Abstract
We study stochastic static teams with countably infinite number of decision makers (DMs), 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 DMs. We show that under uniform integrability and uniform convergence conditions, an optimal policy for static teams with countably infinite number of DMs can be established as the limit of sequences of optimal policies for static teams with $N$ DMs as $N \to \infty$ . Under the presence of a symmetry condition, we relax the conditions and this leads to optimal 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 DMs or an infinite-horizon stochastic control problem reduced to a static team.
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