§ Analysing simple games

I found the clear articulation of these ideas quite nice.
  1. In a game with symmetry, a symmetric move can be blocked or prevented onlyby the previous move an opponent has just made.
  2. The symmetry in many games can be written as some kind of equality, whereat each turn, one first player breaks the symmetry, and the other player(who has the winning strategy) restores it.

§ Example game

Consider a game where two players take turns placing bishops on a chessboard, so that the pieces cannot capture each other. The player who cannot win loses.

§ Winning strategy

place the bishop symmetrically about the line passing between the fourth and fifth column (file). Note that the only way this bishop could be blocked is if the move just made by the other player can block it.

§ References