Optimal Stochastic Teams with Infinitely Many Decision Makers and Mean-Field Teams

Abstract

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 a limit of a sequence of optimal policies for static teams with $N$ decision makers as $N\rightarrow\infty$ . 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). We consider a number of illustrative examples where the theory is applied to setups with either infinitely many decision makers or an infinite-horizon classical stochastic control problem reduced to a static team.