Linear Quadratic Mean Field Games under Heterogeneous Erroneous Initial Information

TL;DR

This work addresses the robustness of linear quadratic mean field games to heterogeneous erroneous initial information when no direct mean-field observations are available. It introduces an error-propagation framework that yields linear relationships between initial errors and deviations in the predicted mean field, controls, and actual trajectories, applicable both deterministically and stochastically. It then develops two practical remedies: a one-time error-correction mechanism that recovers the true mean field at a chosen time and reinitializes the equilibrium, and a real-time estimation framework that updates predictions and strategies as new information arrives. Simulations corroborate a linear dependence of deviations on initial errors and demonstrate that the proposed correction and estimation procedures effectively align the system with the correct-information equilibrium, offering a path toward robust large-scale strategic planning under imperfect information.

Abstract

In this paper, linear quadratic mean field games (LQMFGs) under heterogeneous erroneous initial information are investigated, focusing on how to achieve error correction by calculation based on the agents' own actual state and interactions in the game, rather than process observations. First, we establish a mathematic model for initial information error propagation in LQMFGs, several all-agents-known linear relationships between initial errors and deviations of agents' strategies and MF from those under correct information are given. Next, we investigate the error correction and strategy modification behavior of an agent and corresponding methods that only requires it own states. Under deterministic situation, a sufficient condition is provided for agents to compute actual MF and optimal strategies by one-time error correction, which is only related to modification time and parameters of the system. Under stochastic situation, the mathematical model of agents' real-time estimations for MF and corresponding strategies are given, and estimation error affections are analysed. Finally, simulations are performed to verify above conclusions.

Figures (7)

• Figure 1: At the initial moment, agents get different information about the initial mean field state. The correct initial mean field state is marked by red.
• Figure 2: MF-S in predictions under correct information and that under erroneous information, $\bar{E}=(0,0)^T$. (a) Predicted MF-S under correct information. (b) Predicted MF-S under erroneous information.
• Figure 3: Deviations $\Delta \bar{z}(t)$ under mean error $\bar{E}=k*(0.1,-0.1)^T$, the relationship between $k$ and $\Delta \bar{z}(t)$ for fixed $t=0.25,1,1.75$. (a) $\Delta\bar{z}(t)$ under different mean error. (b) $\Delta\bar{z}(t)$ for fixed $t$.
• Figure 4: Actual trajectories under correct information and erroneous information, where $\bar{E}=k*(0.1,-0.1)^T$. (a) Trajectories under correct information. (b) Actual trajectories under erroneous information, $k=0$. (c) Actual trajectories under erroneous information, $k=2$. (d) Actual trajectories under erroneous information, $k=4$.
• Figure 5: Deviations $x^{(N)}(t)-z^c(t)$, $\Delta z^A(t)$ under $\bar{E}=k*(0.1,-0.1)^T, k=1,2,3,4$, the relationship between $k$ and deviations $x^{(N)}(t)-z^c(t)$ for fixed $t=0.25,1,1.75$. (a) $x^{(N)}(t)-z^c(t)$ under different $\bar{E}$. (b) $\Delta z^A(t)$ under different $\bar{E}$. (c) $x^{(N)}(t)-z^c(t)$ for fixed $t$.