How do I solve max min (x − y) and min max (x − y) such that y≥0 and x≥0?

Question

solve max min (x − y) and min max (x − y) such that y≥0 and x≥0

I don't have a clue where to start.

Answer 1

Start from the inside and work your way out. $$\min_{y\ge 0} x-y$$ is a function of $x$. However, regardless of what $x$ is, the minimum is $-\infty$ as $y$ may be arbitrarily large. Hence, $$\max_{x\ge 0}\min_{y\ge 0}x-y=\max_{x\ge 0} -\infty$$ which is just $-\infty$.