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Decentralized Strategies for Backward Linear-Quadratic Mean Field Games and Teams

Yu Si, Jingtao Shi

TL;DR

The paper addresses a new class of linear-quadratic mean field games and social optima in finite populations whose dynamics are governed by $N$ weakly coupled linear backward stochastic differential equations (BSDEs) with the $z_i$-component entering the dynamics and costs. It uses the stochastic maximum principle and optimal filtering to derive a fully coupled Hamiltonian forward–backward SDE and then decouples it via Riccati equations, an SDE, and a BSDE to obtain decentralized feedback strategies. The key contributions include the introduction of backward separation for backward MFGs/teams with $z$-driven coupling and an explicit Riccati/SDE/BSDE framework for both Nash and social-optimal problems, enabling exact (or near-exact in large populations) decentralized control laws. A numerical example demonstrates feasibility and illustrates the evolution of states, adjoints, and controls under the proposed schemes.

Abstract

This paper studies a new class of linear-quadratic mean field games and teams problem, where the large-population system satisfies a class of $N$ weakly coupled linear backward stochastic differential equations (BSDEs), and $z_i$ (a part of solution of BSDE) enter the state equations and cost functionals. By virtue of stochastic maximum principle and optimal filter technique, we obtain a Hamiltonian system first, which is a fully coupled forward-backward stochastic differential equation (FBSDE). Decoupling the Hamiltonian system, we derive a feedback form optimal strategy by introducing Riccati equations, stochastic differential equation (SDE) and BSDE. Finally, we provide a numerical example to simulate our results.

Decentralized Strategies for Backward Linear-Quadratic Mean Field Games and Teams

TL;DR

The paper addresses a new class of linear-quadratic mean field games and social optima in finite populations whose dynamics are governed by weakly coupled linear backward stochastic differential equations (BSDEs) with the -component entering the dynamics and costs. It uses the stochastic maximum principle and optimal filtering to derive a fully coupled Hamiltonian forward–backward SDE and then decouples it via Riccati equations, an SDE, and a BSDE to obtain decentralized feedback strategies. The key contributions include the introduction of backward separation for backward MFGs/teams with -driven coupling and an explicit Riccati/SDE/BSDE framework for both Nash and social-optimal problems, enabling exact (or near-exact in large populations) decentralized control laws. A numerical example demonstrates feasibility and illustrates the evolution of states, adjoints, and controls under the proposed schemes.

Abstract

This paper studies a new class of linear-quadratic mean field games and teams problem, where the large-population system satisfies a class of weakly coupled linear backward stochastic differential equations (BSDEs), and (a part of solution of BSDE) enter the state equations and cost functionals. By virtue of stochastic maximum principle and optimal filter technique, we obtain a Hamiltonian system first, which is a fully coupled forward-backward stochastic differential equation (FBSDE). Decoupling the Hamiltonian system, we derive a feedback form optimal strategy by introducing Riccati equations, stochastic differential equation (SDE) and BSDE. Finally, we provide a numerical example to simulate our results.
Paper Structure (8 sections, 14 theorems, 103 equations, 4 figures)

This paper contains 8 sections, 14 theorems, 103 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Mean field game problem
  4. Mean field social optima problem
  5. Optimal strategy of mean field social optima problem
  6. Comparison with classical mean field social optima problem
  7. Numerical examples
  8. Conclusion

Key Result

Theorem 3.1

Assume (A1) and (A2 of game) hold. Then Problem (game problem) has a Nash equilibrium strategy $u^*(\cdot)\equiv(u^*_1(\cdot),\cdots,u^*_N(\cdot))$, $u^*_k(\cdot) \in \mathcal{U}^{ad}_k$ if and only if the following Hamiltonian system admits a set of solutions $(x^*_k(\cdot),z^*_k(\cdot),p^*_k(\cdot and the optimal strategy $u^*_k(\cdot)$ of agent $\mathcal{A}_k$ satisfies the stationary condition

Figures (4)

  • Figure 1: The solution curve of $\Sigma(\cdot)$, $K(\cdot)$, $\Pi(\cdot)$ and $M(\cdot)$
  • Figure 2: The solution curve of $\zeta^*_i(\cdot)$, $i=1,\cdots,30$
  • Figure 3: The solution curve of $x^*_i(\cdot)$, $i=1,\cdots,30$
  • Figure 4: The solution curve of $u^*_i(\cdot)$, $i=1,\cdots,30$

Theorems & Definitions (25)

  • Theorem 3.1
  • proof
  • Lemma 3.1
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Remark 3.1
  • Proposition 3.3
  • ...and 15 more