Semi-Explicit Solutions to some Non-Linear Non-Quadratic Mean-Field-Type Games: A Direct Method

TL;DR

This paper introduces a direct method to find semi-explicit solutions for a broad class of non-linear, non-quadratic mean-field-type games, including complex dynamics and payoff functions, expanding solvable models beyond classical cases.

Contribution

It presents a simple direct approach to solve complex mean-field games with non-linear dynamics and payoffs, including jump-diffusion and regime switching processes.

Findings

Derived semi-explicit solutions for various non-quadratic mean-field games.

Extended solvability to models with complex dynamics like log-state and hyperbolic functions.

Provided solutions for cooperative, noncooperative, and adversarial game settings.

Abstract

This article examines mean-field-type game problems by means of a direct method. We provide various solvable examples beyond the classical linear-quadratic game problems. These include quadratic-quadratic games and games with power, logarithmic, sine square, hyperbolic sine square payoffs. Non-linear state dynamics such as log-state, control-dependent regime switching, quadratic state, cotangent state and hyperbolic cotangent state are considered. We identify equilibrium strategies and equilibrium payoffs in state-and-conditional mean-field type feedback form. It is shown that a simple direct method can be used to solve broader classes of non-quadratic mean-field-type games under jump-diffusion-regime switching Gauss-Volterra processes which include fractional Brownian motions and multi-fractional Brownian motions. We provide semi-explicit solutions to the fully cooperative, noncooperative nonzero-sum, and adversarial game problems.

Equations

$$\left\{\begin{array}[]{lll}dx=b(t,x,y,z,u,\bar{x},\bar{y},\bar{z},\bar{u},m_{1},m_{2},s)dt\\ +\sigma(t,x,y,z,u,\bar{x},\bar{y},\bar{z},\bar{u},m_{1},m_{2},s)d{B}\\ +\sigma_{gv}(t,x,y,z,u,\bar{x},\bar{y},\bar{z},\bar{u},m_{1},m_{2},s)d{B}_{gv}\\ +\int_{\Theta}\ \mu(t,x,y,z,u,\bar{x},\bar{y},\bar{z},\bar{u},m_{1},m_{2},s,\theta)\tilde{N}(dt,d\theta,s)\\ \\ +\sigma_{o}(t,x,y,z,u,\bar{x},\bar{y},\bar{z},\bar{u},m_{1},m_{2},s)d{B}_{o}\\ +\sigma_{o,gv}(t,x,y,z,u,\bar{x},\bar{y},\bar{z},\bar{u},m_{1},m_{2},s)d{B}_{o,gv}\\ +\int_{\Theta_{o}}\ \mu_{o}(t,x,y,z,u,\bar{x},\bar{y},\bar{z},\bar{u},m_{1},m_{2},s,\theta)\tilde{N}_{o}(dt,d\theta,s)\\ \\ x_{i}(t)=x_{i0}(t),\ \ \ t\in[-\tau_{i},0],\ i\in\mathcal{I},\\ s(t)\in\mathcal{S}=\{1,2,\ldots,S\},\\ \mathbb{P}(s(t+\epsilon)=s^{\prime}|s,u)=\int_{t}^{t+\epsilon}\tilde{q}_{ss^{\prime}}dt^{\prime}+o(\epsilon),\ s^{\prime}\neq s,\ \epsilon>0\\ \end{array}\right.$$

