Numerical Solutions to Witsenhausen's Problem by Successive Approximation

The non-classical nature of multi-person stochastic control problems of team problems was first pointed out by Witsenhausen [1] via a simple example, which has since been known as Witsenhausen's counterexample. The example is a simple two-person two-step team problem: The first person 1 observes exactly the system's initial state x
₀
, and based on it, chooses his action x
₁
. Then the second person 2 observes the new state x
₁
= x
₀
+ x
₁
corrupted by an independent zero-mean noise v as y = x
₁
+ v and chooses his action x
₂
. The team objective is to minimize the cost J ≜
²
(x
₁
)
²
+ (x
₁
- x
₂
)
²
, over pairs of admissible (measurable) strategies. This short paper is concerned with one typical case of this problem, restated as in [1]: Problem: Given a pair (σ,κ) of positive real numbers, find a pair (f,g) of measurable functions which minimizes (κ
²
/σ) ∫ (x - f(x))
²
ψ(z/σ)dx + ∫ (f(z) - g (f(z)+v))
²
ψ(v)dv where ψ(ϵ)(2πexp(z
²
))
^-1/2
. Despite the fact that this very simple-looking problem has stimulated much important literature on control and information (See e.g. [2]), the problem itself has remained unsolved for nearly twenty years. There have been several attempts to solve the problem numerically but generally with little success. Recently, it was pointed out that the difficulty resides in the particular form of the objective function as well as the non-classical information pattern [3].