This method of alternating moves in a branching tree matches both our intuitive thought processes during a chess game (“Okay, if I do this, then Black's going to do this, and then I'd do this, and then...”) and the foundation of the algorithms chess computers like Deep Blue use.

Could you include a reference to alpha-beta pruning, since that is precisely what you're describing? Some readers may be more familiar with that subject domain and appreciate explicitly linking game theory to an established search algorithm.

game theorists call this a gnash equilibrium

Did you mean "Nash equilibrium"? If it was a deliberate pun, you might want to indicate it for those who are looking up new concepts, this being an introductory series.