  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.
