Risk-sensitive mean-field games

Hamidou Tembine, Quanyan Zhu, Tamer Başar

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Article number6656891
Pages (from-to)835-850
Number of pages16
JournalIEEE Transactions on Automatic Control
Issue number4
StatePublished - Apr 2014


  • Decentralized control
  • H infinity control

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Computer Science Applications
  • Electrical and Electronic Engineering


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