scholarly journals Short time solution to the master equation of a first order mean field game

2020 ◽  
Vol 268 (10) ◽  
pp. 6251-6318
Author(s):  
Sergio Mayorga
Keyword(s):  
Master Equation ◽  
Mean Field ◽  
Mean Field Game ◽  
First Order ◽  
Time Solution ◽  
Short Time

The Second-Order Master Equation

2019 ◽  
pp. 128-158
Author(s):  
Pierre Cardaliaguet ◽  
François Delarue ◽  
Jean-Michel Lasry ◽  
Pierre-Louis Lions
Keyword(s):  
Master Equation ◽  
Mean Field ◽  
Second Order ◽  
Well Posedness ◽  
Mean Field Game ◽  
First Order ◽  
Common Noise ◽  
System P ◽  
The Mean

This chapter investigates the second-order master equation with common noise, which requires the well-posedness of the mean field game (MFG) system. It also defines and analyzes the solution of the master equation. The chapter explains the forward component of the MFG system that is recognized as the characteristics of the master equation. The regularity of the solution of the master equation is explored through the tangent process that solves the linearized MFG system. It also analyzes first-order differentiability and second-order differentiability in the direction of the measure on the same model as for the first-order derivatives. This chapter concludes with further description of the derivation of the master equation and well-posedness of the stochastic MFG system.

scholarly journals Viability analysis of the first-order mean field games

2020 ◽  
Vol 26 ◽  
pp. 33
Author(s):  
Yurii Averboukh
Keyword(s):  
Mean Field ◽  
Initial Distribution ◽  
Initial Time ◽  
Mean Field Games ◽  
Sufficient Condition ◽  
Mean Field Game ◽  
First Order ◽  
Viability Analysis ◽  
A Value ◽  
Object Of Study

In the paper, we examine the dependence of the solution of the deterministic mean field game on the initial distribution of players. The main object of study is the mapping which assigns to the initial time and the initial distribution of players the set of expected rewards of the representative player corresponding to solutions of mean field game. This mapping can be regarded as a value multifunction. We obtain the sufficient condition for a multifunction to be a value multifunction. It states that if a multifunction is viable with respect to the dynamics generated by the original mean field game, then it is a value multifunction. Furthermore, the infinitesimal variant of this condition is derived.

Presentation of the Main Results

2019 ◽  
pp. 28-47
Author(s):  
Pierre Cardaliaguet ◽  
François Delarue ◽  
Jean-Michel Lasry ◽  
Pierre-Louis Lions
Keyword(s):  
Master Equation ◽  
Classical Solution ◽  
Mean Field ◽  
Approximate Solutions ◽  
Unique Classical Solution ◽  
Mean Field Game ◽  
The Mean ◽  
Derivatives Of

This chapter collects the main results of the master equation for the convergence of the Nash system. It explains the notation used, specifies the notion of derivatives in the space of measures, and describes the assumptions on the data. One of the striking features of the master equation is that it involves derivatives of the unknown with respect to the measure. This chapter also discusses the link between the two notions of derivatives, which have been used in the mean field game (MFG) theory. The main result states that the master equation has a unique classical solution under the regularity and monotonicity assumptions on H, F, and G. Once the master equation has a solution, this solution can be used to build approximate solutions for the Nash system with N-players.

scholarly journals Long time behavior of the master equation in mean field game theory

2019 ◽  
Vol 12 (6) ◽  
pp. 1397-1453 ◽  
Author(s):  
Pierre Cardaliaguet ◽  
Alessio Porretta
Keyword(s):  
Game Theory ◽  
Master Equation ◽  
Mean Field ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Mean Field Game ◽  
Long Time

A Starter: The First-Order Master Equation

2019 ◽  
pp. 48-84
Author(s):  
Pierre Cardaliaguet ◽  
François Delarue ◽  
Jean-Michel Lasry ◽  
Pierre-Louis Lions
Keyword(s):  
Master Equation ◽  
Preliminary Analysis ◽  
Initial Conditions ◽  
Mean Field ◽  
Mean Field Games ◽  
Lipschitz Continuous ◽  
First Order ◽  
Common Noise ◽  
The Mean ◽  
Uniformly Lipschitz

This chapter contains a preliminary analysis of the master equation in the simpler case when there is no common noise. Some of the proofs given in this chapter consist of a sketch only. One of the reasons is that some of the arguments used to investigate the mean field games (MFGs) system have been already developed in the literature. Another reason is that the chapter constitutes a starter only, specifically devoted to the simpler case without common noise. It provides details of the global Lipschitz continuity of H. The solutions of the MFG system are uniformly Lipschitz continuous, which are independently of initial conditions.

scholarly journals Mean Field Game with Delay: A Toy Model

Risks ◽  
2018 ◽  
Vol 6 (3) ◽  
pp. 90
Author(s):  
Jean-Pierre Fouque ◽  
Zhaoyu Zhang
Keyword(s):  
Master Equation ◽  
Dimensional Space ◽  
Mean Field ◽  
Kolmogorov Equation ◽  
Linear Quadratic ◽  
Quadratic Mean ◽  
Mean Field Game ◽  
Infinite Dimensional ◽  
Player Game ◽  
Hamilton Jacobi Bellman

We study a toy model of linear-quadratic mean field game with delay. We “lift” the delayed dynamic into an infinite dimensional space, and recast the mean field game system which is made of a forward Kolmogorov equation and a backward Hamilton-Jacobi-Bellman equation. We identify the corresponding master equation. A solution to this master equation is computed, and we show that it provides an approximation to a Nash equilibrium of the finite player game.

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