I live by the rule that I never follow an "after you" with a "no, after you" without good reason. If they want demonstrate their politeness, why compete?
The obvious reason is because you get to play rock-paper-scissors.
The less obvious reason is that after a certain number of back-and-forths, you can interrupt the other person saying "aft-", cut them in line sideways so you both sort of swoop through the doorway, which then leads to a warp zone.
