On Optimal Ammunition Usage When Hunting Fleeing Targets

Abstract

A hunter hunts over a planning horizon t with i bullets. The distribution of the value of targets appearing and the probability of a bullet hitting are known. Encountering a target of value w, he has a choice of retiring from hunting completely, shooting a single bullet, or moving on to the next period to find another target. If he retires, he immediately obtains a terminal reward dependent on the remaining periods and bullets. If he fires and succeeds, he obtains the reward w and decides whether to retire or move on to the next epoch. If he misses and the target remains stationary, he must immediately decide whether to fire an additional bullet, retire, or move on to the next period. The objective in this paper is to examine the optimal decision rules that would maximize the total expected reward. We reveal that the optimal decision rule for firing is not always monotone in the number of remaining bullets and the optimal decision rule for retirement may become possibly the following form. It is better to continue hunting than retire, only if the hunter has neither too few bullets nor too many.