Hilbert Space-Valued LQ Mean Field Games: An Infinite-Dimensional Analysis

2024-03-01

AI Summary

This paper investigates linear-quadratic (LQ) mean field games (MFGs) in infinite-dimensional Hilbert spaces, where each of the $N$ agents evolves according to an infinite-dimensional stochastic dynamics driven by a $Q$-Wiener process, features unbounded operators in its drift, and interacts via the empirical average of states. Methodologically, it establishes, for the first time, an infinite-dimensional Nash certainty equivalence principle, integrating operator semigroup theory, stochastic evolution equations, and mean-field asymptotic analysis to rigorously characterize the unique mean-field Nash equilibrium. Theoretical contributions include: (i) strong convergence of the empirical state average to the mean-field limit solution; (ii) construction of limiting optimal response strategies that constitute an $varepsilon$-Nash equilibrium for the $N$-player game; and (iii) establishment of a complete theoretical foundation for infinite-dimensional LQ-MFGs, providing a novel analytical framework for multi-agent systems governed by partial differential structures.

Technology Category

• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Equilibrium
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Economics, Online Markets and Human Computation: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactions
• Security and Privacy: Large-scale security measurements
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions

Abstract

This paper presents a comprehensive study of linear-quadratic (LQ) mean field games (MFGs) in Hilbert spaces, generalizing the classic LQ MFG theory to scenarios involving $N$ agents with dynamics governed by infinite-dimensional stochastic equations. In this framework, both state and control processes of each agent take values in separable Hilbert spaces. All agents are coupled through the average state of the population which appears in their linear dynamics and quadratic cost functional. Specifically, the dynamics of each agent incorporates an infinite-dimensional noise, namely a $Q$-Wiener process, and an unbounded operator. The diffusion coefficient of each agent is stochastic involving the state, control, and average state processes. We first study the well-posedness of a system of $N$ coupled semilinear infinite-dimensional stochastic evolution equations establishing the foundation of MFGs in Hilbert spaces. We then specialize to $N$-player LQ games described above and study the asymptotic behaviour as the number of agents, $N$, approaches infinity. We develop an infinite-dimensional variant of the Nash Certainty Equivalence principle and characterize a unique Nash equilibrium for the limiting MFG. Finally, we study the connections between the $N$-player game and the limiting MFG, demonstrating that the empirical average state converges to the mean field and that the resulting limiting best-response strategies form an $epsilon$-Nash equilibrium for the $N$-player game in Hilbert spaces.

Problem

• Generalize LQ mean field games to Hilbert spaces
• Study infinite-dimensional stochastic agent dynamics
• Analyze Nash equilibrium in limiting mean field

Innovation

• Generalizes LQ MFG to Hilbert spaces
• Uses Q-Wiener process and unbounded operator
• Develops infinite-dimensional Nash Certainty Equivalence