Linear-quadratic-Gaussian mean-field games driven by Poisson jumps with major and minor agents

1.   Introduction

Mean-field games of a large population system have attracted consistent and intense attention in recent years (see, e.g., ^{[1,2,3,4,5,6,7,8,9,10]}) due to their wide applicability in many fields such as finance, economics, engineering, biological science, and social science. The agents in mean-field games are individually insignificant, while their aggregated behavior has a substantial effect on each agent. This collective influence can be captured by the mean-field couplings in their individual dynamics and/or individual cost functionals. For mean-field games, it is unrealistic for a given agent to collect detailed state information of all agents due to the highly complex interactions among its peers. To tackle the dimensionality difficulty caused by the highly complex interactions among the agents in mean-field games, Huang, Caines, and Malhamé ^[11], Huang ^[12], and Nourian and Caines ^[13] developed a powerful approach—the Nash certainty equivalence (NCE) methodology. The key idea of this methodology is to establish a consistency relationship between the individual strategies and the mass effect (i.e., the asymptotic limit of state-average) as the population size goes to infinity. Based on this effective analytical tool, one can construct a set of decentralized strategies for each agent in the mean-field game, and verify the asymptotic Nash equilibrium property (namely, $\epsilon$-Nash equilibrium) of the decentralized strategies where the individual optimality loss level $\epsilon$ depends on the population size $N$. A closely related method for solving mean-field games was independently developed by Lasry and Lions ^[14,15,16]. For a comprehensive survey of the theory of the mean-field game and its applications, one is referred to ^{[11,12,14,16,17,18,19,20,21]} and the references therein.

The consideration of major and minor a agent game problems under a large population framework has been well studied in ^{[3,12,13,21,22]}. Huang ^[12] investigated a kind of stochastic dynamic linear-quadratic-Gaussian mean-field games model involving a major agent interacting with a large number of minor agents. The major agent has a significant influence in affecting minor agents, while the minor agents individually have negligible impact on others, but their collective behavior will impose a significant impact on all agents through mean-field coupling terms in the individual dynamics and costs. Applications of this type of mean-field game appear in many socio-economic problems such as economic and social opinion models with an influential leader (e.g., ^[23]), such as the charging control of plug-in electric vehicles ^[24]. Xu and Wu ^[21] studied large-population dynamic games involving a LQG system with an exponential cost functional, and the parameter in the cost functional can describe an investor's risk attitude. Moreover, in the game, there is a major agent and a population of $N$ minor agents where $N$ is very large. Wang and Xu ^[22] investigated a time-inconsistent linear-quadratic game involving a major agent as well as numerous minor agents.

Motivated by the absence of relevant theory and some practical applications, this paper studies mean-field LQG games with random jumps involving a major agent and plenty of minor agents. Specifically, we consider mean-field games with agents of the following mixed types: (ⅰ) a major agent and (ⅱ) a large population of $N$ minor agents where $N$ is very large. The dynamic of each agent follows a linear stochastic differential equation driven by both Brownian motions and Poisson random measures. Moreover, the present study considers the mean-field LQG mixed games in which the diffusion term depends on the major agent's and the minor agent's states as well as the individual control strategy. Stochastic processes with random jumps can be used to model fluctuations in the financial market, both for option pricing purposes and risk management (see ^{[20,25,26,27]}). As for mean-field LQG games with random jumps, Benazzoli, Campi, and Di Persio ^[1] studied a symmetric $n$-player nonzero-sum stochastic differential game with jump-diffusion dynamics and mean-field type interaction among the players, and they constructed an approximate Nash equilibrium for the $n$-player game with $n$ sufficiently large. Xu and Shi ^[20] investigated LQG games of a stochastic large population system with jump diffusion processes. It is worth noting that in existing research on mean-field games of a stochastic large population system driven by jump-diffusion processes, all agents are comparably small and may be regarded as peers.

