A Unified Approach to Linear-Quadratic-Gaussian Mean-Field Team: Homogeneity, Heterogeneity and Quasi-Exchangeability

This paper aims to systematically solve stochastic team optimization of a large-scale system, in a linear-quadratic-Gaussian framework. Concretely, the underlying large-scale system involves considerable weakly coupled co­operative agents for which the individual admissible controls: (i) enter the diffusion terms, (ii) are constrained in some closed-convex subsets and (iii) subject to a general partial decentralized information structure. A more im­portant but serious feature: (iv) all agents are heterogenous with continuum instead of finite diversity. Combination of (i)-(iv) yields a quite general mod­eling of stochastic team-optimization, but on the other hand, also fails current existing techniques of team analysis. In particular, classical team consistency with continuum heterogeneity collapses because of (i). As the resolution, a novel unified approach is proposed under which the intractable continuum heterogeneity can be converted to a more tractable homogeneity. As a trade­off, the underlying randomness is augmented, and all agents become (quasi) weakly exchangeable. Such an approach essentially involves a subtle bal­ance between homogeneity v.s. heterogeneity, and left (prior-sampling)- v.s. right (posterior-sampling) information filtration. Subsequently, the consis­tency condition (CC) system takes a new type of forward-backward stochastic system with double-projections (due to (ii), (iii)), along with spatial mean on continuum heterogenous index (due to (iv)). Such a system is new in team literature and its well-posedness is also challenging. We address this issue under mild conditions. Related asymptotic optimality is also established.