A unified approach to mean-field team: homogeneity, heterogeneity and quasi-exchangeability

This paper aims to systematically solve stochastic team optimization of large-scale system, in a rather general framework. Concretely, the underlying large-scale system involves considerable weakly-coupled cooperative agents for which the individual admissible controls: ($\mathbf{\text{i}}$) enter the diffusion terms, ($\mathbf{\text{ii}}$) are constrained in some closed-convex subsets, and ($\mathbf{\text{iii}}$) subject to a general \emph{partial decentralized information} structure. A more important but serious feature: ($\mathbf{\text{iv}}$) all agents are heterogenous with \emph{continuum} instead \emph{finite} diversity. Combination of ($\mathbf{\text{i}}$)-($\mathbf{\text{iv}}$) 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 ($\mathbf{\text{i}}$). As the resolution, a novel \emph{unified approach} is proposed under which the intractable \emph{continuum} \emph{heterogeneity} can be converted to a more tractable \emph{homogeneity}. As a trade-off, the underlying randomness is augmented, and all agents become (quasi) weakly-exchangeable. Such 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 \emph{double-projections} (due to ($\mathbf{\text{ii}}$), ($\mathbf{\text{iii}}$)), along with \emph{spatial mean} on continuum heterogenous index (due to ($\mathbf{\text{iv}}$)). Such 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.