Let's say you have a distribution that is either symmetric or positively skewed (and defined over 0-1). Call it F.
Then, you find the distribution of the minimum of n>1 draws from F. Call it Fmin.
Clearly the mean of Fmin is less than the mean of F.
Now, find the distribution of the maximum of the same number n draws from Fmin. Call it Fminimax.
The mean of Fminimax is of course greater than the mean of Fmin. However, the increase from Fmin to Fminimax is less in magnitude than the original decrease from F to Fmin. Meaning, that the "round trip" is not complete, and the minimax operation ends up lowering the mean.
What I meant to ask, was, does this single operation lower the mean for any non-negatively skewed distribution?
I deleted a bunch of stuff from the question that I think I made a mess out of. But, I can always revert the changes if anyone objects.