Constructive design of open-loop Nash equilibrium strategies that admit a feedback synthesis in LQ games

Open-loop Nash equilibrium strategies that admit a feedback synthesis in Linear–Quadratic (LQ) games are studied. A characterization alternative to the classic system of coupled (asymmetric) Riccati equations – one for each player – is provided by relying on a fixed-point argument based on the composition of flows of the underlying state/costate dynamics. As a result, it is shown that in competitive games, namely games in which the players influence the shared state via linearly independent input channels, the characterization of Nash equilibrium strategies hinges upon the solution to a single (regardless of the number of players), sign-definite Riccati equation, with coefficients described by polynomial functions of the feedback gains. The structure of the latter equation is computationally appealing since it naturally allows for gradient-descent algorithms on matrix manifolds, thus ensuring (local) guaranteed convergence to the equilibrium strategy. In the case of antagonistic games, namely games in which the players may share linearly dependent input directions, the fixed-point condition above is combined with a geometric requirement involving the largest invariant subspace contained in the kernel of an auxiliary output matrix. Finally, by building on the latter characterization it is shown that closed-form expressions for the equilibrium strategy for a class of dynamic games can be given.