# Mean Field Game with Delay: A Toy Model

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^{†}

## Abstract

**:**

## 1. Introduction

## 2. A Differential Game with Delay

#### 2.1. The Model

#### 2.2. Construction of a Nash Equilibrium

## 3. The Mean Field Game System

## 4. The Master Equation

#### 4.1. Derivatives

**Definition**

**1.**

**Definition**

**2.**

**Remark**

**1.**

**Example**

**1.**

#### 4.2. The Master Equation

**Theorem**

**1.**

**Proof.**

#### 4.3. Explicit Solution of the Master Equation

## 5. Convergence of the Nash System

**Proposition**

**1.**

**Proof.**

- $$\begin{array}{cc}& \sum _{k=1}^{N}\frac{1}{2}{\sigma}^{2}{\partial}_{{z}_{0}^{k}{z}_{0}^{k}}{u}^{i}(t,{z}_{0},{z}_{1})\hfill \\ \hfill =& \frac{1}{2}{\sigma}^{2}{\partial}_{{z}_{0}^{i}{z}_{0}^{i}}{u}^{i}(t,{z}_{0},{z}_{1})+\sum _{k\ne i}\frac{1}{2}{\sigma}^{2}{\partial}_{{z}_{0}^{k}{z}_{0}^{k}}{u}^{i}(t,{z}_{0},{z}_{1})\hfill \\ \hfill =& \frac{1}{2}{\sigma}^{2}{\partial}_{{z}_{0}^{i}{z}_{0}^{i}}U(t,{z}_{0}^{i},{z}_{1}^{i},{\nu}^{i})+\frac{1}{2}{\sigma}^{2}\sum _{k\ne i}\frac{1}{N-1}{\partial}_{{z}_{0}^{k}}\left[{D}_{{\mu}_{0}^{i}}U\right](t,{z}_{0}^{i},{z}_{1}^{i},{\nu}^{i},{z}_{0}^{k})\hfill \\ & \text{\hspace{1em}}+\frac{1}{2}{\sigma}^{2}\sum _{k\ne i}\frac{1}{{(N-1)}^{2}}{D}_{{\mu}_{0}^{i}{\mu}_{0}^{i}}U(t,{z}_{0}^{i},{z}_{1}^{i},{\nu}^{i},{z}_{0}^{k},{z}_{0}^{k})\hfill \\ \hfill =& \frac{1}{2}{\sigma}^{2}{\partial}_{{z}_{0}^{i}{z}_{0}^{i}}U(t,{z}_{0}^{i},{z}_{1}^{i},{\nu}^{i})+\frac{1}{2}{\sigma}^{2}{\int}_{\mathbb{R}}{\partial}_{{y}_{0}}\left[{D}_{{\mu}_{0}^{i}}U\right](t,{z}_{0}^{i},{z}_{1}^{i},{\nu}^{i},{y}_{0})d{\mu}_{0}^{i}({y}_{0})\hfill \\ & \text{\hspace{1em}}+\frac{1}{2}{\sigma}^{2}\frac{1}{N-1}{\int}_{\mathbb{R}}{D}_{{\mu}_{0}^{i}{\mu}_{0}^{i}}U(t,{z}_{0}^{i},{z}_{1}^{i},{\nu}^{i},{y}_{0},{y}_{0})d{\mu}_{0}^{i}({y}_{0}).\hfill \end{array}$$
- $$\begin{array}{cc}& \sum _{k=1}^{N}{\int}_{-\tau}^{0}{z}_{1}^{k}\frac{d}{ds}({\partial}_{{z}_{1}^{k}}{u}^{i})ds\hfill \\ \hfill =& {\int}_{-\tau}^{0}{z}_{1}^{i}\frac{d}{ds}({\partial}_{{z}_{1}^{i}}{u}^{i})ds+\sum _{k\ne i}{\int}_{-\tau}^{0}{z}_{1}^{k}\frac{d}{ds}({\partial}_{{z}_{1}^{k}}{u}^{i})ds\hfill \\ \hfill =& {\int}_{-\tau}^{0}{z}_{1}^{i}\frac{d}{ds}({\partial}_{{z}_{1}^{i}}U)ds+\sum _{k\ne i}{\int}_{-\tau}^{0}{z}_{1}^{k}\frac{d}{ds}\left(\right)open="["\; close="]">\frac{1}{N-1}{D}_{{\mu}_{1}^{i}}U(t,{z}_{0}^{i},{z}_{1}^{i},{\nu}^{i},{z}_{1}^{k})ds\hfill \end{array}$$