To obtain an asymptotic Nash equilibrium property (i.e., $\epsilon$-Nash equilibrium) for the original mean-field game, we apply the NCE approach to establish a certain consistency relationship between all minor agents and the mass effect. First, we construct two auxiliary stochastic control problems driven by stochastic differential equations driven by Poisson jumps (SDEPs) which depict the state of the major agent and a generic minor agent, and obtain the corresponding optimal control in feedback form. Next, to devise the decentralized strategies of individual agents, we formulate a kind of fully coupled forward-backward stochastic differential equation driven by Poisson jumps which is called a consistency condition (CC) system. Then, a set of decentralized strategies are constructed by using the solution of the CC system, which are demonstrated to be the $\epsilon$-Nash equilibrium.

The main contributions of this paper can be summarized as follows:

● A new class of LQG mean-field games involving major and minor agents is investigated. The dynamics of each agent follows a linear stochastic differential equation driven by both Brownian motions and Poisson random measures, in which the diffusion terms of the major and minor agents depend on their states and control strategy.

● The average state of all minor agents $x^{(N)}(\cdot)$ appears in the drift term and diffusion term of the state equations for both the major agent and all the minor agents, as well as in their cost functionals.

● The consistency condition system called the NCE equation is represented through a fully coupled two-point boundary value problem, and based on this equation, we design a set of decentralized feedback control strategies for the $N+1$ agents by use of two limiting control systems.

● By the approximation relationship between the closed-loop mean-field game system and the limiting systems, the set of NCE-based decentralized control strategies is shown to be an $\epsilon$-Nash equilibrium for a finite $N+1$ population system where $\epsilon = O(1/\sqrt{N})$.

This paper is organized as follows. In Section 2, we formulate the LQG mean-field games driven by Poisson random jumps involving a major agent and many minor agents. Section 3 introduces two auxiliary optimization problems for the major agent and each minor agent, respectively, and the consistency condition system is derived. Section 4 aims to present the $\epsilon$-Nash equilibrium property of the decentralized control strategies. A numerical example is given in Section 5. Finally, Section 6 concludes the paper.

2.   Formulation of the problem

2.1.   Notations

Throughout this paper, we denote by $\mathbb{R}^{n}$ the $n$-dimensional Euclidean space. For a given Euclidean space, we denote by $|\cdot|$ (respectively, $\langle \cdot, \cdot\rangle$) the standard Euclidean norm (respectively, inner product). The transpose of a matrix (or vector) $\mathbb{X}$ is denoted by $\mathbb{X}^T$. Let $(\Omega, \mathcal{F}, \{\mathcal{F}_{t}\}_{0\leq t\leq T}, \mathbb{P})$ be a complete filtered probability measure space for fixed time $T > 0$, and let the number $N$ represent the population size of minor agents. Denote by $\mathcal{N}$ the index set $\{1, 2, \cdots, N\}$. Let $\mathcal{F}_t$ be the filtration generated by the following mutually independent processes:

(ⅰ) $(N+1)$ independent one-dimensional standard Brownian motions $\{W_{i}(t), \ \ i = 0, 1, \cdots, N\}_{0\leq t\leq T}$;

(ⅱ) $(N+1)$ independent Poisson random measures $\{\widetilde{G}_i, i = 0, 1, \cdots, N\}$ on $E_i\times \mathbb{R}^{+}$, where $E_i\subset \mathbb{R}$ is a nonempty open set equipped with its Borel field $\mathcal{B}(E_i)$, with compensator $\widehat{G} _i(dedt) = \pi_i(de)dt$, such that $G_i (S\times [0, t]) = (\widetilde{G}_i-\widehat{G}_i)(S\times [0, t])_{t\geq 0}$ is a martingale for all $S \in \mathcal{B}(E_i)$. $\pi_i$ is a $\sigma$-finite measure on $(E_i, \mathcal{B}(E_i))$ and is called the characteristic measure. Moreover, $\forall S\in \mathcal{B}(E_i)$, $\mathbb{C}_0: = \sup\limits_{0\leq i \leq N}\pi_i(S) < +\infty$ is a positive constant independent of the number $N$.