$$\left\{\begin{array}[]{lll}f(T,x(T),s(T))\\ =f(0,x_{0},s_{0})+\int_{0}^{T}[f_{t}+\langle f_{x},b\rangle]dt\\ +\frac{1}{2}\int_{0}^{T}\langle f_{xx}\sigma,\sigma\rangle dt\\ +\frac{1}{2}\int_{0}^{T}\langle f_{xx}\sigma_{cogv},\sigma_{cogv}\rangle dt\\ +\int_{0}^{T}\int_{\Theta}[f(x+\mu)-f(x)-\langle\mu,f_{x}\rangle]\nu(\theta)dt\\ +\frac{1}{2}\int_{0}^{T}\langle f_{xx}\sigma_{o},\sigma_{o}\rangle dt\\ +\frac{1}{2}\int_{0}^{T}\langle f_{xx}\sigma_{o,cogv},\sigma_{o,cogv}\rangle dt\\ +\int_{0}^{T}\int_{\Theta}[f(x+\mu_{o})-f(x)-\langle\mu_{o},f_{x}\rangle]\nu_{o}(\theta)dt\\ +\int_{0}^{T}\sum_{s^{\prime}\neq s}[f(.,x,s^{\prime})-f(.,x,s)]\tilde{q}_{ss^{\prime}}dt\\ \\ +\int_{0}^{T}\langle f_{x},\sigma dB\rangle\\ +\int_{0}^{T}\langle f_{x},\sigma_{gv}dB_{gv}\rangle\\ +\int_{0}^{T}\int_{\Theta}[f(x+\mu)-f(x)\rangle]\tilde{N}(dt,d\theta)\\ \\ +\int_{0}^{T}\langle f_{x},\sigma_{o}dB_{o}\rangle\\ +\int_{0}^{T}\langle f_{x},\sigma_{o,gv}dB_{o,gv}\rangle\\ +\int_{0}^{T}\int_{\Theta}[f(x+\mu_{o})-f(x)\rangle]\tilde{N}_{o}(dt,d\theta).\\ \end{array}\right.$$

$$\begin{array}[]{ll}\begin{cases}L_{i}(x,u)=-q_{i}(T,s(T))\ln(x(T))+\int_{0}^{T}\left(-q_{i}\ln(x)+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}r_{i}u_{i}^{2k}}\right)dt,\\ \inf_{u_{i}}~{}\mathbb{E}[L_{i}(x,u)],\\ \text{subject~{}to}\\ \mathrm{d}x=\left(b_{1}x\ln(x)+\sum_{j\in\mathcal{I}}b_{2j}xu_{j}\right)dt+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}x[\sigma\mathrm{d}B+\int\mu d\tilde{N}]},\\ \mathbb{P}(s(t+\epsilon)=s^{\prime}|s,u)=\int_{t}^{t+\epsilon}\tilde{q}_{ss^{\prime}}dt^{\prime}+o(\epsilon),\ s^{\prime}\neq s\\ \end{cases}\end{array}$$

$$u_i^{\ast }$$

$$\mathbb{E}[L_i(x,u^{\ast })]$$

$$\begin{array}[]{ll}\dot{\alpha}_{i}+q_{i}+\alpha_{i}b_{1}+\sum_{s^{\prime}}[{\alpha}_{i}(t,s^{\prime})-{\alpha}_{i}(t,s)]\tilde{q}_{ss^{\prime}}=0,\\ \dot{\delta}_{i}+\alpha_{i}[\frac{\sigma^{2}}{2}-\int_{\theta\in\Theta}[\ln(1+\mu(\theta))-\mu(\theta)]\nu(d\theta)]\\ +\sum_{s^{\prime}}[{\delta}_{i}(t,s^{\prime})-{\delta}_{i}(t,s)]\tilde{q}_{ss^{\prime}}-(2k-1)r_{i}(\frac{1}{2kr_{i}}b_{2i}\alpha_{i})^{\frac{2k}{2k-1}}\\ -\alpha_{i}\sum_{j\neq i}b_{2j}[\frac{1}{2k}\frac{\alpha_{j}b_{2j}}{r_{j}}]^{\frac{1}{2k-1}}=0,\end{array}$$

