Risk-Sensitive Mean Field Games via the Stochastic Maximum Principle

Jun Moon, Tamer Başar

Research output: Contribution to journalArticlepeer-review


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 discuss extensions to heterogeneous (non-symmetric) risk-sensitive mean field games.

Original languageEnglish (US)
Pages (from-to)1100-1125
Number of pages26
JournalDynamic Games and Applications
Issue number4
StatePublished - Dec 1 2019


  • Decentralized control
  • Forward–backward stochastic differential equations
  • Mean field game theory
  • Risk-sensitive optimal control

ASJC Scopus subject areas

  • Statistics and Probability
  • Economics and Econometrics
  • Computer Science Applications
  • Computer Graphics and Computer-Aided Design
  • Computational Theory and Mathematics
  • Computational Mathematics
  • Applied Mathematics


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