We also set

where $\bigvee_{\alpha}\mathcal{F}_{\alpha}: = \sigma\left(\bigcup_{\alpha}\mathcal{F}_{\alpha}\right)$. Here, $\{\mathcal{F}_t^{0}\}_{0\leq t\leq T}$ represents the information of the major agent, whereas for the given $i\in \mathcal{N}$, $\{\mathcal{F}_t^{i}\}_{0\leq t\leq T}$ stands the individual information of the $i$th minor agent.

Denote by $\mathcal{S}^n$ the set of symmetric $n\times n$ matrices with real elements. If $M\in \mathcal{S}^n$ is positive (semi) definite, we write $M > (\geq) \ 0$. We also introduce the following spaces:

2.2.   Major-minor mean-field game problems

Let us consider an LQG mean-field game involving a major agent $\mathcal{A}_0$ and a population of $N$ minor agents $\{\mathcal{A}_i, i = 1, 2, \cdots, N\}$. For the major agent $\mathcal{A}_0$, $\mathcal{U}^{c, 0}_{ad}: = \{u(\cdot)|u(\cdot)\in L_{\mathcal{F}}^{2}([0, T]; \mathbb{R}^k)\}$ denotes the centralized admissible control set, and $\mathcal{U}^{0}_{ad}: = \{u(\cdot)|u(\cdot)\in L_{\mathcal{F}^{0}}^{2}([0, T]; \mathbb{R}^k)\}$ represents the corresponding decentralized admissible control set. For each $i\in \mathcal{N}$, we define the centralized admissible control set for the minor agent $\mathcal{A}_i$ as $\mathcal{U}^{c, i}_{ad}: = \{u_i(\cdot)|u_i(\cdot)\in L_{\mathcal{F}}^{2}([0, T]; \mathbb{R}^k)\}$, while the corresponding decentralized admissible control set is $\mathcal{U}^{i}_{ad}: = \{u_i(\cdot)|u_i(\cdot)\in L_{\mathcal{F}^{0, i}}^{2}([0, T]; \mathbb{R}^k)\}$. Note that we have $\mathcal{U}^{i}_{ad}\subset \mathcal{U}^{c, i}_{ad}$ for $i = 0, 1, \cdots, N$.

The dynamics of the major agent $\mathcal{A}_0$ is given as follows:

and the state of the minor agent $\mathcal{A}_i$ is described by

where $x^{(N)}(t) = \frac{1}{N}\sum_{j = 1}^{N}x_{j}(t)$ represents the average state of all minor agents. Here, $A_{0}\in\mathbb{R}^{n\times n}, B_{0}\in\mathbb{R}^{n\times k}, C_{0}\in\mathbb{R}^{n\times n}, D_{0}\in\mathbb{R}^{n\times k}$, $b_{0}\in\mathbb{R}^{n\times n}, l_{0}\in\mathbb{R}^{n\times n}, F_{0}\in\mathbb{R}^{n}, A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{n\times k}, C\in\mathbb{R}^{n\times n}, D\in\mathbb{R}^{n\times k}$, $b_{1}\in\mathbb{R}^{n\times n}, b_{2}\in\mathbb{R}^{n\times n}, H\in\mathbb{R}^{n\times n},$ and $F\in\mathbb{R}^{n}$ are given constants, and $f_{0}(\cdot)\in\mathbb{R}^{n}, \sigma_{0}(\cdot)\in\mathbb{R}^{n}, f(\cdot)\in\mathbb{R}^{n},$ and $\sigma(\cdot)\in\mathbb{R}^{n}$ are given deterministic functions. For given admissible control $u_0$ and $u_i$, it follows that the systems (1) and (2) admit a unique solution $x_0(\cdot), x_i(\cdot) \in S_{\mathcal{F}}^{2}([0, T]; \mathbb{R}^n)$.

