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 → ∞. 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 personby-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.

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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. We first establish a de Finetti type representation theorem for exchangeable decentralized policies, that is, for the probability measures induced by admissible relaxed control policies under decentralized information structures. 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 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.

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