  We study stochastic team (known also as decentralized stochastic control or
identical interest stochastic dynamic game) problems with large or countably
infinite number of decision makers, and characterize existence and structural
properties for (globally) optimal policies. We consider both static and dynamic
non-convex team problems where the cost function and dynamics satisfy an
exchangeability condition. To arrive at existence and structural results on
optimal policies, we first introduce a topology on control policies, which
involves various relaxations given the decentralized information structure.
This is then utilized to arrive at a de Finetti type representation theorem for
exchangeable policies. This leads to a representation theorem for policies
which admit an infinite exchangeability condition. For a general setup of
stochastic team problems with $N$ decision makers, under exchangeability of
observations of decision makers and the cost function, we show that without
loss of global optimality, the search for optimal policies can be restricted to
those that are $N$-exchangeable. Then, by extending $N$-exchangeable policies
to infinitely-exchangeable ones, establishing a convergence argument for the
induced costs, and using the presented de Finetti type theorem, we establish
the existence of an optimal decentralized policy for static and dynamic teams
with countably infinite number of decision makers, which turns out to be
symmetric (i.e., identical) and randomized. In particular, unlike prior work,
convexity of the cost in policies is not assumed. Finally, we show near
optimality of symmetric independently randomized policies for finite
$N$-decision maker team problems and thus establish approximation results for
$N$-decision maker weakly coupled stochastic teams.
