This paper studies a system security problem in the context of observability based on a two-person noncooperative infinitely repeated game. Both the attacker and the defender have means to modify the dimension of the unobservable subspace, which is set as the value function. Utilizing tools from geometric control, the authors construct the best response sets considering one-step and two-step optimality respectively to maximize or minimize the value function. The authors establish a unified necessary and sufficient condition for Nash equilibrium that holds for both one-step and two-step optimizations. The proposed analysis further uncovers two evolutionary patterns, lock and loop modes, and shows an asymmetry between defense and attack. The defender can lock the game into equilibrium, whereas the attacker can disrupt the equilibrium by sacrificing short-term utility for longer-term advantage. Six representative numerical examples corroborate the theoretical results and highlight the complexity of possible game patterns.