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

Mean field game systems consisting of a major agent and a large population of minor agents were introduced in (Huang, 2010) in an LQG setup. In the past years several approaches towards major-minor mean field games have been developed, principally (i) the Nash certainty equivalence (Huang, 2010), (ii) master equations, (iii) asymptotic solvability, and (iv) the probabilistic approach. In a recent work (Huang, 2020), for the LQG case the equivalence of the solutions obtained via approaches (i)-(iii) was established. In this work we demonstrate that the closed-loop Nash equilibrium derived in the infinite-population limit through approaches (i) and (iv) are identical

Large Banks and Systemic Risk: Insights from a Mean-Field Game Model

This paper aims to investigate the impact of large banks on the financial system stability. To achieve this, we employ a linear-quadratic-Gaussian (LQG) mean-field game (MFG) model of an interbank market, which involves one large bank and multiple small banks. Our approach involves utilizing the MFG methodology to derive the optimal trading strategies for each bank, resulting in an equilibrium for the market. Subsequently, we conduct Monte Carlo simulations to explore the role played by the large bank in systemic risk under various scenarios. Our findings indicate that while the major bank, if its size is not too large, can contribute positively to stability, it also has the potential to generate negative spillover effects in the event of default, leading to increased systemic risk. We also discover that as banks become more reliant on the interbank market, the overall system becomes more stable but the probability of a rare systemic failure increases. This risk is further amplified by the presence of a large bank, its size, and the speed of interbank trading. Overall, the results of this study provide important insights into the management of systemic risk

LQG Risk-Sensitive Mean Field Games with a Major Agent: A Variational Approach

Risk sensitivity plays an important role in the study of finance and economics as risk-neutral models cannot capture and justify all economic behaviors observed in reality. Risk-sensitive mean field game theory was developed recently for systems where there exists a large number of indistinguishable, asymptotically negligible and heterogeneous risk-sensitive players, who are coupled via the empirical distribution of state across population. In this work, we extend the theory of Linear Quadratic Gaussian risk-sensitive mean-field games to the setup where there exists one major agent as well as a large number of minor agents. The major agent has a significant impact on each minor agent and its impact does not collapse with the increase in the number of minor agents. Each agent is subject to linear dynamics with an exponential-of-integral quadratic cost functional. Moreover, all agents interact via the average state of minor agents (so-called empirical mean field) and the major agent's state. We develop a variational analysis approach to derive the 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. We conclude the paper with an illustrative example