# Structural Results and Explicit Solution for Two-Player LQG Systems on a Finite Time Horizon

arXiv: https://arxiv.org/abs/1303.3256 · HTML: https://mlsanity.com/paper/1303.3256/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Mar 2013.

## Built on by (most cited first, 2 of 2)

- [Solving Common-Payoff Games with Approximate Policy Iteration](https://mlsanity.com/paper/2101.04237/) (2021)
- [Abstracting Imperfect Information Away from Two-Player Zero-Sum Games](https://mlsanity.com/paper/2301.09159/) (2023)

## Abstract (arXiv)

It is well-known that linear dynamical systems with Gaussian noise and quadratic cost (LQG) satisfy a separation principle. Finding the optimal controller amounts to solving separate dual problems; one for control and one for estimation. For the discrete-time finite-horizon case, each problem is a simple forward or backward recursion. In this paper, we consider a generalization of the LQG problem in which there are two controllers. Each controller is responsible for one of two system inputs, but has access to different subsets of the available measurements. Our paper has three main contributions. First, we prove a fundamental structural result: sufficient statistics for the controllers can be expressed as conditional means of the global state. Second, we give explicit state-space formulae for the optimal controller. These formulae are reminiscent of the classical LQG solution with dual forward and backward recursions, but with the important difference that they are intricately coupled. Lastly, we show how these recursions can be solved efficiently, with computational complexity comparable to that of the centralized problem.