Ok, let's rework this a bit. Let's say instead of doors, there are three boxes that are identical. You can either keep your first pick, which was 1/3 chance of winning, or give up the box you picked, they get shuffled, and one without the prize in it, is removed. You then pick one of the two remaining boxes. Do you do this or not?
In this case, the repicking makes the odds 1/2 of being a winner, as opposed to 1 out 3. So, I would say you do that. I understand this to be this way, because the conditions when picking have been reset so it is like a different choice. The odds are set based on the conditions that were around when the person picked. These odds do remain consistent. However, if things change in the environment to the OTHER choices, and those conditions improve, it is possible to get something more favorable by switching.