Let $u = (u_{0}, u_{1}, \ldots, u_{i}, \ldots, u_{N})$ be the set of control strategies for all $N+1$ agents, and $u_{-i} = (u_{0}, u_{1}, \ldots, u_{i-1}, u_{i+1}, \ldots, u_{N})$ for $i = 0, 1\cdots, N$. The cost functional for the major agent $\mathcal{A}_{0}$ is

The cost functional for minor agent $\mathcal{A}_{i}, 1\leq i\leq N$, is

The coefficients of cost functionals satisfy that $Q_{0}, Q\in \mathcal{S}^n, Q_{0}\geq 0, Q\geq 0, \beta_{0}, \beta_{1}, \beta_{2}\in\mathbb{R}^n, R_{0} > 0, R > 0, R_{0}, R\in \mathcal{S}^k$ and $M_{0}\geq 0, M\geq 0, M_{0}, M\in \mathcal{S}^n$.

Parallel to (2), the cost functional (4) contains the term $\beta_{2}x_{0}(t)$ to capture the strong influence of the major agent. Note that the state dynamics (1) and (2), and the cost functionals (3) and (4), indicate that the major agent $\mathcal{A}_0$ has a significant influence on minor agents, while each minor agent $\mathcal{A}_i, i\in\mathcal{N}$, has a negligible impact on other agents in a large $N$ population system.

Now, we propose the following LQG mean-field games.

Problem (LP): Find an admissible strategy $\bar{u} = (\bar{u}_{0}, \bar{u}_{1}, \ldots, \bar{u}_{i}, \ldots, \bar{u}_{N})$ where $\bar{u}_{i}(\cdot)\in\mathcal{U}_{ad}^{c, i}, i = 0, 1, \cdots, N$, such that

We call $\bar{u}$ a Nash equilibrium strategy for Problem (LP).

Remark 2.1. It should be noted that this paper only addresses the existence of Nash equilibrium strategies and does not involve whether the Nash equilibrium is unique. The study of the uniqueness of Nash equilibrium strategies is also an active research topic. The variational inequality approach proposed in He and Wang ^[28] provides a feasible methodology for studying the uniqueness of Nash equilibrium strategies.

3.   Closed-loop behavior of the agents

In this section, we first construct two auxiliary stochastic optimal control problems, which are called limiting systems, for the major and a generic minor agent in Sections 3.1 and 3.2, respectively. Then we present the approximations between the limiting systems and the corresponding mean-field system in Section 3.3.

3.1.   Optimal control of the major agent

For any $v_0(\cdot)\in\mathcal{U}_{ad}^{0}$, the state $y_{0}(\cdot)$ of agent $\mathcal{A}_{0}$ satisfies the following stochastic differential equation:

where function $x^{(0)}(\cdot)$ will be given later.

The corresponding cost functional is given by

Problem (LM1): The objective is to find $\bar{v}_{0}(\cdot)\in\mathcal{U}_{ad}^{0}$ such that

Let $P_{0}(\cdot)$ be the solution of the following Riccati equation:

Let $\eta_{0}(\cdot)$ denote the solution of

The following result presents the optimal control of Problem (LM1).

Theorem 3.1. Suppose that

Then the optimal control strategy of Problem (LM1) is

where $\bar{y}_{0}(\cdot)$ satisfies

Proof. Let $\hat{b}(t): = b_{0}x^{(0)}(t)+f_{0}(t), \hat{\sigma}(t): = l_{0}x^{(0)}(t)+\sigma_{0}(t)$. Then the state equation (5) can be written as

