  In this paper, we consider finite model approximations of a large class of
static and dynamic team problems where these models are constructed through
uniform quantization of the observation and action spaces of the agents. The
strategies obtained from these finite models are shown to approximate the
optimal cost with arbitrary precision under mild technical assumptions. In
particular, quantized team policies are asymptotically optimal. This result is
then applied to Witsenhausen's celebrated counterexample and the Gaussian relay
channel problem. For the Witsenhausen's counterexample, our approximation
approach provides, to our knowledge, the first rigorously established result
that one can construct an $\varepsilon$-optimal strategy for any $\varepsilon >
0$ through a solution of a simpler problem.
