We consider the problem of safely controlling an unknown stochastic linear dynamical system subject to an infinite-horizon discounted quadratic cost.
Many of the existing model-based approaches for handling the corresponding robust optimal control problem resort to game theoretic formulations of the uncertainty, either explicit- or implicitly. It is widely known that in practice the corresponding control laws can be rather conservative. In this work, we give further theoretical evidence that this is an inherent property of the underlying game theoretic formulation. We show that the most common uncertainty sets, for example resulting from linear least-squares identification, are almost surely different from the geometry a game theoretic adversary samples from. Nevertheless, we provide theoretical- and empirical evidence that a game theoretic control law has favourable properties over the nominal control law when the estimated model is obtained using regularized linear least-squares.