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Unique determination of cost functions in a multipopulation mean field game model

Kui Ren, Nathan Soedjak, Kewei Wang

TL;DR

This work addresses the inverse problem of uniquely determining running costs $\mathbf F$ and terminal costs $\mathbf G$ in a multipopulation mean field game. It develops a multilinearization framework based on a holomorphic dependence of the forward solution on initial data to reduce the nonlinear inverse problem to a sequence of linearized problems and to compare Taylor coefficients. The authors prove uniqueness for both multipopulation data and, under two structured cost-function classes, single-population data, with substantial use of orthogonality relations recovered from adjoint-type testing and Fourier analysis. The results illuminate when cost-function recovery is possible from limited observations and lay groundwork for extending these ideas to more general domains and coupling structures, with implications for parameter identification in large-scale interacting-population models.

Abstract

This paper studies an inverse problem for a multipopulation mean field game (MFG) system where the objective is to reconstruct the running and terminal cost functions of the system that couples the dynamics of different populations. We derive uniqueness results for the inverse problem with different types of available data. In particular, we show that it is possible to uniquely reconstruct some simplified forms of the cost functions from data measured only on a single population component under mild additional assumptions on the coupling mechanism. The proofs are based on the standard multilinearization technique that allows us to reduce the inverse problems into simplified forms.

Unique determination of cost functions in a multipopulation mean field game model

TL;DR

This work addresses the inverse problem of uniquely determining running costs and terminal costs in a multipopulation mean field game. It develops a multilinearization framework based on a holomorphic dependence of the forward solution on initial data to reduce the nonlinear inverse problem to a sequence of linearized problems and to compare Taylor coefficients. The authors prove uniqueness for both multipopulation data and, under two structured cost-function classes, single-population data, with substantial use of orthogonality relations recovered from adjoint-type testing and Fourier analysis. The results illuminate when cost-function recovery is possible from limited observations and lay groundwork for extending these ideas to more general domains and coupling structures, with implications for parameter identification in large-scale interacting-population models.

Abstract

This paper studies an inverse problem for a multipopulation mean field game (MFG) system where the objective is to reconstruct the running and terminal cost functions of the system that couples the dynamics of different populations. We derive uniqueness results for the inverse problem with different types of available data. In particular, we show that it is possible to uniquely reconstruct some simplified forms of the cost functions from data measured only on a single population component under mild additional assumptions on the coupling mechanism. The proofs are based on the standard multilinearization technique that allows us to reduce the inverse problems into simplified forms.
Paper Structure (18 sections, 6 theorems, 100 equations)

This paper contains 18 sections, 6 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Setup of notations
  3. The forward problem and its linearization
  4. Well-posedness of the forward problem
  5. Multi-linearizations
  6. Reconstructing from multipopulation data
  7. Reconstructions from single-population data
  8. Population-independent cost functions
  9. State-independent cost functions
  10. Assumption \ref{['ASS:m']}.
  11. Case 1: $\alpha_1=1$.
  12. Case 2: $\alpha_1 = 0$.
  13. Assumption \ref{['ASS:Laplacian u']}.
  14. Assumption \ref{['ASS:u']}.
  15. Case 1: $\alpha_1=1$.
  16. ...and 3 more sections

Key Result

Lemma 3.1

Suppose $f,\tilde{f}\in C^{\alpha,\frac{\alpha}{2}}(Q)$, $F_k\in C^{\alpha}(\mathbb T^d)$, and $g,\tilde{g},G_k\in C^{2+\alpha}(\mathbb T^d)$ for some $\alpha\in(0,1)$, then the system admits a unique solution $(\mathbf u,\bm)\in [C^{2+\alpha,1+\frac{\alpha}{2}}(Q)]^{2n}$.

Theorems & Definitions (13)

  • Definition 2.1
  • Lemma 3.1
  • Theorem 3.2
  • Lemma 4.1
  • proof
  • Theorem 4.2
  • proof
  • Theorem 5.1
  • proof
  • Remark 5.2
  • ...and 3 more