OK, two qualifiers before I lay this out.
One: I just awoke from a nap and a dream in which this came to me, so it is not fully flushed out. I want to get it in writing before it leaks out of my memory.
Two: This is my original thought, as far as I know. I am not consciously borrowing, stealing, or modeling this after any other thought experiment or game, so there is no ulterior motive or hidden trick or anything like that.
With that out of the way, here is my puzzle/game.
Using game theory, what would be the best strategy in this elevator tag scenario?
Scenario: You are on the 6th floor of a 6-story building with two elevators. They are not particularly fast elevators; they move from one floor to another about as fast as someone running up or down the stairs can move from one floor to the other. There are also stairs on one side of the elevators. While getting into one of the elevators, you notice the Texas Chainsaw Massacre Dude running after you, suddenly revving it up as he runs towards you. You assume he's decided you're his next intended victim. The elevator door closes. You assume he's going to jump in the other elevator and chase you. There is no one else in the building as far as you know (it's late at night) or any building or standby security, so there is no help to call or depend on. You are unarmed.
What button(s) do you press upon entering the elevator? You cannot pull the emergency button and get the elevator stuck between floors. You have to choose a floor to get off the elevator. Which floor button(s) do you push, and which floor would you get off the elevator on, assuming your escape plan is to eventually run out of the building and to your car parked two blocks away in a parking lot or deck?
If you are the Texas Chainsaw Massacre Dude, and your goal is to kill me, what floor would you head to and how (in the other elevator, or on foot down the stairs)? How would you catch me and prevent me from escaping?
How would your strategy or plan differ, if at all, if you start on the 6th floor, but it's a 10-story building?
It's just a thought experiment and I don't have a methodical plan yet. I'm just posting it and soliciting input and ideas. Feel free to tell me it's just a fever dream.