$$f_i(t,x,s)=-{\alpha }_i\ln (x)+{\delta }_i.$$

$$\begin{array}[]{ll}\mathbb{E}[L_{i}(x,u)-f_{i}(0,x_{0},s_{0})]\\ =\mathbb{E}\left(-q_{i}(T,s(T))+\alpha_{i}(T,s(T))\right)\ln(x(T))+\delta_{i}(T,s(T))\\ \mathbb{E}\int_{0}^{T}-\{\dot{\alpha}_{i}+q_{i}+\alpha_{i}b_{1}+\sum_{s^{\prime}}[{\alpha}_{i}(t,s^{\prime})-{\alpha}_{i}(t,s)]\tilde{q}_{ss^{\prime}}\}ln(x)dt\\ +\int_{0}^{T}\dot{\delta}_{i}+\frac{\sigma^{2}}{2}\alpha_{i}+\sum_{s^{\prime}}[{\delta}_{i}(t,s^{\prime})-{\delta}_{i}(t,s)]\tilde{q}_{ss^{\prime}}dt\\ +\int_{0}^{T}-(2k-1)r_{i}(\frac{1}{2kr_{i}}b_{2i}\alpha_{i})^{\frac{2k}{2k-1}}-\alpha_{i}\sum_{j\neq i}b_{2j}[\frac{1}{2k}\frac{\alpha_{j}b_{2j}}{r_{j}}]^{\frac{1}{2k-1}}\\ -\int_{0}^{T}\alpha_{i}\int_{\theta\in\Theta}[\ln(1+\mu(\theta))-\mu(\theta)]\nu(d\theta)dt\\ +\mathbb{E}\int_{0}^{T}[-b_{2i}\alpha_{i}u_{i}+r_{i}u_{i}^{2k}+(2k-1)r_{i}(\frac{1}{2kr_{i}}b_{2i}\alpha_{i})^{\frac{2k}{2k-1}}]dt,\end{array}$$

$$[-b_{2i}{\alpha }_i{u}_i+r_i{u}_i^{2k}+(2k-1)r_i(\frac{1}{2k{r}_i}{b}_{2i}{\alpha }_i{)}^{\frac{2k}{2k-1}}]\ge 0$$

$$\begin{array}[]{ll}u_{i}^{*}=[\frac{1}{2}\frac{\alpha_{i}b_{2i}}{r_{i}}],\\ \mathbb{E}[L_{i}(x,u^{*})]=\mathbb{E}[\alpha_{i}(0)\ln(x_{0})]+\delta_{i}(0)],\\ \dot{\alpha}_{i}+q_{i}+\alpha_{i}b_{1}+\sum_{s^{\prime}}[{\alpha}_{i}(t,s^{\prime})-{\alpha}_{i}(t,s)]\tilde{q}_{ss^{\prime}}=0,\\ \dot{\delta}_{i}+\frac{\sigma^{2}}{2}\alpha_{i}+\sum_{s^{\prime}}[{\delta}_{i}(t,s^{\prime})-{\delta}_{i}(t,s)]\tilde{q}_{ss^{\prime}}\\ -\frac{1}{4r_{i}}b_{2i}^{2}\alpha_{i}^{2}-\frac{1}{2}\alpha_{i}\sum_{j\neq i}b^{2}_{2j}\frac{\alpha_{j}}{r_{j}}\\ -\alpha_{i}\int_{\theta\in\Theta}[\ln(1+\mu(\theta))-\mu(\theta)]\nu(d\theta)=0\end{array}$$

$$\begin{array}[]{ll}\begin{cases}L_{i}(x,u)=q_{i}(T,s(T))\ln^{2}(x(T))+\int_{0}^{T}\left(q_{i}\ln^{2}(x)+r_{i}u_{i}^{2}\right)dt,\\ \inf_{u_{i}}~{}\mathbb{E}[L_{i}(x,u)],\\ \text{subject~{}to}\\ \mathrm{d}x=\left(b_{1}x\ln(x)+\sum_{j\in\mathcal{I}}b_{2j}xu_{j}\right)dt,\\ x(0)\triangleq x_{0}>>e,\\ \mathbb{P}(s(t+\epsilon)=s^{\prime}|s,u)=\int_{t}^{t+\epsilon}\tilde{q}_{ss^{\prime}}dt^{\prime}+o(\epsilon),\ s^{\prime}\neq s\\ \end{cases}\end{array}$$

$$u_i^{\ast }$$

$$\mathbb{E}[L_i(x,u^{\ast })]$$

$${\dot{\alpha }}_i=-q_i-2b_1{\alpha }_i+2{\alpha }_i\sum _{j\in \mathcal{I}\setminus \{i\}}{\alpha }_j\frac{b_{2j}^2}{r_j}$$

