A Lagrangian approach to constrained potential games: Theory and examples

In this paper, we use a Lagrangian approach to solve for Nash equilibrium in a continuous non-cooperative game with coupled constraints. We discuss the necessary and the sufficient conditions to characterize the equilibrium of the constrained games. In addition, we discuss the existence and uniqueness of the equilibrium. We focus on the class of potential games and point out a relation between potential games and centralized optimization. Based on these results, we illustrate the Lagrangian approach with symmetric quadratic games and briefly discuss the notion of game duality. In addition, we discuss two engineering potential game examples from network rate control and wireless power control, for which the Lagrangian approach simplifies the solution process.