11,306 research outputs found
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
Quadratic Mean Field Games
Mean field games were introduced independently by J-M. Lasry and P-L. Lions,
and by M. Huang, R.P. Malham\'e and P. E. Caines, in order to bring a new
approach to optimization problems with a large number of interacting agents.
The description of such models split in two parts, one describing the evolution
of the density of players in some parameter space, the other the value of a
cost functional each player tries to minimize for himself, anticipating on the
rational behavior of the others.
Quadratic Mean Field Games form a particular class among these systems, in
which the dynamics of each player is governed by a controlled Langevin equation
with an associated cost functional quadratic in the control parameter. In such
cases, there exists a deep relationship with the non-linear Schr\"odinger
equation in imaginary time, connexion which lead to effective approximation
schemes as well as a better understanding of the behavior of Mean Field Games.
The aim of this paper is to serve as an introduction to Quadratic Mean Field
Games and their connexion with the non-linear Schr\"odinger equation, providing
to physicists a good entry point into this new and exciting field.Comment: 62 pages, 4 figure
On the existence of oscillating solutions in non-monotone Mean-Field Games
For non-monotone single and two-populations time-dependent Mean-Field Game
systems we obtain the existence of an infinite number of branches of
non-trivial solutions. These non-trivial solutions are in particular shown to
exhibit an oscillatory behaviour when they are close to the trivial (constant)
one. The existence of such branches is derived using local and global
bifurcation methods, that rely on the analysis of eigenfunction expansions of
solutions to the associated linearized problem. Numerical analysis is performed
on two different models to observe the oscillatory behaviour of solutions
predicted by bifurcation theory, and to study further properties of branches
far away from bifurcation points.Comment: 24 pages, 10 figure
An Extended Mean Field Game for Storage in Smart Grids
We consider a stylized model for a power network with distributed local power
generation and storage. This system is modeled as network connection a large
number of nodes, where each node is characterized by a local electricity
consumption, has a local electricity production (e.g. photovoltaic panels), and
manages a local storage device. Depending on its instantaneous consumption and
production rates as well as its storage management decision, each node may
either buy or sell electricity, impacting the electricity spot price. The
objective at each node is to minimize energy and storage costs by optimally
controlling the storage device. In a non-cooperative game setting, we are led
to the analysis of a non-zero sum stochastic game with players where the
interaction takes place through the spot price mechanism. For an infinite
number of agents, our model corresponds to an Extended Mean-Field Game (EMFG).
In a linear quadratic setting, we obtain and explicit solution to the EMFG, we
show that it provides an approximate Nash-equilibrium for -player game, and
we compare this solution to the optimal strategy of a central planner.Comment: 27 pages, 5 figures. arXiv admin note: text overlap with
arXiv:1607.02130 by other author
Mean Field Games models of segregation
This paper introduces and analyses some models in the framework of Mean Field
Games describing interactions between two populations motivated by the studies
on urban settlements and residential choice by Thomas Schelling. For static
games, a large population limit is proved. For differential games with noise,
the existence of solutions is established for the systems of partial
differential equations of Mean Field Game theory, in the stationary and in the
evolutive case. Numerical methods are proposed, with several simulations. In
the examples and in the numerical results, particular emphasis is put on the
phenomenon of segregation between the populations.Comment: 35 pages, 10 figure
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