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LQG Risk-Sensitive Single-Agent and Major-Minor Mean-Field Game Systems: A Variational Framework

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Bibliographic Details
Title: LQG Risk-Sensitive Single-Agent and Major-Minor Mean-Field Game Systems: A Variational Framework
Authors: Hanchao Liu, Dena Firoozi, Michèle Breton
Source: SIAM Journal on Control and Optimization. 63:2251-2281
Publication Status: Preprint
Publisher Information: Society for Industrial & Applied Mathematics (SIAM), 2025.
Publication Year: 2025
Subject Terms: FOS: Economics and business, Quantitative Finance - Mathematical Finance, Optimization and Control (math.OC), Risk Management (q-fin.RM), Probability (math.PR), FOS: Mathematics, FOS: Electrical engineering, electronic engineering, information engineering, Systems and Control (eess.SY), Mathematics - Optimization and Control, Electrical Engineering and Systems Science - Systems and Control, Mathematical Finance (q-fin.MF), Mathematics - Probability, Quantitative Finance - Risk Management
Description: We develop a variational approach to address risk-sensitive optimal control problems with an exponential-of-integral cost functional in a general linear-quadratic-Gaussian (LQG) single-agent setup, offering new insights into such problems. Our analysis leads to the derivation of a nonlinear necessary and sufficient condition of optimality, expressed in terms of martingale processes. Subject to specific conditions, we find an equivalent risk-neutral measure, under which a linear state feedback form can be obtained for the optimal control. It is then shown that the obtained feedback control is consistent with the imposed condition and remains optimal under the original measure. Building upon this development, we (i) propose a variational framework for general LQG risk-sensitive mean-field games (MFGs) and (ii) advance the LQG risk-sensitive MFG theory by incorporating a major agent in the framework. The major agent interacts with a large number of minor agents, and unlike the minor agents, its influence on the system remains significant even with an increasing number of minor agents. We derive the Markovian closed-loop best-response strategies of agents in the limiting case where the number of agents goes to infinity. We establish that the set of obtained best-response strategies yields a Nash equilibrium in the limiting case and an $\varepsilon$-Nash equilibrium in the finite-player case.
Document Type: Article
Language: English
ISSN: 1095-7138
0363-0129
DOI: 10.1137/23m1595734
DOI: 10.48550/arxiv.2305.15364
Access URL: http://arxiv.org/abs/2305.15364
Rights: arXiv Non-Exclusive Distribution
Accession Number: edsair.doi.dedup.....7f9c735d46cfe79c14ee08d6acae2bc4
Database: OpenAIRE
Description
Abstract:We develop a variational approach to address risk-sensitive optimal control problems with an exponential-of-integral cost functional in a general linear-quadratic-Gaussian (LQG) single-agent setup, offering new insights into such problems. Our analysis leads to the derivation of a nonlinear necessary and sufficient condition of optimality, expressed in terms of martingale processes. Subject to specific conditions, we find an equivalent risk-neutral measure, under which a linear state feedback form can be obtained for the optimal control. It is then shown that the obtained feedback control is consistent with the imposed condition and remains optimal under the original measure. Building upon this development, we (i) propose a variational framework for general LQG risk-sensitive mean-field games (MFGs) and (ii) advance the LQG risk-sensitive MFG theory by incorporating a major agent in the framework. The major agent interacts with a large number of minor agents, and unlike the minor agents, its influence on the system remains significant even with an increasing number of minor agents. We derive the Markovian closed-loop best-response strategies of agents in the limiting case where the number of agents goes to infinity. We establish that the set of obtained best-response strategies yields a Nash equilibrium in the limiting case and an $\varepsilon$-Nash equilibrium in the finite-player case.
ISSN:10957138
03630129
DOI:10.1137/23m1595734