3 research outputs found
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
-Nash equilibrium in the finite player case. We conclude the paper
with an illustrative example
Risk-Sensitive Mean Field Games via the Stochastic Maximum Principle
In this paper, we consider risk-sensitive mean field games via the risk-sensitive maximum principle. The problem is analyzed through two sequential steps: (i) risk-sensitive optimal control for a fixed probability measure, and (ii) the associated fixed-point problem. For step (i), we use the risk-sensitive maximum principle to obtain the optimal solution, which is characterized in terms of the associated forward-backward stochastic differential equation (FBSDE). In step (ii), we solve for the probability law induced by the state process with the optimal control in step (i). In particular, we show the existence of the fixed point of the probability law of the state process determined by step (i) via Schauder???s fixed-point theorem. After analyzing steps (i) and (ii), we prove that the set of N optimal distributed controls obtained from steps (i) and (ii) constitutes an approximate Nash equilibrium or ϵ -Nash equilibrium for the N player risk-sensitive game, where ϵ???0 as N?????? at the rate of O(1N1/(n+4)) . Finally, we discuMean field game theory Risk-sensitive optimal control Forward-backward stochastic differential equations Decentralized control ss extensions to heterogeneous (non-symmetric) risk-sensitive mean field games