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

Abstract

We study stochastic teams (known also as decentralized stochastic control problems or identical interest stochastic dynamic games) with large or countably infinite numbers of decision makers and characterize the existence and structural properties of (globally) optimal policies. We consider both static and dynamic nonconvex teams where the cost function and dynamics satisfy an exchangeability condition. To arrive at existence and structural results for optimal policies, we first introduce a topology on control policies, which involves various relaxations given the decentralized information structure. This is then utilized to arrive at a de Finetti–type representation theorem for exchangeable policies. This leads to a representation theorem for policies that 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 numbers of decision makers, which turns out to be symmetric (i.e., identical) and randomized. In particular, unlike in prior work, convexity of the cost in policies is not assumed. Finally, we show the near optimality of symmetric independently randomized policies for finite N-decision-maker teams and thus establish approximation results for N-decision-maker weakly coupled stochastic teams.