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2013 | Buch

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Mean Field Games and Mean Field Type Control Theory

verfasst von: Alain Bensoussan, Jens Frehse, Phillip Yam

Verlag: Springer New York

Buchreihe : SpringerBriefs in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Mean field games and Mean field type control introduce new problems in Control Theory. The terminology “games” may be confusing. In fact they are control problems, in the sense that one is interested in a single decision maker, whom we can call the representative agent. However, these problems are not standard, since both the evolution of the state and the objective functional is influenced but terms which are not directly related to the state or the control of the decision maker. They are however, indirectly related to him, in the sense that they model a very large community of agents similar to the representative agent. All the agents behave similarly and impact the representative agent. However, because of the large number an aggregation effect takes place. The interesting consequence is that the impact of the community can be modeled by a mean field term, but when this is done, the problem is reduced to a control problem.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

Mean field games and mean field type control introduce new problems in control theory. The term “games” is appealing, although maybe confusing. In fact, mean field games are control problems, in the sense that one is interested in a single decision maker, who we call the representative agent. However, these problems are not standard, since both the evolution of the state and the objective functional are influenced by terms that are not directly related to the state or to the control of the decision maker.

Alain Bensoussan, Jens Frehse, Phillip Yam

Chapter 2. General Presentation of Mean Field Control Problems

Abstract

Consider a probability space $(\Omega,\mathcal{A},P)$ and a filtration ${\mathcal{F}}^{t}$ generated by a n-dimensional standard Wiener process w(t). The state space is ${\mathbb{R}}^{n}$ with the generic notation x and the control space is ${\mathbb{R}}^{d}$ with generic notation v.

Alain Bensoussan, Jens Frehse, Phillip Yam

Chapter 3. The Mean Field Games

Abstract

Let us set

$$\displaystyle{ a(x) = \frac{1} {2}\sigma {(x)\sigma }^{{\ast}}(x), }$$

(3.1)

and introduce the second-order differential operator

$$\displaystyle{ A\varphi (x) = -\text{tr }a(x){D}^{2}\varphi (x). }$$

(3.2)

We define the dual operator

$$\displaystyle{ {A}^{{\ast}}\varphi (x) = -\sum _{ k,l=1}^{n} \frac{{\partial }^{2}} {\partial _{x_{k}}\partial _{x_{l}}}(a_{kl}(x)\varphi (x)). }$$

(3.3)

Since m(t) is the probability distribution of $\hat{x}(t)$, it has a density with respect to the Lebesgue measure denoted by m(x, t), which is the solution of the Fokker–Planck equation

$$\displaystyle\begin{array}{rcl} \frac{\partial m} {\partial t} + {A}^{{\ast}}m + \text{div }(g(x,m,\hat{v}(x))m)& =& 0, \\ m(x,0)& =& m_{0}(x).{}\end{array}$$

(3.4)

We next want the feedback $\hat{v}(x)$ to solve a standard control problem, in which m appears as a parameter. We can thus readily associate an HJB equation with this problem, parametrized by m.

Alain Bensoussan, Jens Frehse, Phillip Yam

Chapter 4. The Mean Field Type Control Problems

Abstract

We need to assume that the

$$\displaystyle\begin{array}{lll}\,\, m \rightarrow f(x,m,v),\;g(x,m,v),\;h(x,m) \\ \text{are} \text{ differentiable in}\;m \in {L}^{2}({\mathbb{R}}^{n}){}\end{array}$$

(4.1)

and we use the notation $\dfrac{\partial f} {\partial m}(x,m,v)(\xi )$ to represent the derivative, so that

$$\displaystyle{\frac{d} {d\theta }f(x,m +\theta \tilde{ m},v)_\vert {}_{\theta =0} =\int _{{\mathbb{R}}^{n}} \dfrac{\partial f} {\partial m}(x,m,v)(\xi )\tilde{m}(\xi )\,d\xi.}$$

Here, x, v are simply parameters. Coming back to the definition (2.9)–(2.11), consider a feedback v(x) and the corresponding trajectory defined by (2.9). The probability distribution m _{v(. )}(t) of x _{v(. )}(t) is a solution of the FP equation

$$\displaystyle\begin{array}{rcl} \frac{\partial m_{v(.)}} {\partial t} + {A}^{{\ast}}m_{ v(.)} + \text{div }(g(x,m_{v(.)},v(x))m_{v(.)})& =& 0, \\ m_{v(.)}(x,0)& =& m_{0}(x){}\end{array}$$

(4.2)

and the objective functional J(v(. ), m _{v(. )}) can be expressed as follows

$$\displaystyle\begin{array}{rcl} J(v(.),m_{v(.)}(.))& =& \int _{0}^{T}\int _{{ \mathbb{R}}^{n}}f(x,m_{v(.)}(x),v(x))m_{v(.)}(x)dxdt\; \\ & & \quad +\int _{{\mathbb{R}}^{n}}h(x,m_{v(.)}(x,T))m_{v(.)}(x,T)dx.{}\end{array}$$

(4.3)

Consider an optimal feedback $\hat{v}(x)$ and the corresponding probability density $m_{\hat{v}(.)}(x,t) = m(x,t)$.

