  We consider a non-zero-sum linear quadratic Gaussian (LQG) dynamic game with
asymmetric information. Each player observes privately a noisy version of a
(hidden) state of the world $V$, resulting in dependent private observations.
We study perfect Bayesian equilibria (PBE) for this game with equilibrium
strategies that are linear in players' private estimates of $V$. The main
difficulty arises from the fact that players need to construct estimates on
other players' estimate on $V$, which in turn would imply that an infinite
hierarchy of estimates on estimates needs to be constructed, rendering the
problem unsolvable. We show that this is not the case: each player's estimate
on other players' estimates on $V$ can be summarized into her own estimate on
$V$ and some appropriately defined public information. Based on this finding we
characterize the PBE through a backward/forward algorithm akin to dynamic
programming for the standard LQG control problem. Unlike the standard LQG
problem, however, Kalman filter covariance matrices, as well as some other
required quantities, are observation-dependent and thus cannot be evaluated
off-line through a forward recursion.
