Equilibrium for Time-inconsistent Mean Field Games: A Systematic Analysis by Entropy Regularization

TL;DR

This paper develops a systematic entropy regularization method to analyze and approximate equilibria in complex time-inconsistent mean field games, ensuring existence and convergence of solutions.

Contribution

It introduces a novel entropy regularization approach for time-inconsistent MFGs, characterizes regularized equilibria, and proves convergence to true equilibria.

Findings

Established global existence of regularized equilibria.

Proved convergence of regularized solutions to original equilibria.

Proposed and analyzed a policy iteration algorithm for MFGs.

Abstract

This paper studies the existence and approximation of equilibria for general time-inconsistent mean field game (MFG) problems in the continuous-time setting. To handle the intricate nonlocal equilibrium Hamilton-Jacobi-Bellman (EHJB) system arising from initial-time dependence, such as non-exponential discounting, we develop a vanishing entropy regularization approach for solving the MFG. With entropy regularization, we first characterize the regularized equilibrium via a coupled exploratory equilibrium HJB (EEHJB) equation and a law-dependent stochastic differential equation. By exploiting Schauder fixed-point arguments and tailored parabolic regularity estimates in a suitable functional space involving both value functions and measure flows, we establish the global existence of regularized equilibria under mild assumptions. We next analyze convergence as the entropy regularization vanishes. By employing compactness arguments, Young measure techniques, and a duality tool for divergence-form Fokker-Planck equations, we prove that the regularized equilibria converge, up to subsequences, to an equilibrium of the original time-inconsistent MFG. Furthermore, under entropy regularization, we propose a policy iteration algorithm and establish its convergence under a short time horizon and weak terminal interaction conditions.