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  • Gabriel Ng

In pursuit of ultimate performance in sport and other fields

Updated: May 11, 2021

How the ‘interconnectedness’ of factors can lead to different ways of maximising performance

Gabriel Ng, Department of Economics, University of Oxford

Ox Pers Med J 2021; 1(1): 24-28

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This paper is an exploration into how maximum performance in sport is achieved. A simple mathematical model is presented, which illustrates how there are different potential routes to reach the top. The idea behind the model is that there is a subordinate factor that feeds into the performance of the main factor. The interaction between the factors (modelled by parameter α which I call the interconnectedness parameter) determines the optimal training schedule. The interconnectedness parameter α can also be a proxy for talent, as athletes with different α are able to achieve different levels of maximum performance. Wider potential applications of the model include being applied in medicine or in modelling social groups. The model has great affinity with personalised medicine, as it affirms the heterogeneity of human bodies, and shows how different people require different treatment (training) to achieve the very best performance.


This paper is an exploration into how maximum performance in sport is achieved. We often hear that to become the very best at something, one must focus on that particular task or activity over long periods of practice or training. No doubt there is truth to this, but that does not explain why at the very top, there are different training regimes and routes to get there. In this model, the validity of different training regimes to achieve maximum performance is confirmed.

A simple mathematical model will be presented. The idea behind the model is that there are subordinate factors that feed into the level of the performance of the target endeavour. How the factors affect each other determines the optimal training schedule for performance. The model postulates that the heterogeneity of different training schemes and talent is based on how “interconnected” the factors of the system are. This is encapsulated by the parameter α, which I call the interconnectedness parameter.

In this model, intrinsic talent plays a big part in determining the level of maximum performance that can be achieved. The interconnectedness parameter α can be viewed as a measure of how talented an athlete is, as certain values of α enable athletes to achieve better performance than others.

The model shows that different athletes require different training regimes to achieve their maximum performance. This has similarities with the tenets of personalised medicine, namely that people have heterogenous bodies, and thus treatment should be different to achieve the best results. Furthermore, the model provides a possible framework for modelling complex diseases for “21st century medicine” as described by Whitcomb (1). This will be explored in the “Other Applications” section, along with how the model could be used to model social interaction.


I will construct the model using a simple two-factor system that reduces all athletic ability into two traits: speed and strength. The choice of these factors is arbitrary and could be replaced by other factors.

Imagine that at the end of the year, there is a competition to find the fastest athlete. All participants hope to win this competition by being the one with the highest level of speed.

Speed is thus the main factor and is the only factor that is assessed. Speed is increased by running outdoors. Strength, on the other hand, is not assessed and is the subordinate factor. As a participant, what is the optimum training schedule? It all depends on the effect of strength on the level of speed. If the level of strength increases speed, it may make sense to spend some time training strength (e.g. going to gym). However, daily training time is constrained, in this case to 6 hours. Would it not be optimal to spend all the time training speed directly? In this model, the answer depends on how interconnected the two factors are.


I have modelled the two factors with a symmetrical two-equation system. This is the athlete’s maximisation problem:

where L1 = level of speed, L2 = level of strength, α is the level of interconnectedness (0 ≤ α ≤1), t1, t2 = time spent training speed and strength respectively.

The problem to be solved comes in two parts. Firstly, given a limited amount of daily training time (6 hours), what is the optimum amount of time that should be devoted to training speed (t1) and strength (t2) for a given value of α to maximise the level of speed (L1)? Secondly, what is the maximum level of speed (max(L1)) that can be achieved for each value of α, given this optimum training schedule?

The intuition behind the interconnectedness parameter α is as follows. When the level of speed increases, strength also increases by a fraction of the initial increase in speed, much like how running improves fitness and leg strength. At the same time, when your strength increases, your speed also increases, since stronger muscles can give you power and acceleration. The model is symmetrical such that the fraction of the initial increase passed on to the other trait is the same. This fraction is denoted by the interconnectedness parameter α. The interconnectedness parameter α here can be thought of as a specific athlete’s body configuration.

