I'm a bit sleepy and ate far too much seafood to process all you've written here presently -- but the specific thing I was referring to is called Zermelo's Theorem. Which is up there for top 10 theories with cool-sounding names, as far as I'm concerned -- here it is in case you're interested (I periodically forget about this theorem and it's always neat to re-find it -- which you could call an... optimally suboptimal move)
https://en.wikipedia.org/wiki/Zermelo%27s_theorem_(game_theory)