For simplicity, we denote $\hat{{R}}_0(t): = R_{0}+D_{0}^{\top}P_{0}(t)D_{0}, \ $ $\hat{{B}}_0(t): = B_{0}^{\top}P_{0}(t)+D_{0}^{\top}P_{0}(t)C_{0}$. Applying Itô's formula to $\Big(\frac{1}{2}y_{0}^{\top}(t)P_{0}(t)y_{0}(t)+y_{0}^{\top}(t)\eta_{0}(t)\Big)$, we obtain

Combing the above equation with the definition of $\widetilde{J}_{0}(v_{0})$, it follows that

Hence we obtain the optimal control

The proof is therefore complete. □

3.2.   Optimal control of the minor agent

For any $i\in\mathcal{N}$, the limiting state of minor agent $\mathcal{A}_{i}$ is

The limiting cost functional is given by

Problem (LM2): Find a control strategy $\bar{v}_{i}(\cdot)\in\mathcal{U}_{ad}^{i}, 1\leq i\leq N$, such that

Let $P_{1}(\cdot)$ be the solution of the following Riccati equation:

$\eta_{1}(\cdot)$ satisfies

Denote

Using a similar proof as in Theorem 3.1, we have the following result.

Theorem 3.2. The optimal control strategy of Problem (LM2) is

where $\bar{y}_{i}(\cdot)$ satisfies

3.3.   Approximation for the closed-loop system

In this subsection, we design a closed-loop mean-field system, and show the approximations between the limiting system and the corresponding closed-loop system.

Based on the feedback formulation of the optimal control for major agent $\mathcal{A}_{0}$ and minor agents $\mathcal{A}_{i}, 1\leq i\leq N$, we obtain

and

By $\bar{x}^{(N)}(t) = \frac{1}{N}\sum\limits_{k = 1}^{N}\bar{x}_i(t)$, the function $x^{(0)}(t)$ fulfills

Now, we introduce the following NCE equation:

which can be written as

where

The above NCE equation is a kind of coupled two-point boundary value problem, whose well-posedness can be found in Theorem 4.2 of Hu et al. ^[3] under some monotonicity assumptions. We will not repeat them here for simplicity.

Next, we establish the approximation relationship between the closed-loop mean-field game system and the limiting system.

Proposition 3.3. The following estimates hold:

Proof. Let $\bar{z}(t): = \bar{x}^{(N)}(t)-x^{(0)}(t)$, $\bar{z}_{0}(t): = \bar{x}_{0}(t)-\bar{y}_{0}(t)$, $\bar{z}_{i}(t): = \bar{x}_{i}(t)-\bar{y}_{i}(t)\ \ (1\leq i\leq N).$ Combining (9) with (10), we derive

Define $\chi(t): = b_{2}\bar{x}^{(N)}(t)+H\bar{x}_{0}(t)+\sigma(t)$. Applying Itô's formula to $\bar{z}^{2}(t)$, we obtain

According to Gronwall's inequality, it follow that

For (ii), according to Hölder's inequality, we have

By (12) and the boundedness of $|x^{(0)}(t)|$, one has

We now prove (iii). According to (6) and (8), it follows that

Applying Itô's formula to $\bar{z}_{0}^{2}(t)$, we obtain

By (12) and Gronwall's inequality, we have

Note that

According to (13) and the boundedness of $|\bar{y}_{0}(t)|$, we obtain

Next, we prove (v). Combining (7) with (9), we have

Applying Itô's formula to $\bar{z}_{i}^{2}(t)$, we obtain

By Gronwall's inequality, and estimates (12) and (13), we obtain

Finally, we prove (ⅵ). Since

According to (14) and the boundedness of $|\bar{y}_{i}(t)|$, we get

The proof is then complete. □

Define the control strategy for the major agent as

and the control strategy for minor agents as

Based on the approximation relationship between the closed-loop mean-field systems and the limiting system, the following approximation relationship between cost functionals can be derived.

