Is there a known family of "functors" for games like this, e.g. veto or having one opportunity to swap positions with your opponent etc.? It would be cool to see what you could say about the rule modification in a general sense before applying to a particular game.
There is some literature on this, yes. I don't know quite how general it gets.
See for example several books by Elwyn Berlekamp.
One outcome of this work was Berlekamp (IIRC) solving a small class of endgame problem that has eluded professional (full-time) go players for literally hundreds of years.
