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We study stochastic static teams with countably infinite number of decisionmakers, with the goal of obtaining (globally) optimal policies under adecentralized information structure. We present sufficient conditions toconnect the concepts of team optimality and person by person optimality forstatic teams with countably infinite number of decision makers. We show thatunder uniform integrability and uniform convergence conditions, an optimalpolicy for static teams with countably infinite number of decision makers canbe established as the limit of sequences of optimal policies for static teamswith $N$ decision makers as $N \to \infty$. Under the presence of a symmetrycondition, we relax the conditions and this leads to optimality results for alarge class of mean-field optimal team problems where the existing results havebeen limited to person-by-person-optimality and not global optimality (understrict decentralization). In particular, we establish the optimality ofsymmetric (i.e., identical) policies for such problems. As a further condition,this optimality result leads to an existence result for mean-field teams. Weconsider a number of illustrative examples where the theory is applied tosetups with either infinitely many decision makers or an infinite-horizonstochastic control problem reduced to a static team.

Optimal Solutions to Infinite-Player Stochastic Teams and Mean-Field Teams