Proposition 3.4. For any $i = 0, 1, \cdots, N$, we have

Proof. Based on the definitions of the cost functionals, we obtain

where $L(t): = \Big[(\bar{x}_{i}(t)-\bar{y}_{i}(t))-\beta_{1}(\bar{x}^{(N)}(t)-x^{(0)}(t))-\beta_{2}(\bar{x}_{0}(t)-\bar{y}_{0}(t))\Big]$. Obviously, according to Proposition 3.3, we have $\mathbb{E}\Big[\Big|L(t)\Big|^{2}\Big] = O(\frac{1}{N})$. Therefore, it follows that $\Big|J_{i}(\bar{u}_{i}, \bar{u}_{-i})-\widetilde{J_{i}}(\bar{v}_{i})\Big| = O(\frac{1}{\sqrt{N}})$. The proof is then complete. □

4.   $\varepsilon$-Nash equilibrium for Problem (LP)

This section will verify the asymptotic Nash equilibrium property of the decentralized control strategies $\bar{u} = (\bar{u}_{0}, \bar{u}_{1}, \cdots, \bar{u}_{N})$ specified by (15) and (16).

4.1.   Major agent's perturbation

Let the major agent $\tilde{\mathcal{A}}_{0}$ take an alternative control strategy $u_{0}$, and let the minor agent $\tilde{\mathcal{A}}_{i}$ take the control law (16). Then the state system with the major agent's perturbation is

where $\tilde{x}^{(N)}(t) = \frac{1}{N}\sum\limits_{k = 1}^{N}\tilde{x}_{k}(t)$. The cost functional for major agent $\tilde{\mathcal{A}}_{0}$ is

The corresponding limiting state equation with the major agent's perturbation control is

The cost functional is

The following result presents an approximation relationship between two perturbation systems.

Proposition 4.1. We have the following conclusion:

Proof. We only need to prove the first approximation relationship, and a other three approximation relationships can be obtained by a similar proof as in Proposition 3.3.

Define $\Phi(t): = \tilde{x}^{(N)}(t)-x^{(0)}(t)$. Combining (10) with (17), we have

Define $L_{k}(t): = [(C+D\Lambda_{1}(t))\tilde{x}_{k}(t)+D\bar{\Theta}_{1}(t)+b_{2}\tilde{x}^{(N)}(t)+H\tilde{x}_{0}(t)+\sigma(t)]$. Therefore

where

are constants independent of $N$.

Furthermore,

By Grownwall's inequality, we have

Then, the proof is complete. □

Similarly to the proof of Proposition 3.4, we can obtain the following result.

Proposition 4.2. For any $u_{0}(\cdot)\in\mathcal{U}_{ad}^{c, 0}$, we have

4.2.   Minor agent's perturbation

Now, let us consider the following case: a given minor agent $\tilde{\mathcal{A}}_{i}$ takes an alternative control strategy $u_{i}(\cdot)\in\mathcal{U}_{ad}^{c, i}$, the major agent uses the optimal control strategy $\bar{u}_{0}(\cdot)$ defined by (15), while other minor agents $\tilde{\mathcal{A}}_{j}$ take the control strategy $\bar{u}_{j}(\cdot), j\neq i, \ 1\leq j\leq N$, defined by (16). Then the dynamics of the agents with the given minor agent's perturbation can be written in the form

where $\hat{x}^{(N)}(t) = \frac{1}{N}\sum\limits_{k = 1}^{N}\hat{x}_{k}(t)$. The cost functional is

The corresponding limiting system with the minor agent's perturbation strategy is

The cost functional is

Now, we are in a position to state the following approximation results.

Proposition 4.3. For the fixed $i$, we have

Proof. We prove only the first approximation relationship, and the other three approximation relationships can be obtained by a similar proof as in Proposition 3.3.

Define $\tilde{z}(t): = \hat{x}^{(N)}(t)-x^{(0)}(t)$. According to (10) and (18), we get

where

Since

we get

Note that

Thus

Applying Itô's formula to $\tilde{z}^{2}(t)$, we obtain

Combining (19) and (20) with Gronwall's inequality, we get

This completes the proof. □

By using similar arguments as in Proposition 3.4, we can obtain the following conclusion.

