Logical Dynamic Games: Models, Equilibria, and Potentials

Logical dynamic games (LDGs) are a class of dynamic games that incorporate logical dynamics to describe the evolution of external states. Such games can be found in a wide range of natural and engineered systems, such as the Boolean network of lactose operon in
Escherichia coli
. However, little attention has been paid to LDGs in the control community. This paper aims at developing a framework for the analysis and synthesis of LDGs under finite-horizon criteria. First, a general mathematical model of LDGs is constructed. Using dynamic programming theory, we prove that an LDG can be decomposed as a series of time-sliced static games, and the existence of pure dynamic Nash equilibrium (DNE) is proved to be equivalent to the existence of pure Nash equilibria of the decomposed time-sliced static games. To disentangle the circular dependency in the DNE calculation, a backward recursive method is proposed. Second, the concept of logical dynamic potential games (LDPGs) is proposed, and the connection between an LDG and its corresponding optimal control problem is established. Three verification conditions for a given LDG to be LDPG are presented, including time-sliced condition, closed-path condition, and potential equation condition. And a recursive algorithm is further designed for the verification of LDPGs via potential equation conditions. Third, to seek time-independent verification conditions, LDGs with action-independent transition properties are investigated. We prove that, if the auxiliary game constructed by the stage cost function is a state-based potential game, then the LDG is an LDPG. Finally, the effectiveness of the theoretical results is demonstrated by some numerical examples.