Alain Bensoussan, Jens Frehse, Phillip Yam

Chapter 5. Approximation of Nash Games with a Large Number of Players

Abstract

We first assume that the functions f(x, m, v), g(x, m, v), and h(x, m)—as functions of m—can be extended to Dirac measures and the sum of Dirac measures that are probabilities. Since m is no more in L ^p, the reference topology will be the weak * topology of measures. That will be sufficient for our purpose, but we refer to [14] for metric space topology. At any rate, the vector space property is lost.

We consider N players.

Alain Bensoussan, Jens Frehse, Phillip Yam

Chapter 6. Linear Quadratic Models

Abstract

The linear quadratic model has been developed in [10]. See also [2, 20, 22]. We highlight here the results. We take

$$\displaystyle\begin{array}{rcl} f(x,m,v)& =& \dfrac{1} {2}\left [{x}^{{\ast}}Qx + {v}^{{\ast}}Rv +{ \left (x - S\int \xi m(\xi )d\xi \right )}^{{\ast}}\bar{Q}\left (x - S\int \xi m(\xi )d\xi \right )\right ]{}\end{array}$$

(6.1)

$$\displaystyle\begin{array}{rcl} g(x,m,v)& = Ax +\bar{ A}\int \xi m(\xi )d\xi + Bv&{}\end{array}$$

(6.2)

$$\displaystyle\begin{array}{rcl} h(x,m)& = \dfrac{1} {2}\left [{x}^{{\ast}}Q_{T}x +{ \left (x - S_{T}\int \xi m(\xi )d\xi \right )}^{{\ast}}\bar{Q_{T}}\left (x - S_{T}\int \xi m(\xi )d\xi \right )\right ].&{}\end{array}$$

(6.3)

Alain Bensoussan, Jens Frehse, Phillip Yam

Chapter 7. Stationary Problems

Abstract

We shall consider only mean field games, but mean field type control can also be considered. To obtain stationary problems, Lasry and Lions [27] consider ergodic situations. This introduces an additional difficulty. It is, however, possible to motivate stationary problems that correspond to infinite horizon discounted control problems. The price to pay concerns the N player differential game associated with the mean field game. It is less natural than the one used in the time-dependent case. However, other interpretations are possible, which do not lead to the same difficulty.

Alain Bensoussan, Jens Frehse, Phillip Yam

Chapter 8. Different Populations

Abstract

In preceding chapters, we considered a single population composed of a large number of individuals with identical behavior. In real situations, we will have several populations. The natural extension to the preceding developments is to obtain mean field equations for each population. A much more challenging situation will be to consider competing populations. This will be addressed in the next chapter.

Alain Bensoussan, Jens Frehse, Phillip Yam

Chapter 9. Nash Differential Games with Mean Field Effect

Abstract

The mean field game and mean field type control problems introduced in Chap. 2 are both control problems for a representative agent, with mean field terms influencing both the evolution and the objective functional of this agent. The terminology game comes from the fact that the optimal feedback of the representative agent can be used as an approximation for a Nash equilibrium of a large community of agents that are identical. In Sect. 8.2 we have shown that the theory extends to a multi-class of representative agents. However, each class still has its individual control problem.

Alain Bensoussan, Jens Frehse, Phillip Yam

Chapter 10. Analytic Techniques

Abstract

We consider here the system

$$\displaystyle\begin{array}{rcl} -\dfrac{\partial {u}^{i}} {\partial t} + A{u}^{i}& =& {H}^{i}(x,Du) + f_{ 0}^{i}(x,m(t)) \\ {u}^{i}(x,T)& =& {h}^{i}(x,m(T)) {}\end{array}$$

(10.1)

Alain Bensoussan, Jens Frehse, Phillip Yam

Backmatter

Titel: Mean Field Games and Mean Field Type Control Theory
verfasst von: Alain Bensoussan
Jens Frehse
Phillip Yam
Verlag: Springer New York
Electronic ISBN: 978-1-4614-8508-7
Print ISBN: 978-1-4614-8507-0
DOI: https://doi.org/10.1007/978-1-4614-8508-7