On Logical Dynamic Potential Games

Logical dynamic games are dynamic games with logical dynamics describing the external state evolutionary process, which exist widely in real systems like the Boolean network of lactose operon in Escherichia coli. To the best of our knowledge, there is little attention on LDGs. In this paper, we aim at developing a framework for the analysis of dynamic games with logical dynamics under finite-horizon criteria. First, mathematical model of logical dynamic games is provided. A necessary and sufficient condition for the existence of pure feedback Nash equilibrium in a logical dynamic game is derived. Second, rigorous mathematical model of logical dynamic potential game is proposed, which establishes the relationship between logical dynamic games and corresponding optimal control problems. Third, we proved that a logical dynamic game is a logical dynamic potential game, if and only if, all the static sub-games are potential games. Finally, an example is provided to illustrate the theoretical results.