- From the solution (40) of the master equation, ${\partial}_{z}U$ is Lipschitz with respect to the measures. Namely,$$\begin{array}{c}|{\partial}_{{z}_{0}}U(t,{z}^{k},{\nu}^{i})-{\partial}_{{z}_{0}}U(t,{z}^{k},{\nu}^{k})|\le {C}_{1}({d}_{MK}({\mu}_{0}^{i},{\mu}_{0}^{k})+{d}_{MK}({\mu}_{1}^{i},{\mu}_{1}^{k}))\le \frac{{C}_{1}}{N-1},\hfill \\ \parallel {\partial}_{{z}_{1}}U(t,{z}^{k},{\nu}^{i})-{\partial}_{{z}_{1}}U(t,{z}^{k},{\nu}^{k}){\parallel}_{\mathbb{H}}\le {C}_{2}({d}_{MK}({\mu}_{0}^{i},{\mu}_{0}^{k})+{d}_{MK}({\mu}_{1}^{i},{\mu}_{1}^{k}))\le \frac{{C}_{2}}{N-1}.\hfill \end{array}$$Thus,$$\begin{array}{cc}& \sum _{k\ne i}({\partial}_{{z}_{0}^{k}}{u}^{k}-\left[{\partial}_{{z}_{1}^{k}}{u}^{k}\right](-\tau ))({\partial}_{{z}_{0}^{k}}{u}^{i}-\left[{\partial}_{{z}_{1}^{k}}{u}^{i}\right](-\tau ))\hfill \\ \hfill =& \sum _{k\ne i}{\partial}_{{z}_{0}^{k}}U(t,{z}_{0}^{k},{z}_{1}^{k},{\nu}^{k})\left(\right)open="("\; close=")">\frac{1}{N-1}{D}_{{\mu}_{0}^{i}}U(t,{z}_{0}^{i},{z}_{1}^{k},{\nu}^{i},{z}_{0}^{k})-\frac{1}{N-1}\left[{D}_{{\mu}_{1}^{i}}U(t,{z}_{0}^{i},{z}_{1}^{i},{\nu}^{i},{z}_{1}^{k})\right](-\tau )\hfill \end{array}\hfill =& \sum _{k\ne i}{\partial}_{{z}_{0}^{k}}U(t,{z}_{0}^{k},{z}_{1}^{k},{\nu}^{i})\left(\right)open="("\; close=")">\frac{1}{N-1}{D}_{{\mu}_{0}^{i}}U(t,{z}_{0}^{i},{z}_{1}^{k},{\nu}^{i},{z}_{0}^{k})-\frac{1}{N-1}\left[{D}_{{\mu}_{1}^{i}}U(t,{z}_{0}^{i},{z}_{1}^{i},{\nu}^{i},{z}_{1}^{k})\right](-\tau )\hfill $$Then,$$\begin{array}{cc}& {\partial}_{t}{u}^{i}+\sum _{k=1}^{N}\frac{1}{2}{\sigma}^{2}{\partial}_{{z}_{0}^{k}{z}_{0}^{k}}{u}^{i}+\sum _{k=1}^{N}{\int}_{-\tau}^{0}{z}_{1}^{k}\frac{d}{ds}({\partial}_{{z}_{1}^{k}}{u}^{i})ds\hfill \\ & \text{\hspace{1em}}-\sum _{k\ne i}^{N}({\partial}_{{z}_{0}^{k}}{u}^{k}-\left[{\partial}_{{z}_{1}^{k}}{u}^{k}\right](-\tau ))({\partial}_{{z}_{0}^{k}}{u}^{i}-\left[{\partial}_{{z}_{1}^{k}}{u}^{i}\right](-\tau ))-\frac{1}{2}{\left(\right)}^{{\partial}_{{z}_{0}^{i}}}2+\frac{\u03f5}{2}{({\overline{z}}_{0}-{z}_{0}^{i})}^{2}\hfill \end{array}& \text{\hspace{1em}}-{\int}_{\mathbb{R}\times \mathbb{H}}\left(\right)open="("\; close=")">{\partial}_{{y}_{0}}U-\left[{\partial}_{{y}_{1}}U\right](-\tau )(t,{y}_{0},{y}_{1},{\nu}^{i})\xb7\left(\right)open="("\; close=")">{D}_{{\mu}_{0}^{i}}U-\left[{D}_{{\mu}_{1}^{i}}U\right](-\tau )\hfill & (t,{z}_{0}^{i},{z}_{1}^{i},{\nu}^{i},{y}_{0},{y}_{1})d{\nu}^{i}({y}_{0},{y}_{1})$$

**Theorem**

**2.**

**Proof.**

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

SDDE | Stochastic Delayed Differential Equation |

HJB | Hamilton-Jacobi-Bellman |

MFG | Mean Field Game |

PDE | Partial Differential Equation |

LLN | Law of Large Numbers |

NSF | National Science Foundation |

## Appendix A. Adjoint Operator

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**MDPI and ACS Style**

Fouque, J.-P.; Zhang, Z.
Mean Field Game with Delay: A Toy Model. *Risks* **2018**, *6*, 90.
https://doi.org/10.3390/risks6030090

**AMA Style**

Fouque J-P, Zhang Z.
Mean Field Game with Delay: A Toy Model. *Risks*. 2018; 6(3):90.
https://doi.org/10.3390/risks6030090

**Chicago/Turabian Style**

Fouque, Jean-Pierre, and Zhaoyu Zhang.
2018. "Mean Field Game with Delay: A Toy Model" *Risks* 6, no. 3: 90.
https://doi.org/10.3390/risks6030090