Stochastic teams with nonclassical information revisited: When is an affine law optimal?

In this note we consider a parameterized family of two-stage stochastic control problems with nonclassical information patterns, which includes the well-known 1968 counterexample of Witsenhausen. We show that whenever the performance index does not contain a product term between the decision variables, the optimal solution is linear in the observation variables. The parameter space can be partitioned into two regions in one of which the optimal solution is linear, whereas in the other it is inherently nonlinear. Extensive computations using two-point piecewise constant policies and linear plus piecewise constant policies provide numerical evidence that nonlinear policies may indeed outperform linear policies when the product term is present.