3,987 research outputs found
Translation invariant mean field games with common noise
This note highlights a special class of mean field games in which the
coefficients satisfy a convolution-type structural condition. A mean field game
of this type with common noise is related to a certain mean field game without
common noise by a simple transformation, which permits a tractable construction
of a solution of the problem with common noise from a solution of the problem
without
Linear-Quadratic -person and Mean-Field Games with Ergodic Cost
We consider stochastic differential games with players, linear-Gaussian
dynamics in arbitrary state-space dimension, and long-time-average cost with
quadratic running cost. Admissible controls are feedbacks for which the system
is ergodic. We first study the existence of affine Nash equilibria by means of
an associated system of Hamilton-Jacobi-Bellman and
Kolmogorov-Fokker-Planck partial differential equations. We give necessary and
sufficient conditions for the existence and uniqueness of quadratic-Gaussian
solutions in terms of the solvability of suitable algebraic Riccati and
Sylvester equations. Under a symmetry condition on the running costs and for
nearly identical players we study the large population limit, tending to
infinity, and find a unique quadratic-Gaussian solution of the pair of Mean
Field Game HJB-KFP equations. Examples of explicit solutions are given, in
particular for consensus problems.Comment: 31 page
Mean field games with common noise
A theory of existence and uniqueness is developed for general stochastic
differential mean field games with common noise. The concepts of strong and
weak solutions are introduced in analogy with the theory of stochastic
differential equations, and existence of weak solutions for mean field games is
shown to hold under very general assumptions. Examples and counter-examples are
provided to enlighten the underpinnings of the existence theory. Finally, an
analog of the famous result of Yamada and Watanabe is derived, and it is used
to prove existence and uniqueness of a strong solution under additional
assumptions
On the convergence problem in Mean Field Games: a two state model without uniqueness
We consider N-player and mean field games in continuous time over a finite
horizon, where the position of each agent belongs to {-1,1}. If there is
uniqueness of mean field game solutions, e.g. under monotonicity assumptions,
then the master equation possesses a smooth solution which can be used to prove
convergence of the value functions and of the feedback Nash equilibria of the
N-player game, as well as a propagation of chaos property for the associated
optimal trajectories. We study here an example with anti-monotonous costs, and
show that the mean field game has exactly three solutions. We prove that the
value functions converge to the entropy solution of the master equation, which
in this case can be written as a scalar conservation law in one space
dimension, and that the optimal trajectories admit a limit: they select one
mean field game soution, so there is propagation of chaos. Moreover, viewing
the mean field game system as the necessary conditions for optimality of a
deterministic control problem, we show that the N-player game selects the
optimizer of this problem
- …