Rational inattention in scalar LQG control

Motivated in part by the “rational inattention” framework of information-constrained decision-making by economic agents, we have recently introduced a general model for average-cost optimal control of Markov processes subject to mutual information constraints [1]. The optimal information-constrained control problem reduces to an infinite-dimensional convex program and admits a decomposition based on the Bellman error, which is the object of study in approximate dynamic programming. In this paper, we apply our general theory to an information-constrained variant of the scalar linear-quadratic-Gaussian (LQG) control problem. We give an upper bound on the optimal steady-state value of the quadratic performance objective and present explicit constructions of controllers that achieve this bound. We show that the obvious certainty-equivalent control policy is suboptimal when the information constraints are very severe, and exhibit another policy that performs better in this low-information regime. In the two extreme cases of no information (open-loop) and perfect information, these two policies coincide with the optimum.