Substitutability vs complementarity of efforts Suppose $n$ players are in joint production to produce a common output $v(a_1,a_2,...,a_n)$ with their individual actions $a_i$. Does the convexity or concavity of $v(a_1,...,a_n)$ in the actions say something about the substitutability or complementarity of their actions?
In problems like this, complementarity and substitutability are often defined with respect to the cross-derivatives.

Write $v_i(\cdot)$ for the derivative of $v$ with respect to its $i$th argument. Thus, $v_i(a_1,\ldots,a_n)$ measures how much output increases if $i$ increases his effort slightly.

Now let's think about the derivative of $v_i$ with respect to $a_j$: $v_{ij}(a_1,\ldots,a_n)$. If $v_{ij}&gt;0$ then an extra unit of effort from $i$ yields more output when $a_j$ is high (and vice-versa). We say that $i$'s and $j$'s efforts are complements because extra effort from $j$ makes $i$'s effort more effective (and vice versa).

Conversely, if $v_{ij}&lt;0$ then extra effort from $j$ makes $i$'s effort less effective, and this is often taken as a definition of substitutability.