Optimal Pilfering Policies for Dynamic Continuous Thieves

This paper considers the problems of a dynamic continuous thief, such as a habitual shoplifter or a gas siphoner, who must choose the pilfering rate (which increases the probability of his arrest over time) to maximize the present value of his total expected gain over a given finite or infinte horizon. The gain function incorporates the amount of pilfering, the one-shot penalty at the time of arrest and the continuous punishment subsequent to arrest. A thief is classified as a profit-maximizer, a kleptomaniac, or a risk-averter depending on whether he considers his gain function to be linear, convex, or concave in the pilfering rate, respectively. Optimal pilfering rates are obtained for each of these different types of thieves by using the methods of optimal control. It is shown that the optimal pilfering policy of a profit-maximizing thief is bang-bang, i.e., either to pilfer at the maximum possible rate or not to pilfer at all. In the particular case of a finitely-lived profit-maximizing thief, the optimal policy is not to pilfer while the thief is young and then to pilfer at the maximum rate toward the end of his life so that the continuous punishment subsequent to his arrest will not last too long. In the infinite horizon cases, both the profit-maximizing thieves and kleptomaniacs will either never pilfer or pilfer at the maximum rate depending on whether the parameters of the problem they are facing are unfavorable or favorable. The risk-averse thief, on the other hand, would settle at an intermediate pilfering rate throughout the infinite horizon. Finally, it is expected that knowledge of these optimal policies will have implications for the allocation of money between police, courts and penal institutions.