In economics jargon, both traits are complementary. The greater the α parameter, the greater the complementary effect, and the more interconnected the two traits. It should be noted that there is the possibility of factors being substitutes, where increasing one trait decreases the other (e.g. becoming less agile if you become too heavy), but this is not the case in the present model.

There are two important features of the model stemming from the fact that speed and strength are both logarithmic functions of training time. The first is that the logarithmic function exhibits decreasing returns to scale. This is a central assumption, and the intuition is that the marginal gains to more training decreases as more time is spent on that activity. The initial time spent training a particular trait is very productive, but the productivity goes down as time spent on it increases. It is precisely because there are decreasing returns to training speed that it makes sense to train strength at all.

The second important feature is that from values 0<t2<1, the value of ln(t2) is, in fact, negative. This is a crucial point that assumes that if you do not spend time training your strength, and 0<t2<1, your level of strength can actually be negative, and can therefore exert a negative effect on your level of speed.

Another thing to note is that the equations are symmetrical: training strength increases speed and vice versa. This results in a positive feedback loop and highlights the interconnectedness of the traits. In this model, the equations are solved simultaneously so the feedback effect is not important. A more complicated model could include a dynamic system where there are transition dynamics.


Before the findings are revealed, let us examine some of our intuitions about the results. It makes sense that the more interconnected the traits are (the higher the α), the more time should be spent training the subordinate trait, strength. To give an extreme example, suppose the level of interconnectedness α is 0. Then, strength has no impact whatsoever on speed so all training time should be solely devoted to speed.

Figure 1 graphs the optimum training schedule (t1 and t2), to achieve the maximum level of speed (max(L1)) for each level of α. As expected, when α is 0, all time should be spent training speed. As α increases, the t1 steadily decreases and t2 decreases, as more time is spent training strength. These results make sense, as the larger the impact strength has on speed (higher α), the more time should be spent training strength. In the case of α = 1, time should be split equally between the two. The full results are found in Table 1.

The next question is trickier and more interesting. Which values of α yield the highest maximum level of speed given the optimal training schedule for each α? Note that because the interconnectedness parameter α here refers to a specific athlete’s body configuration, α is exogenous.

We might expect higher levels of interconnectedness (higher α) to always yield a higher maximum performance. This is true for α ≥ 0.3 but for values of α < 0.3, the maximum performance that can be achieved in these values is lower than when α = 0.

This is shown in Figure 2, where the max(L1) (maximum level of speed that can be achieved) decreases at first as α is increased but goes up again in a U-shaped fashion. This may seem counterintuitive but can be explained by the fact that the level of strength can be negative at low levels of time spent strength training (t2). Thus, when we increase α, the negative effect can increase. If a particular individual has a higher α, neglecting strength training actually results in a higher negative effect on their speed compared to an individual with a lower value of α.

However, for athletes that have α above 0.3, the potential gains from increased strength training seem to outweigh the initial negative effect. If we continue increasing interconnectedness from 0.3, the maximum level of performance increases exponentially and shoots off to infinity. This result is obviously unrealistic, hence an important question when the model is applied is whether there should be a limit set on the possible α values, for example from 0 < α < 0.5.

The interconnectedness parameter can be viewed as a measure of an athlete’s talent. This is assuming an athlete is considered more talented if their maximum performance with optimal training is higher. Since α is exogenous and cannot be changed, those athletes with little to no interconnectedness (α close to 0), or a high level of interconnectedness α > 0.3, are more talented than those with 0 < α < 0.3, because they can achieve a higher maximum level of speed.

Significantly, the model affirms the major role talent has in achieving top performance. Even if athletes with 0 < α < 0.3 work optimally hard, they cannot surpass more talented athletes who work optimally with α = 0 or α > 0.3. However, it must be noted that this is only in the case when everybody is training optimally. If more talented athletes do not train optimally, less talented athletes can surpass them by training optimally.