Proposition 4.4. For any $u_{i}(\cdot)\in\mathcal{U}_{ad}^{c, i}, 1\leq i\leq N$, one has

4.3.   $\epsilon$-Nash equilibrium

In this subsection, we will verify the $\epsilon$-Nash equilibrium property of the decentralized control strategies (15) and (16).

Before presenting the main result, we give the definition of $\epsilon$-Nash equilibrium in the following manner.

Definition 4.5. A set of control strategies $\bar{u} = (\bar{u}_0, \bar{u}_1, \cdots, \bar{u}_N)$ where $\bar{u}_i(\cdot)\in \mathcal{U}^{c, i}_{a, d}, \ i = 0, 1\cdots, N$, is called an $\epsilon$-Nash equilibrium with respect to costs $J_i, i = 0, 1\cdots, N$, if there exists an $\epsilon\geq0$, such that for any $i = 0, 1\cdots, N$, we have

when any alternative strategy $u_i(\cdot)\in \mathcal{U}^{c, i}_{a, d}$ is applied by agent $\mathcal{A}_i$.

Based on the above results, we obtain the following main result.

Theorem 4.6. Suppose that $\bar{x}_{i}(\cdot), i = 0, 1, \cdots, N$, is the solution to the equation systems (8) and (9). Then the set of control strategy profiles $\bar{u} = (\bar{u}_0, \bar{u}_1, \cdots, \bar{u}_N)$ defined by (15) and (16) is an $\epsilon$-Nash equilibrium of $\text{ Problem (LP)}$, where $\epsilon = O(\frac{1}{\sqrt{N}})\rightarrow 0$ as $N\rightarrow +\infty$.

Proof. Combining Propositions 3.4 and 4.2 with Proposition 4.4, we obtain

Therefore, the conclusion holds with $\epsilon = O(\frac{1}{\sqrt{N}})$. □

5.   Numerical examples

This section demonstrates the consistency of mean-field estimation as well as the influence of the population's collective behavior $\bar{x}^{(N)}(\cdot)$ on the state trajectories of the agents through a numerical example.

Consider a mean-field game system with one major agent and $N = 500$ minor agents. For any $u_j\in\mathcal{U}^{c, j}_{ad}, j = 0, 1, \cdots, N$, the dynamics of the major agent and minor agents are given by

where $t\in[0, T]$ with $T = 1$. Let the initial states of the agents $\{a_{i}, i = 1, \cdots, N\}$ be independent and identically distributed random variables with the normal distribution $N(-5, 1)$.

The cost functional of the major agent $\mathcal{A}_{0}$ is

and the cost functional of the minor agent $\mathcal{A}_{i}, i = 1, \cdots, 500$, is

It is easy to check that $\{P_{0}(t)\equiv 3, \forall t\in[0, 1]\}$ is a unique solution of the following Riccati equation:

Suppose that $P_{1}(\cdot)$ fulfills

Then the NCE Eq (11) turns out to be

where $\mathbb{P}(t) = -(1+P_{1}(t))^{-1}P_1(t)$.

According to Theorem 4.6, the set of control strategies $\bar{u} = (\bar{u}_0, \bar{u}_1, \cdots, \bar{u}_N)$ defined by

is an $\epsilon$-Nash equilibrium of the mean-field systems (24) and (25), where $\bar{x}_{0}(\cdot)$ and $\bar{x}_{i}(\cdot)$ satisfy

where $\bar{x}^{(500)}(t) = \frac{1}{500}\sum\limits_{j = 1}^{500}\bar{x}_{j}(t)$.

