Risk-Sensitive Mean Field Games via the Stochastic Maximum Principle

Jun Moon () and Tamer Başar ()
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Jun Moon: Ulsan National Institute of Science and Technology (UNIST)
Tamer Başar: University of Illinois at Urbana-Champaign

Dynamic Games and Applications, 2019, vol. 9, issue 4, No 11, 1100-1125

Abstract: Abstract 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 $$\epsilon $$ϵ-Nash equilibrium for the N player risk-sensitive game, where $$\epsilon \rightarrow 0$$ϵ→0 as $$N \rightarrow \infty $$N→∞ at the rate of $$O(\frac{1}{N^{1/(n+4)}})$$O(1N1/(n+4)). Finally, we discuss extensions to heterogeneous (non-symmetric) risk-sensitive mean field games.

Keywords: Mean field game theory; Risk-sensitive optimal control; Forward–backward stochastic differential equations; Decentralized control (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1007/s13235-018-00290-z

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