This morning I was walking into my building and a man was walking parallel with me. He was going towards one set of doors; I was going towards a pair adjacent to them. He smiled at me and gestured me over to his side with a head nod. “Let me get the door for you.” (He meant let me get these doors over here for you.) I capitulated, averted my path over to his doors, smiled and said thank you. Then, as we approached the second set of doors into the lobby, he yelled out, “I’ll get those, too.” He wanted to do it. So I waited for him to catch up. And then out of obligation I thanked him again, but I was annoyed.
I think that as the two of you approached the second set of doors, after he yelled out, I’ll get those, too”, you should have stepped to the door of your choice and replied, “My turn” while holding it for him to catch up.
Potentially, this tit-for-tat maneuver (and I do not mean this in an offensive or sexist manner) may have evened the playing field! Alas, if I were in your place, I would have likely thought of this only in hindsight.