  We study stochastic static teams with countably infinite number of decision
makers, with the goal of obtaining (globally) optimal policies under a
decentralized information structure. We present sufficient conditions to
connect the concepts of team optimality and person by person optimality for
static teams with countably infinite number of decision makers. We show that
under uniform integrability and uniform convergence conditions, an optimal
policy for static teams with countably infinite number of decision makers can
be established as the limit of sequences of optimal policies for static teams
with $N$ decision makers as $N \to \infty$. Under the presence of a symmetry
condition, we relax the conditions and this leads to optimality results for a
large class of mean-field optimal team problems where the existing results have
been limited to person-by-person-optimality and not global optimality (under
strict decentralization). In particular, we establish the optimality of
symmetric (i.e., identical) policies for such problems. As a further condition,
this optimality result leads to an existence result for mean-field teams. We
consider a number of illustrative examples where the theory is applied to
setups with either infinitely many decision makers or an infinite-horizon
stochastic control problem reduced to a static team.
