A Differential Games Solution to a Model of Competition Between a Thief and the Police

This paper contains an extension of an optimal control model considered by Sethi (Sethi, S. P. 1979. Optimal Pilfenng policies for dynamic continuous thieves. Management Sci. 25 (6, June) 535--542.) to a differential game situation. It is assumed that Sethi's concave thief, i.e., a risk-averter, plays against the police, whose objective function incorporates convex costs of law enforcement, a one-shot utility at the time of arrest and a continuous utility or cost rate after the thief is arrested. The probability that the thief is caught by time t is influenced not only by the pilfering rate (as is the case in Sethi's model) but also by the instrumental variable of the police, namely its rate of law enforcement. Our aim is to analyze noncooperative Nash solutions of the differential game sketched above. Due to the special structure of the Hamiltonians, a system of two nonlinear differential equations for the control variables of both players can be derived. The system is solved explicitly under special assumptions. It is shown that the optimal rate of law enforcement of police increases monotonically whereas for the Nash-optimal pilfering rate three cases may be characterized in which increase, decrease or constancy occur. Thus the Nash-optimal solutions show in each case monotonic behavior, i.e., changes in trend never occur. Moreover, the dependency of the terminal values of control can be described. Finally, an interpretation of the model as market entrance game is sketched.