Mitigation of targeted and non-targeted covert attacks as a timing game

We consider a strategic game in which a defender wants to maintain control over a resource that is subject to both targeted and non-targeted covert attacks. Because the attacks are covert, the defender must choose to secure the resource in real time without knowing who controls it. Each move by the defender to secure the resource has a one-time cost and these defending moves are not covert, so that a targeted attacker may time her attacks based on the defender's moves. The time between when a targeted attack starts and when it succeeds is given by an exponentially distributed random variable with a known rate. Non-targeted attackers are modeled together as a single attacker whose attacks arrive following a Poisson process. We find that in this regime, the optimal moving strategy for the defender is a periodic strategy, so that the time intervals between consecutive moves are constant.