This is a two step control problem with the simplest linear system. The
initial condition xo is a standard Gaussian random variable
N(0,sigma2) which is perfectly observed by decision maker
DM1 (Decision Maker 1). It is possible for DM1 to directly cancel
xo via control u1. However it is costly
for DM1 to act (the k u12 term in the performance
criterion). DM2 acts after DM1. DM2 has no cost to act but s/he does not
know what DM1 knows. Instead DM2 knows a noisy observation of
x1 = xo + u1, the result
of DM1 acting on xo, through z2 =
x1 + v, v ~ N(0,1). Thus, we have a
tradeoff between a DM with perfect information but costly control and another
DM with noisy information but inexpensive control. The classical version of
the problem in which the DM who acts later always knows what the previous DM
knows, the so-called perfect memory information structure is trivial to solve.
But the twist here is that the problem is decentralized. DM2 does not know
z1 the information of DM1. It is also known that if one of
DM uses a linear strategy, then the optimal strategy of the other DM must also
be linear. In fact, this linear strategy pair is also easy to solve and
yields the performance of J = 0.9 (for k = 0.2 and sigma = 5).
But surprisingly, a nonlinear strategy exist which out-performs the best
linear strategy. The basic idea here is that of signaling. DM1 can use
his/her control variable u1 not so much to reduce
xo but to signal to DM2 his knowledge of
x1 so that s/he can cancel it with no cost. One such
control found by Witsenhausen is
The action of DM1 converts xo to ±sigma. Then DM2 only
needs to figure out whether or not x1 is positive or
negative. For large sigma, this can be done with negligible error and thus
insure the canceling of x1. The performance of such a
strategy is J = 0.404, considerably better than the best linear
strategy. However, the optimal nonlinear strategy is not known even after
28 years. The problem is also known to be NP-hard.