Invariant equilibria of polytopic games via optimized robust control invariance

In the context of non-cooperative linear discrete-time games, invariant equilibria are introduced as an extension to many players of the corresponding concept for two-player games (doubly invariant equilibria). It is established that the set of equilibrium control amplitudes is convex. An efficient computational procedure for the computation of piecewise affine equilibrium strategies based on the recent concept of optimized robust control invariance is proposed. The procedure makes use of a simple linear programming formulation when constraints are polytopic, thus avoiding recursive projection of convex sets. A numerical example for three player game is presented.