LQG Mean Field Games with a Major Agent: Nash Certainty Equivalence versus Probabilistic Approach (2012.04866v3)

Abstract: Mean field game (MFG) systems consisting of a major agent and a large number of minor agents were introduced in (Huang, 2010) in an LQG setup. The Nash certainty equivalence was used to obtain a Markovian closed-loop Nash equilibrium for the limiting system when the number of minor agents tends to infinity. In the past years several approaches to major--minor mean field game problems have been developed, principally (i) the Nash certainty equivalence and analytic approach, (ii) master equations, (iii) asymptotic solvability, and (iv) the probabilistic approach. For the LQG case, the recent work (Huang, 2021) establishes the equivalency of the Markovian closed-loop Nash equilibrium obtained via (i) with those obtained via (ii) and (iii). In this work, we demonstrate that the Markovian closed-loop Nash equilibrium of (i) is equivalent to that of (iv) for the LQG case. These two studies answer the long-standing questions about the consistency of the solutions to major-minor LQG MFG systems derived using different approaches.

• The paper establishes that Nash certainty equivalence and probabilistic approaches yield equivalent Nash equilibria in MM LQG MFG systems.
• It employs detailed Riccati equations and FBSDEs analysis to derive optimal control laws for major and minor agents.
• The findings offer insights into strategic decision-making in economic and engineering domains under complex agent interactions.

Summary of "LQG Mean Field Games with a Major Agent: Nash Certainty Equivalence versus Probabilistic Approach"

Introduction

The paper investigates Linear Quadratic Gaussian (LQG) Mean Field Games (MFG) systems involving a major agent along with numerous minor agents. These systems pose challenges in finding Nash equilibria due to their complexity and the asymmetrical influence of major and minor agents on the system dynamics. The research primarily compares two methodologies for deriving Nash equilibria in such setups: the Nash certainty equivalence approach and the probabilistic approach. The outcomes demonstrate the equivalency of these procedures in terms of yielding consistent Nash equilibria for LQG MFG systems.

Finite-Population and Infinite-Population Dynamics

The paper differentiates between finite-population and infinite-population setups. In finite-population models, interactions among agents are discretely managed, whereas, in infinite-population models, the mean field approximation significantly influences the analysis. The paper presents detailed formulations of the systems and cost functionals for major and minor agents. In finite populations, minor agents collectively influence the system dynamics, causing challenges in maintaining the Markovian consistency of the Nash equilibria as the population tends towards infinity.

Nash Certainty Equivalence Approach

The Nash certainty equivalence approach involves deriving an a priori mean field and extending the agent states to encapsulate this mean field. For the major agent, this extends its state to a composite including the mean field dynamics, allowing for traditional optimal control methods to determine the best-response strategy. Similarly, minor agents' systems are extended by incorporating the major agent's state and the mean field, thereby simplifying the control problem to classical control derivations through Riccati equations. A series of consistency equations then solve for the mean field evolution, ensuring the optimal control actions are in equilibrium.

Probabilistic Approach

The probabilistic approach, in contrast, utilizes stochastic maximum principles to derive the control laws. Here, the system dynamics of minor agents treat the major agent's state and the mean field as exogenous factors. Through a fixed-point method in the space of control maps, it solves for Nash equilibria. The method involves detailed FBSDEs analysis with McKean-Vlasov dynamics and adopts linear adjoint process ansatzes to determine best response strategies.

Equivalence and Comparison

The paper establishes that both the Nash certainty equivalence and probabilistic approaches yield equivalent Nash equilibria for MM LQG MFG systems. The authors demonstrate the structural similarity of the control laws derived from both approaches, despite the methodological differences in solving the underlying stochastic differential equations. By aligning the consistency equations with respect to the major and minor agents' dynamics among the two approaches, the equivalency is mathematically substantiated.

Conclusion

This paper serves as a substantive analysis in reconciling varying methodologies for MM LQG MFG systems, confirming that despite the methodological diversity, the Nash equilibria derived remain consistent. Future research might focus on extending these findings to more generalized settings or exploring other types of interactions that deviate from the assumptions made in classical LQG frameworks. The implications are critical for advancing strategic decision-making frameworks in economic and engineering domains where game-theoretical considerations among large populations are prevalent.