Maxmin and minmax strategies
The question was clarified in the comments as


  How to calculate $\min\limits_q \max\limits_p E(p,q)$ for a given function $E(p,q)$?


What this notation means is that $p$ is chosen first, and $q$ is chosen afterwards.

Assume that $p$ is a given parameter. Then $E(p,\cdot)$ is a function over $q$ only. So you take the minimum of the function $E(p,\cdot)$. In economics this usually means taking the first derivative with respect to $q$. From this you get the optimal value of $q$, given $p$. This is basically a function $q^*(p)$. So you know how $q$ will be chosen given the choice of $p$. This gives you the function $E(q^*(p),p)$. You choose $p$ to maximize this function. Again in economics this usually means taking the first derivative w.r.t. $p$.

With $\max\limits_p \min\limits_q$ the procedure is essentially the same, but there you first derive the maximum for $p$ given $q$. This yields $p^*(q)$. Using this, you minimize $E(p^*(q),q)$ w.r.t. $q$.

A sidenote:
If $E(p,q)$ is linear in both $p$ and $q$ the functions $q^*(p)$ and $p^*(q)$ are usually not continuous. To be more precise they are not functions but mappings. As such, they upper hemicontinuous but that is very technical and you may not need to know what it means.