Speed and strength are only two possible factors that might be interconnected in an athlete. In fact, there may be many combinations of factors possible. The limits set on the range of α could also depend on how intrinsically related the two factors are. If the factors are closely related, for example running speed and running stamina, then higher α values should be possible (e.g. α above 0.5), compared to relatively unrelated factors like speed and strength.


Whitcomb (1) describes personalised medicine as being based on two “facts” – that most uncured disorders are complex processes, and that each person is fundamentally different from the average of the population. He adds that disease does not result from a single cause but from a myriad of “genetic, environmental and structural factors, or a combination of these”. Whitcomb states that 21st century medicine must begin with a mechanistic and predictive modelling of normal biological systems to determine which of these factors result in disease.

Although this model is a very simple one, it could potentially be expanded to be used in modelling in personalised medicine. This is because it acknowledges that there are not only multiple factors contributing to disease but also that they are interconnected. Furthermore, it is in line with the assertion of personalised medicine that the population is heterogenous.

The core concept in the model, the interconnectedness parameter α, is powerful because it not only can model the interactions between factors, but it can also explain the heterogeneity of cases. Cases can be different, even if their underlying factors are the same, if their interactions—given by the interconnectedness parameter—are different. The model could be used in several ways, for example to model the factors and causes leading up to the disease, or to model the effects of different treatments on diseases.

The model could also be applied to endeavours requiring interaction or collaboration. Take the decision of whether to work with a friend or to go alone. In this example, the only goal is to maximise your marks (L1) in the year end exam. You can choose between studying by yourself (t1) or helping your friend (t2). This example is particularly helpful in explaining the U-shaped result of Figure 2.

Your friend’s final marks exert an effect on your own marks (as they can help you back), and the level of the effect depends on the level of α. In this case, α can be seen as the level of chemistry between the two of you. If your relationship has zero chemistry (α = 0), you will be very happy working alone. However, if you had a slight chemistry with your friend (0 < α < 0.3), you might feel obliged to help them in some way, but that will reduce the time studying for yourself and lead to worse results. Of course, if you had great chemistry with them (α > 0.3), the synergy that results from your interactions can help you achieve even better marks than studying alone. The best results are achieved if you have little to no relationship with your friend (α = 0) or if you have a great relationship with them (α > 0.3). The mediocre relationship may actually weigh you down.


In this essay, I have proposed a simple framework that allows us to assess the effect of interconnectedness (α) between a main factor and a subordinate factor. These factors feed into each other, leading to different routes to maximising performance in the main factor.

The results show the validity of different training regimes in achieving maximal performance. In addition, because of the unique feature of the logarithmic function, a rather surprising result occurs regarding the maximal performance that can be achieved. Either low levels of interconnectedness or high levels of interconnectedness are optimal, with the values in between suboptimal.

This model of an interconnectedness parameter can potentially be applied to personalised medicine because it shares many of its core assumptions, namely that diseases and treatments are complex systems with multi-pronged, interconnected factors and that every case is different. The model can also be applied to other examples, including teamwork.

I have only presented a simple two-factor model. In theory, there could potentially be multi-factor models, each with a different interconnectedness parameter. There may be chains of factors where subordinate traits do not affect the main trait directly but through other subordinate traits. This goes into the realm of the study of causal maps. The results hint that when aiming for maximal performance, not only must all the potential subordinate factors be considered, but also their level of interconnectedness with the main trait and each other. In addition, the interconnectedness need not be symmetrical – one trait may feed into another trait but not vice versa. These are extensions of the model that can be applied in the future.


  1. Whitcomb, D.C. “What is personalized medicine and what should it replace?”. Nat Rev Gastroenterol Hepatol; 2012 May 22; vol.9, no.7: pp. 419-424. Available from:

Rstudio R-3.5.3 was used in calculations and in Figures.

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