A sequential allocation problem with search cost where the shoot-look-shoot policy is employed

Suppose a hunter starts hunting over t periods with i bullets. A distribution of the value of targets appearing and the hitting probability of a bullet are known. For shooting, he takes a strategy of a shoot-look-shoot scheme. The objective in this paper is to find an optimal policy which maximizes the total expected reward. In the case with no search cost, the optimal policy is monotone in the number of bullets remaining then, but not always monotone in the case with positive search cost. The paper shows such examples of non-monotonicity and examine conditions for the monotonicity of the optimal policy.