Long time average of mean field games

Cardaliaguet, Pierre; Lasry, Jean-Michel; Lions, Pierre-Louis; Porretta, Alessio

doi:10.3934/nhm.2012.7.279

Cited by 130 publications

(151 citation statements)

References 10 publications

Supporting

Mentioning

149

Contrasting

Order By: Relevance

“…makes the model differ from the standard formulation of mean field games in [13] where the terms depending on m and on the control are decoupled. In addition, here the control set is bounded.…”

Section: Model and Problem Set-upmentioning

confidence: 99%

“…Second, even if the optimal feedback exists, it is often discontinuous and so the field f (·, u * (·, ·)) is discontinuous too. We must then consider solutions m in the distributional sense (see for instance [13]). Moreover, in our particular case, as we will see in Section 4, we need to consider distributions m which are not absolutely continuous with respect to the Lebesgue measure, that is, distributions that cannot be defined through a function.…”

Section: Weak Solutions In the System (5)mentioning

confidence: 99%

“…In addition, the provided game differs from a standard mean field game in [13] in at least two other aspects: first both controls and states are bounded, and second we have an additional cross-term on distribution and controls in the objective function, which is not monotonic on distributions.…”

Section: Introductionmentioning

confidence: 99%

“…While in the former case, the resulting mean field game should be justified by a limit procedure of Nash equilibria of games with a finite number of players, in the latter case such a limit procedure is no longer needed. On one hand, this simplifies the tractability as, except for some special cases, such a procedure is still far from being understood and satisfactory, see [13] for a comprehensive account. On the other hand, the formation of atomic parts in the distribution requires a suitably modified weak solution concept, which is formally defined in Section 3.…”

Section: Introductionmentioning

confidence: 99%

“…The second PDE is the Fokker-Planck equation which describes the density of the players and is solved forwards in time with boundary conditions on the initial population distribution (see, e.g., the Fokker-Planck-Kolmogorov equation in [2,25,32,36] and in the lecture notes [13] and M.S. thesis [27]).…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Mean-Field Games and Dynamic Demand Management in Power Grids

Bagagiolo

Bauso

2013

Dyn Games Appl

View full text Add to dashboard Cite

This paper applies mean field game theory to dynamic demand management. For a large population of electrical heating or cooling appliances (called agents), we provide a mean field game that guarantees desynchronization of the agents thus improving the power network resilience. Second, for the game at hand, we exhibit a mean field equilibrium, where each agent adopts a bang-bang switching control with threshold placed at a nominal temperature. At the equilibrium, through an opportune design of the terminal penalty, the switching control regulates the mean temperature (computed over the population) and the mains frequency around the nominal value. To overcome Zeno phenomena we also adjust the bang-bang control by introducing a thermostat. Third, we show that the equilibrium is stable in the sense that all agents' states, initially at different values, converge to the equilibrium value or remain confined within a given interval for an opportune initial distribution.

show abstract

Section: Model and Problem Set-upmentioning

confidence: 99%

Section: Weak Solutions In the System (5)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Mean-Field Games and Dynamic Demand Management in Power Grids

Bagagiolo

Bauso

2013

Dyn Games Appl

View full text Add to dashboard Cite

show abstract

Variational Mean Field Games for Market Competition

Graber¹,

Mouzouni²

2018

Springer INdAM Series

View full text Add to dashboard Cite

An Introduction to Mean Field Game Theory

Cardaliaguet

Porretta

2020

Lecture Notes in Mathematics

Self Cite

View full text Add to dashboard Cite

We study the behavior of solutions to the first-order mean field games system with a local coupling, when the initial density is a compactly supported function on the real line. Our results show that the solution is smooth in regions where the density is strictly positive, and that the density itself is globally continuous. Additionally, the speed of propagation is determined by the behavior of the cost function near small values of the density. When the coupling is entropic, we demonstrate that the support of the density propagates with infinite speed. On the other hand, for a power-type coupling, we establish finite speed of propagation, leading to the formation of a free boundary. We prove that under a natural nondegeneracy assumption, the free boundary is strictly convex and enjoys C 1,1 regularity. We also establish sharp estimates on the speed of support propagation and the rate of long time decay for the density. Moreover, the density and the gradient of the value function are both shown to be Hölder continuous up to the free boundary. Our methods are based on the analysis of a new elliptic equation satisfied by the flow of optimal trajectories. The results also apply to mean field planning problems, characterizing the structure of minimizers of a class of optimal transport problems with congestion.

show abstract

Long time average of mean field games

Cited by 130 publications

References 10 publications

Mean-Field Games and Dynamic Demand Management in Power Grids

Mean-Field Games and Dynamic Demand Management in Power Grids

Variational Mean Field Games for Market Competition

An Introduction to Mean Field Game Theory

Contact Info

Product

Resources

About