Abstract
In this paper, we study a class of risk-sensitive mean-field stochastic differential games. We show that under appropriate regularity conditions, the mean-field value of the stochastic differential game with exponentiated integral cost functional coincides with the value function satisfying a Hamilton -Jacobi- Bellman (HJB) equation with an additional quadratic term. We provide an explicit solution of the mean-field best response when the instantaneous cost functions are log-quadratic and the state dynamics are affine in the control. An equivalent mean-field risk-neutral problem is formulated and the corresponding mean-field equilibria are characterized in terms of backward-forward macroscopic McKean-Vlasov equations, Fokker-Planck-Kolmogorov equations, and HJB equations. We provide numerical examples on the mean field behavior to illustrate both linear and McKean-Vlasov dynamics.
Highlights
We consider a class of n−person stochastic differential games, where Player j’s individual state, xnj, evolves according to the Itostochastic differential equation (S) as follows: n
Note that because of the symmetry assumption across players, the cost function of Player j is not indexed by j, since it is in the same structural form for all players
We note that the McKean-Vlasov mean field game considered here differs from the model in [16]; in this paper, the volatility term in (SM) is a function of state, control and the mean field, and further, the cost functional is of the risk-sensitive type
Summary
Most formulations of mean-field (MF) models such as anonymous sequential population games [19], [7], MF stochastic controls [17], [15], [36], MF optimization, MF teams [33], MF stochastic games [34], [1], [33], [31], MF stochastic difference games [14], and MF stochastic differential games [23], [13], [32] have been of risk-neutral type where the cost (or payoff, utility) functions to be minimized (or to be maximized) are the expected values of stage-additive loss functions. The particular risk-sensitive mean-field stochastic differential game that we consider in this paper involves an exponential term in the stochastic long-term cost function. This approach was first taken by Jacobson in [18], when considering the risk-sensitive Linear-Quadratic-Gaussian (LQG) problem with state feedback. Jacobson demonstrated a link between the exponential cost criterion and deterministic linear-quadratic differential games He showed that the risk-sensitive approach provides a method for varying the robustness of the controller and noted that in the case of no risk, or risk-neutral case, the well known LQR solution would result (see, for followup work on risk-sensitive stochastic control problems with noisy state measurements, [35], [6], [27]).
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