Optimality of Independently Randomized Symmetric Policies for Exchangeable Stochastic Teams with Infinitely Many Decision Makers.

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 the 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.