Suppose I’m training for an upcoming race. I want to choose the training load that maximises my expected performance on race day. The harder I train, the better my performance will be but the more likely I am to injure myself. How should I balance this trade-off between better performance and greater risk of injury?

We can model this choice problem as follows.
Let `\(t\in[0,1]\)`

represent my training load and `\(a\in\mathbb{R}\)`

my natural ability.^{1}
My performance on race day is some function `\(f(t,a)\)`

of `\(t\)`

and `\(a\)`

.
I assume that this function is increasing and concave in `\(t\)`

(so that there are positive but diminishing returns to training), and increasing in `\(a\)`

.

I can’t compete if I get injured, which occurs with some probability `\(p(t,r)\)`

that depends on my training load and my natural resistance to injury `\(r\in\mathbb{R}\)`

.
I assume that `\(p\)`

is increasing and convex in `\(t\)`

(so that training increases my likelihood of injury at an increasing rate), and decreasing in `\(r\)`

.

My objective is to choose the training load `\(t^*\)`

that maximises my expected performance^{2}
`$$\psi(t)=(1-p(t,r))\,f(t,a).$$`

My assumptions on the shapes of `\(f\)`

and `\(p\)`

imply that `\(\psi\)`

is concave in `\(t\)`

.
Therefore, the unique optimal training load `\(t^*\)`

satisfies the first-order condition (FOC)
`$$\begin{align} 0 &= \psi'(t^*) \\ &= -p_t(t^*,r)\,f(t^*,a)+(1-p(t^*,r))\,f_t(t^*,a), \end{align}$$`

where `\(\psi'\)`

denotes the derivative of `\(\psi\)`

with respect to `\(t\)`

, and
where `\(p_t\)`

and `\(f_t\)`

denote the partial derivatives of `\(p\)`

and `\(f\)`

with respect to `\(t\)`

.
The FOC can be rewritten as
`$$(1-p(t^*,r))\,f_t(t^*,a)=p_t(t^*,r)f(t^*,a),$$`

which shows that I should keep training until the marginal benefit of improved performance (the left-hand side) equals the marginal cost of injury becoming more probable (the right-hand side).

I can’t determine the value of `\(t^*\)`

without further assumptions on `\(f\)`

and `\(p\)`

.
However, I can determine the relationship between `\(t^*\)`

and the parameters `\(a\)`

and `\(r\)`

.
Since `\(\psi''(t)<0\)`

for all feasible `\(t\)`

, the implicit function theorem (IFT) implies that
`$$\mathrm{sign}\frac{\partial t^*}{\partial \theta}=\mathrm{sign}\frac{\partial \psi'(t^*)}{\partial \theta}$$`

for each element `\(\theta\)`

of the symbol set `\(\{a,r\}\)`

.
Now
`$$\frac{\partial \psi'(t^*)}{\partial a}=-p_t(t^*,r)\,f_a(t^*,a)+(1-p(t^*,r))\,f_{ta}(t^*,a),$$`

where `\(f_a\)`

and `\(f_{ta}\)`

denote the partial derivatives of `\(f\)`

and `\(f_t\)`

with respect to `\(a\)`

, and
`$$\frac{\partial \psi'(t^*)}{\partial r}=-p_{tr}(t^*,r)\,f(t^*,a)-p_r(t^*,r)\,f_t(t^*,a),$$`

where `\(p_{tr}\)`

and `\(p_r\)`

denote the partial derivatives of `\(p_t\)`

and `\(p\)`

with respect to `\(r\)`

.
By Young’s theorem, the mixed partials `\(f_{ta}\)`

and `\(p_{tr}\)`

satisfy
`$$f_{ta}(t,a)=\frac{\partial}{\partial t}\left(\frac{\partial f(t,a)}{\partial a}\right)$$`

and
`$$p_{tr}(t,r)=\frac{\partial}{\partial t}\left(\frac{\partial p(t,r)}{\partial r}\right)$$`

for all feasible `\(t\)`

, `\(a\)`

and `\(r\)`

.
Thus, it seems reasonable to assume that `\(f_{ta}(t,a)\le0\)`

and `\(p_{tr}(t,r)\le0\)`

, which mean that training washes out the benefits of natural ability and resistance to injury.
These assumptions, together with the IFT, imply that `\(t^*\)`

is decreasing in `\(a\)`

and increasing in `\(r\)`

—that is, I should train harder if I become less naturally able or more resistant to injury.