A sequential partial information bomber-defender shooting problem

A bomber carrying homogenous weapons sequentially engages ground targets capable of retaliation. Upon reaching a target, the bomber may fire a weapon at it. If the target survives the direct fire, it can either return fire or choose to hold fire (play dead). If the former occurs, the bomber is immediately made aware that the target is alive. If no return fire is seen, the true status of the target is unknown to the bomber. After the current engagement, the bomber, if still alive, can either re-engage the same target or move on to the next target in the sequence. The bomber seeks to maximize the expected cumulative damage it can inflict on the targets. We solve the perfect and partial information problems, where a target always fires back and sometimes fires back respectively using stochastic dynamic programming. The perfect information scenario yields an appealing threshold based bombing policy. Indeed, the marginal future reward is the threshold at which the control policy switches and furthermore, the threshold is monotonic decreasing with the number of weapons left with the bomber and monotonic nondecreasing with the number of targets left in the mission. For the partial information scenario, we show via a counterexample that the marginal future reward is not the threshold at which the control switches. In light of the negative result, we provide an appealing threshold based heuristic instead. Finally, we address the partial information game, where the target can choose to fire back and establish the Nash equilibrium strategies for a representative two target scenario.