Our paper addresses characterizing conditions for a linear quadratic (LQ) game to be a potential game. The desired properties of potential games in finite action settings, such as convergence of learning dynamics to Nash equilibria, and the challenges of learning Nash equilibria in continuous state and action settings motivate us to characterize LQ potential games. Our first contribution is to show that the set of LQ games with full-state feedback that are potential games is very limited, essentially differing only slightly from an identical interest game. Given this finding, we restrict the class of LQ games to those with decoupled dynamics and decoupled state information structure. For this subclass, we show that the set of potential games strictly includes non-identical interest games and characterize conditions for the LQ games in this subclass to be potential. We further derive their corresponding potential function and prove the existence of a Nash equilibrium. Meanwhile, we highlight the challenges in the characterization and computation of Nash equilibrium for this class of potential LQ games.

Accepted for publication in Automatica. 15 pages, 2 figures