Deep Teams: Decentralized Decision Making With Finite and Infinite Number of Agents

Inspired by the concepts of deep learning in artificial intelligence and fairness in behavioral economics, we introduce deep teams in this article. In such systems, agents are partitioned into a few subpopulations so that the dynamics and cost of agents in each subpopulation is invariant to the indexing of agents. The goal of agents is to minimize a common cost function in such a manner that the agents in each subpopulation are not discriminated or privileged by the way they are indexed. Two nonclassical information structures are studied. In the first one, each agent observes its local state as well as the empirical distribution of the states of agents in each subpopulation, called deep state, whereas in the second one, the deep states of a subset (possibly all) of subpopulations are not observed. Novel dynamic programs are developed to identify globally optimal and suboptimal solutions for the first and second information structures, respectively. The computational complexity of finding the optimal solution in both space and time is polynomial (rather than exponential) with respect to the number of agents in each subpopulation and is linear (rather than exponential) with respect to the control horizon. This complexity is further reduced in time by introducing a forward equation, which we call deep Chapman-Kolmogorov equation, described by multiple convolutional layers of binomial probability distributions. Two different prices are defined for computation and communication, and it is shown that under mild conditions they converge to zero as the number of quantization levels and the number of agents tend to infinity. In addition, the main results are extended to infinite-horizon discounted models and arbitrarily asymmetric cost functions. Finally, a service management example with 200 users is presented.