just some pittance as reward.  

Enforcing a strict proportionality in the case of athletics is pointless, given the linear relation of the energetic cost with velocity and length. Thus I opted for deviating a little from athletics. In what follows I will apply the energetic cost approach to two aquatic disciplines, namely swimming and finswimming. 

As I explained in my post on the Rise and Fall of athletic performances, the energetic cost of swimming (and surface finswimming as well) grows like the velocity to a power 2.6. (This is, of course, the dominant term, a more accurate representation necessitating a more complicated relation. However in order to keep the arguments as simple as possible in what follows, only this dominant term will be considered).  The figure below shows the energetic cost of swimming, represented by v^2.6 for the various age groups from junior to master. 

As already pointed out in my previous post the dependence follows the universal curve. This strengthens the argument in favour of using the energetic cost in order to build a scoring table for aquatic disciplines. This is precisely what I did for finswimming. Introducing the  dependence 

between points and velocity I constructed a scoring table for finswimming. By clicking here you can access the interactive website of the performance scoring. Given that the table points for finswimming are proportional to the energetic cost one expects that a plot of points as a function of age will follow the universal curve. This turns out to be the case as one can assess from the dashed lines below obtained from the french finswimming records.

However upon closer inspection one remarks something weird. In the case of the US swimming records the extrapolation of the masters' records to velocity zero leads to an age of circa 115. This is in perfect agreement with the findings in a previous post of mine on Age Factors and their little secret. In the case of the french finswimming records zero velocity is reached at age 85. One may wonder at this point if this is due to the level of french finswimming. Thus I decided to plot the energetic cost for the masters' world records.

As expected the records of the masters age groups do lie on a straight line. However again the extrapolation of that line shows a zero velocity reached at around 85. So, the results for France were not exceptional. 

How can this be explained, in particular given that the rise and fall curve does follow the universal one with a slope ratio very close to 5. The explanation is to be sought in the discipline maturity. Classical swimming is an old, well-established discipline, like athletics. There exist large populations of practitioners in all age groups. Finswimming is a more recent sport and the number of finswimmers, while constantly growing, is still rather small. When the discipline reaches maturity, one would expect a behaviour similar to the one of classical swimming. This is what is represented by the light dotted lines in the figure with the french records. Way to go, finswimming.