A theoretical draw question

[MarkGritter]

Suppose you're playing single-draw lowball (or the last draw in TD) with perfect information about which cards your opponent(s) have in their hands and have discarded. So, it is impossible to make a "mistake" in the Sklansky sense.

Does position still matter? Is there any case in which the order in which draws are announced will influence strategy? Why or why not?

Here's a way of stating it in the two player case: A seeks to increase his EV, B seeks to decrease it. Let DD,DP,PD,PP denote the four cases where A and B, respectively, either Draws one or stands Pat. I am interested in whether there is a case where

max(min(DD,DP), min(PD,PP)) != min(max(DD,PD), max(DP,PP))

I'd also be interested in any answers for more complicated situations, i.e., draw 1 or 2, multiple players, etc.

My guess is that position does not matter with just two players, but it might with three.