Open-Loop Control-Based Linear-Quadratic Stochastic Game with Application to Counter Terror: Farsighted Versus Myopic Policies

Stochastic linear-quadratic problems are frequently found in optimal control and differential game applications. The typical solution to these problems is based on feedback control, which implies that the state dynamics are observable despite their stochastic nature. In real life, however, this is rarely the case. In this chapter, we overcome this unobservability drawback by deriving an open-loop equilibrium control for a linear-quadratic dynamic game with applications to counter-terror activities characterized by stochastic terrorist resource stocks. We derive an open-loop Nash equilibrium solution and its time-dependent feedback representation, which is based on expected terrorist resources rather than on the true state of the resource stock. We contrast the found equilibrium control with myopic behavior in response to resource dynamics by one or both parties and show that a farsighted party always has an advantage over a myopic party (i) under simultaneous commitments and (ii) when a farsighted party’s leader openly commits to actions and the myopic party is a follower responding to the farsighted leader’s actions. Furthermore, uncertainty improves the position of the farsighted party in terms of resource goals. In particular, the greater the resource-related uncertainty, the stronger the resource accumulation when terrorists are farsighted and the government is myopic. Uncertainty is detrimental to the government; it increases the economic damage inflicted by the terrorist organization and thus decreases the cost efficiency of the government’s counter-terror efforts.