For real numbers x and y, compute
max min (x-y) and
(x >=0) (y >=0)
min max (x-y)
Ok. Then let's take the first one. For positive y and any fixed x, the minimum inside the brackets is going to be negative infinity. Taking the maximum of a bunch of negative infinities is just negative infinity. Hence, I'd say the first one is negative infinity.
Using an analogous reasoning method, what do you suppose the second one will be?