Corruption via Mean Field Games

TL;DR

This work introduces a novel retrospective inverse problem for a Mean Field Games system modeling the development of a corrupted hierarchy. It derives a time-reversed MFGS with coupling between the value function $u$ and density $m$, and proves Hölder stability and uniqueness for recovering historical dynamics from terminal data $u(\mathbf{x},T)$ and $m(\mathbf{x},T)$, via three new Carleman estimates for variable-coefficient parabolic operators. The core contributions are the Carleman estimates for $\partial_t\pm L$, their application to a generalized retrospective framework, and the specialization to the specific MFGS of the corruption model, yielding explicit stability bounds and uniqueness. The results pave the way for developing globally convergent numerical methods to reconstruct historical trajectories of corrupted hierarchies from present observations, with potential applications in policy analysis and socio-economic modeling.

Abstract

A new mathematical model governing the development of a corrupted hierarchy is derived. This model is based on the Mean Field Games theory. A retrospective problem for that model is considered. From the applied standpoint, this problem amounts to figuring out the past activity of the corrupted hierarchy using the present data for this community. Three new Carleman estimates are derived. These estimates lead to Hölder stability estimates and uniqueness results for both that retrospective problem and its generalized version. Hölder stability estimates characterize the dependence of the error in the solution of the retrospective problem from the error in the input data.