Linear-Quadratic Mean Field Games

By Alain Bensoussan, Joseph Sung, Phillip S. C. Yam, Siu Pang Yung. Journal of Optimization Theory and Applications, 2014

314 citations recorded by the source corpus.

We provide a comprehensive study of a general class of linear-quadratic mean field games. We adopt the adjoint equation approach to investigate the unique existence of their equilibrium strategies. Due to the linearity of the adjoint equations, the optimal mean field term satisfies a forward–backward ordinary differential equation. For the one-dimensional case, we establish the unique existence of the equilibrium strategy. For a dimension greater than one, by applying the Banach fixed point theorem under a suitable norm, a sufficient condition for the unique existence of the equilibrium strategy is provided, which is independent of the coefficients of controls in the underlying dynamics and is always satisfied whenever the coefficients of the mean field term are vanished, and hence, our theories include the classical linear-quadratic stochastic control problems as special cases. As a by-product, we also establish a neat and instructive sufficient condition, which is apparently absent in the literature and only depends on coefficients, for the unique existence of the solution for a class of nonsymmetric Riccati equations. Numerical examples of nonexistence of the equilibrium strategy will also be illustrated. Finally, a similar approach has been adopted to study the linear-quadratic mean field type stochastic control problems and their comparisons with mean field games.

Subfields: Demography, Economics and Econometrics, Finance

View Paper: https://www.semanticscholar.org/paper/46804ef8e3432e42414739fe0ec5077a0601372d

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Citations in the corpus

Mean Field Games with a Dominating Player (2014, background): “3 in Ma and Young [15] or [1], we have the following proposition.”
A probabilistic weak formulation of mean field games and applications (2013, background): “[11] and [6]) in that the forward SDEs no longer need to be solved in a strong sense.”
A general characterization of the mean field limit for stochastic differential games (2014, methodology): “It is exactly the definition used in the recent papers [15, 29], and it is a generalization of the more standard definition of MFG solution without common noise found in [21, 8, 6], for example.”
Linear Quadratic Optimal Control Problems for Mean-Field Backward Stochastic Differential Equations (2016, background): “Bensoussan–Sung–Yam [3], Bensoussan–Sung–Yam–Yung [4], and Graber [16] analyzed mean-field linearquadratic control problems and games with their strategies.”
Mean field and n‐agent games for optimal investment under relative performance criteria (2017, background): “Beyond the linear quadratic models of [5, 15, 17], such examples are scarce, especially in the presence of common noise.”
Convergence, fluctuations and large deviations for finite state mean field games via the Master Equation (2017, background): “Convergence goes in the opposite direction: are Nash equilibria for the N -player game converging to solutions of the Mean Field Game when the number of players goes to infinity? Results in the first direction are much more common and easier to obtain: for the diffusive case without jumps see for in”
Bellman equation and viscosity solutions for mean-field stochastic control problem (2015, unknown): “We apply our veriﬁcation theorem to the important class of linear quadratic (LQ) McKean − Vlasov control problems, addressed e.g. in [7,38] by maximum principle and adjoint equations, and that we solve by a diﬀerent approach where it turns out that derivations in the space of probability measures…”
Linear Quadratic Mean Field Type Control and Mean Field Games with Common Noise, with Application to Production of an Exhaustible Resource (2016, methodology): “[10], which deals both with mean field games and mean field type control problems.”
Mean Field Games and Systemic Risk (2013, background): “Notice that linear quadratic MFGs are studied in [2] and [6].”
Well-posedness of mean-field type forward–backward stochastic differential equations (2015, unknown): “Mean-field type forward-backward stochastic differential equations (MFTFBSDEs) are forward-backward stochastic differential equations in which the coefficients involved could also depend upon the distribution of the solution triple (X,Y, Z); see [1], [2], [3] [4], and the references therein for more”
Mean field games via controlled martingale problems: Existence of Markovian equilibria (2014, background): “This stochastic approach leads to a notably clean solution of linear-quadratic mean field games, as in [4, 13], as well as an interesting generalization of mean field games involving one dominant agent competing with a mean field of minor agents [27, 8].”
Mean Field Games Models—A Brief Survey (2013, background): “Linear quadratic problems have been considered from distinct points of view, for instance, in [8, 12, 15, 61, 62, 80], and [76].”
Forward-Backward Stochastic Differential Equations and Controlled McKean Vlasov Dynamics (2013, background): “The mean variance portfolio optimization example discussed in [2] and the solution proposed in [3] and [8] of the optimal control of linear-quadratic (LQ) McKean-Vlasov dynamics are based on the general form of the Pontryagin principle proven in this section as applied to the scalar interactions con”
Probabilistic Analysis of Mean-Field Games (2012, methodology): “See also [7] and [2] for the linear quadratic case.”