Independently Randomized Symmetric Policies are Optimal for Exchangeable Stochastic Teams with Infinitely Many Decision Makers

We study stochastic team (known also as decentralized stochastic control or identical interest stochastic game) problems with large or countably infinite number of decision makers, and characterize existence and structural properties for (globally) optimal policies. We consider in particular 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 policies under decentralized information structures. 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 over any convex set of probability measures on 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.