Numerical Evaluation of Exact Person-by-Person Optimal Nonlinear Control Strategies of the Witsenhausen Counterexample

TL;DR

This paper numerically solves the exact nonlinear person-by-person optimal control strategies for Witsenhausen's counterexample, providing insights into their structure and performance through advanced quadrature methods.

Contribution

It introduces a numerical approach using Gauss Hermite Quadrature to solve the nonlinear integral equations defining the optimal strategies, advancing understanding of this classic control problem.

Findings

Numerical solutions for the exact nonlinear strategies are obtained.

Comparison shows the numerical strategies align with or improve upon previous results.

The approach offers a new way to analyze complex decentralized control problems.

Abstract

Witsenhausen's 1968 counterexmaple is a simple two-stage decentralized stochastic control problem that highlighted the difficulties of sequential decision problems with non-classical information structures. Despite extensive prior efforts, what is known currently, is the exact Person-by-Person (PbP) optimal nonlinear strategies, which satisfy two nonlinear integral equations, announced in 2014, and obtained using Girsanov's change of measure transformations. In this paper, we provide numerical solutions to the two exact nonlinear PbP optimal control strategies, using the Gauss Hermite Quadrature to approximate the integrals and then solve a system of non-linear equations to compute the signaling levels. Further, we analyse and compare our numerical results to existing results previously reported in the literature.