Saddle-point solution to zero-sum games subject to noncausal systems

A singular system, assumed to possess both regularity and freedom from impulses, is categorized as a causal system. Noncausal systems (NSs) are a class of singular systems anticipated to exhibit regularity. This study focuses on investigating zero-sum games (ZSGs) in the context of NSs. We introduce recurrence equations grounded in Bellman's optimality principle. The saddle-point solution for multistage two-player ZSGs can be obtained by solving these recurrence equations. This methodology has demonstrated its effectiveness in addressing two-player ZSGs involving NSs. Analytical expressions that characterize saddle-point solutions for two types of two-player ZSGs featuring NSs, encompassing both linear and quadratic control scenarios, are derived in this paper. To enhance clarity, we provide an illustrative example that effectively highlights the utility of our results. Finally, we apply our methodology to analyze a ZSG in the realm of environmental management, showcasing the versatility of our findings.