Risk-Sensitive Mean-Field Games

Risk-Sensitive Mean-Field Games

H. Tembine, Q. Zhu, T. Basar, Risk-Sensitive Mean-Field Games, IEEE Transactions on Automatic Control, volume 59, Issue 4, April 2014.
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H. Tembine, Q. Zhu, T. Ba
Risk-Sensitive, Mean-Field Games, Risk-Seeking, Risk-Averse
2014
n 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 functionalcoincides 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.
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