Introduction:

Within the context of athletic performance, every athlete wants to equal the achievements of the supremum whether this may be a Federer or Zidane. In principle, if we assume that all athletes are equal in ability it is sufficient to systematically follow the training program of these elite performers. However, as this assumption is questionable and the game is constantly evolving any good athlete must embrace the freedom for creative experimentation. Yet such freedom necessarily leads to a diversity of outcomes.

In this hyper-competitive setting, we may analyse the natural tension between freedom and equality.

The challenge of designing a good training program from scratch:

Let’s start by assuming that all athletes have identical physical and cognitive abilities so the main challenge for a coach involves the design of a suitable training program. Without loss of generality, this program will involve a finite number of binary training variables. For concreteness, this may be: \(x_1\)(sleep \(\geq\) 8 hrs), \(x_2\)(daily water consumption \(\leq\) 5 litres) ,…,\(x_N\)(weekly time in sauna \(\leq\) 1 hr).

Now, if we make the reasonable assumption that subsets of variables may interact with each other our coach may need to consider \(3^N\) different training programs since there are distinct subsets of variables and for each binary variable there are two possible settings:

\begin{equation} \sum_{k=0}^N 2^k {N \choose k } = 3^N \end{equation}

Even for small \(N\), the reader may check that this quickly becomes an impractical number from the perspective of data collection. It follows that the coach will need to choose a program based on their incomplete understanding of the mapping:

\begin{equation} f: X \rightarrow Y \end{equation}

\begin{equation} \forall x \in X \exists y \in Y, y = f(x) \end{equation}

where \(\lvert X \rvert = 3^N\), \(y=1\) if the team performs better than they did last year, and \(y=0\) otherwise.

The pursuit of equality or the case for mimetic behaviour:

Given the astronomically large number of options for training programs, most coaches may decide to start with a well-known training program. Or put another way, since \(f\) may be a very complex nonlinear function the coach may adhere to the principle of ‘if it ain’t broke, don’t change it’. This is a reasonable risk-minimizing strategy.

From an evolutionary perspective, we may consider the reproduction number \(R_0\) to be the ratio of the current cohort of coaches adhering to a particular program \(\chi\) relative to the previous cohort of coaches adhering to \(\chi\). Therefore, a risk-averse coach may decide to choose the training program with the greatest value for \(R_0\). This strategy is not equivalent to choosing the most popular training programme.

How freedom may lead to outcomes better than equality, or not

There are two arguments for constructing an original program. By following a well-known program you are unlikely to do better than equal the opposition unless you have an exceptional athlete. But, if you adhere to the principle of ‘better dead than second’ then you may want to pursue victory at any cost. The coach in this case is not merely aiming for progress, but the championship itself.

Due to the size and complexity of the search space for training programs, there are two ways this objective may be satisfied. Either the coach is incredibly lucky or they discovered a non-trivial relationship between several training variables. That said, there’s another quite different principle that may guide a coach in going their own way. This is the belief that the elites simply have it all wrong. Without careful justification, this belief is more often an excuse for taking shortcuts.

Either way, the pursuit of freedom will generally lead to a diversity of outcomes as a freedom allows one to move into unmapped territory. From an evolutionary perspective, these may be considered potentially valuable mutations.

Discussion:

Whenever the discussion of liberty versus equality of outcomes arises there is a tendency to create a false dichotomy where we are forced to choose between extreme positions that are mutually exclusive. However, this is a mistake as the best approach is to find a compromise between different tradeoffs.

A balance may be found by having an honest discussion about the competition landscape and considering the relative merits of different training programs.