Explicit solutions of some linear-quadratic mean field games

Abstract

We consider N -person differential games involving linear systems affected by white noise, running cost quadratic in the control and in the displacement of the state from a reference position, and with long-time-average integral cost functional. We solve an associated system of Hamilton-Jacobi-Bellman and Kolmogorov-Fokker-Plank equations and find explicit Nash equilibria in the form of linear feedbacks. Next we compute the limit as the number N of players goes to infinity, assuming they are almost identical and with suitable scalings of the parameters. This provides a quadratic-Gaussian solution to a system of two differential equations of the kind introduced by Lasry and Lions in the theory of Mean Field Games [19]. Under a natural normalization the uniqueness of this solution depends on the sign of a single parameter. We also discuss some singular limits, such as vanishing noise, cheap control, vanishing discount. Finally, we compare the L-Q model with other Mean Field models of population distribution.

Opinion Dynamics and Stubbornness Via Multi-Population Mean-Field Games

“…Finally, notice that the set (u * i (·, t), m * i (·, t)) t≥0 , solution to the optimization problem (14), is a Nash-Mean Field Equilibrium as described in Appendix…”

“…In the previous section we have derived myopic equilibrium strategies as solution of the optimal control problem (14). These equilibrium strategies govern the microscopic evolution of the players' opinions given in (1).…”

“…The relevance of population j over opinion of player k belonging to population i is given by the coefficient ν ij . 1 1 The choice of a logarithmic cost is commonly used when one wishes to describe a socalled crowd-seeking behavior on the part of the players (see, e.g., [14]). In this context, Note that ν ij = 0 if populations i and j are not directly connected, so that agents of population i are not directly influenced by agents in population j.…”

“…The underlying idea is that a large number of indistinguishable players interact so that the strategy of a single player is influenced by the distribution of the other players. Explicit solutions are available in [14] for the linear-quadratic case, and are extended to more general cases in [15]. However, in [16], it is pointed out that determining the solutions of a mean-field game is often impractical from a computational point of view; moreover, it does not naturally correspond to any reasonable dynamic process that agents are likely to follow in practice.…”

Mean-Field Games and Dynamic Demand Management in Power Grids

“…Explicit solution in terms of mean field equilibria are not common unless the problem has a linear-quadratic structure, see [7]. In this sense, a variety of solution schemes have been recently proposed based on discretization and or numerical approximations.…”