  This paper studies a system security problem in the context of observability
based on a two-party non-cooperative asynchronous dynamic game. A system is
assumed to be secure if it is not observable. Both the defender and the
attacker have means to modify dimension of the unobservable subspace, which is
set as the value function. Utilizing tools from geometric control, we construct
the best response set under one-step or two-step optimality to minimize or
maximize the value function. We find that the best response sets under one-step
optimality are not single-valued maps, resulting in a variety of game outcomes.
In the dynamic game considering two-step optimality, definition and existence
conditions of lock and oscillation game modes are given. Finally, the best
response under two-step optimality and the Stackelberg game equilibrium are
compared.