In this article, Merton's jump model (see Merton ^[29], as well as Platen and Bruti-Liberati [30, pg. 37] is applied to describe the jump-diffusion process. Assume that $\int_{E_{0}} G_{0}(dedt) = \mathbb{Q}_0(\mu_{0}, \sigma_{0})d\Pi_0(\lambda_0)$. $\mathbb{Q}_0(\mu_{0}, \sigma_{0})$ is the jump size with a normally distributed mean $\mu_{0}\sim N(2, 1)$ and a standard deviation $\sigma_{0} = 0.1$. The Poisson process $\Pi_0(\lambda_0)$ has a jump intensity of $\lambda_0 = 2$. For agent $\mathcal{A}_i, i = 1, \cdots, 500$, let $\int_{E_{i}} G_{i}(dedt) = \mathbb{Q}_i(\mu_{1}, \sigma_{1})d\Pi_i(\lambda)$. $\mathbb{Q}_i(\mu_{1}, \sigma_{1})$ is the jump size with a normally distributed mean $\mu_{1}\sim N(1, 1)$ and a standard deviation $\sigma_{1} = 0.05$. The Poisson process $\Pi_i(\lambda)$ has a jump intensity of $\lambda = 5$.

Figure 1 shows the consistency of mean-field estimation, and the interactive influence between mean-field term $\bar{x}^{(500)}(\cdot)$, and the major state $\bar{x}_0(\cdot)$. When the number of minor agents $N = 500$, as shown in Figure 1, the curves of $\bar{x}^{(500)}(\cdot)$ and $x^{(0)}(\cdot)$ coincide well, which illustrates the consistency of the mean-field estimation indicated by Proposition 3.3.

Figure 2 illustrates the state trajectories of the major agent and all the minor agents. As shown in Figure 2, for each fixed $i$, the trajectory $\bar{x}_i(\cdot)$ of $\mathcal{A}_i$, in addition to being influenced by its own initial values and parameters, is also affected by the major agent and the collective behavior of all the minor agents.

To illustrate how the key parameters in the control strategies of Eqs (15) and (16) influence the system's dynamic behavior, we set another set of initial values for $N+1$ agents with $x_{0}(0) = -5$ and the independent and identically distributed random variables $\{a_{i}\sim N(5, 1), \ i = 1, \cdots, 500\}$. Figures 3 and 4 are shown to elaborate the consistency of mean-field estimation and the curves of $\bar{x}_i, i = 0, 1, 2, \cdots, 500$.

6.   Conclusions

Motivated by the lack of theory and some practical applications, this paper in concerned with linear-quadratic-Gaussian mean-field games involving mixed agents of a stochastic large population system with random jumps. There are two mixed types of agents: (ⅰ) a major agent and (ⅱ) a population of $N$ minor agents where $N$ is very large. The coupling of the major and minor agents exists in both their state dynamics and their individual cost functions. To deal with the dimensionality difficulty and obtain decentralized strategies, the NCE methodology is applied to yield a set of decentralized strategies which is verified to be the $\epsilon$-Nash equilibrium. We provide numerical examples to illustrate both the consistency of the mean-field estimation and the impact of the population's collective behavior. In the future, an interesting research direction is to extend the modeling and analysis to the social optima case, which may involve more applications in practice and generate more challenges in theory. Another potential direction is to study the uniqueness of the equilibrium strategy, which may be more valuable and challenging.

Author contributions

Conceptualization and methodology, R. X.; writing-original draft, review and editing, K. D., J. Z. and Y. Z.; supervision, R. X. All authors have read and agreed to the published version of the manuscript.

Use of Generative-AI tools declaration

The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This research is partially supported by the Natural Science Foundation of Shandong Province of China (Grant no. ZR2020MA031, ZR2021MA049), Qilu University of Technology (Shandong Academy of Sciences) Major Innovation Project of Science, Education and Industry Integration Pilot Project (2024ZDZX11), National Natural Science Foundation of China (11971266), and the Colleges and Universities Twenty Terms Foundation of Jinan City (2021GXRC100).

Conflict of interest

All authors declare no conflicts of interest.