$$+{\alpha }_i^2\frac{b_{2i}^2}{r_i}+\sum _{s^{\prime }\in \mathcal{S}}[{\alpha }_i(t,s^{\prime })-{\alpha }_i(t,s)]{\tilde{q}}_{s{s}^{\prime }},$$

$${\alpha }_i(T,s)=q_i(T,s),$$

$$f_i(t,x)={\alpha }_i{\ln }^2(x).$$

$$f_i(T,x(T))-f_i(0,x_0)={\int }_0^T{\dot{\alpha }}_i{\ln }^2(x)dt$$

$$+{\int }_0^{T}2{\alpha }_i\ln (x)\left(b_1\ln (x)+\sum _{j\in \mathcal{I}}{b}_{2j}{u}_j\right)dt$$

$$+{\int }_0^T\sum _{s^{\prime }\in \mathcal{S}}[{\alpha }_i(t,s^{\prime })-{\alpha }_i(t,s)]{\tilde{q}}_{s{s}^{\prime }}{\ln }^2(x)dt$$

$$\mathbb{E}[L_i(x,u)-f_i(0,x_0)]=\mathbb{E}\left(q_i(T)-{\alpha }_i(T)\right){\ln }^2(x(T))$$

$$+\mathbb{E}{\int }_0^T{q}_i{\ln }^2(x)dt+\mathbb{E}{\int }_0^T{\dot{\alpha }}_i{\ln }^2(x)dt$$

$$+\mathbb{E}{\int }_0^T\left(2{\alpha }_i{b}_1{\ln }^2(x)+2{\alpha }_i\ln (x)\sum _{j\in \mathcal{I}\setminus \{i\}}{b}_{2j}{u}_j\right)dt$$

$$+\mathbb{E}{\int }_0^T{r}_i\left(u_i^2+2{\alpha }_i\frac{\ln (x)b_{2i}}{r_i}{u}_i\right)dt$$

$$+{\int }_0^T\sum _{s^{\prime }\in \mathcal{S}}[{\alpha }_i(t,s^{\prime })-{\alpha }_i(t,s)]{\tilde{q}}_{s{s}^{\prime }}{\ln }^2(x)dt$$

$${\left(u_i+{\alpha }_i\frac{\ln (x)b_{2i}}{r_i}\right)}^2-{\alpha }_i^2\frac{{\ln }^2(x)b_{2i}^2}{r_i^2}=u_i^2+2{\alpha }_i\frac{\ln (x)b_{2i}}{r_i}{u}_i,$$

$$\mathbb{E}[L_i(x,u)-f_i(0,x_0)]=\left(q_i(T)-{\alpha }_i(T)\right){\ln }^2(x(T))$$

$$+\mathbb{E}{\int }_0^T{q}_i{\ln }^2(x)dt+\mathbb{E}{\int }_0^T{\dot{\alpha }}_i{\ln }^2(x)dt$$

$$+\mathbb{E}{\int }_0^T\left(2{\alpha }_i{b}_1{\ln }^2(x)-2{\alpha }_i{\ln }^2(x)\sum _{j\in \mathcal{I}\setminus \{i\}}{\alpha }_j\frac{b_{2j}^2}{r_j}\right)dt$$

$$+\mathbb{E}{\int }_0^T{r}_i{\left(u_i+{\alpha }_i\frac{\ln (x)b_{2i}}{r_i}\right)}^{2}dt$$

$$-\mathbb{E}{\int }_0^T{\alpha }_i^2\frac{{\ln }^2(x)b_{2i}^2}{r_i}dt$$

$$+\mathbb{E}{\int }_0^T\sum _{s^{\prime }\in \mathcal{S}}[{\alpha }_i(t,s^{\prime })-{\alpha }_i(t,s)]{\tilde{q}}_{s{s}^{\prime }}{\ln }^2(x)dt$$

$$\{\begin{array}{l}{L}_i=q_{iT}{l}_1(x_T)+{\int }_0^T{q}_i{l}_1(x)+{\sum }_{j\in \mathcal{I}}{r}_{ij}{l}_2(u_j)dt,\\ {\inf }_{u_i}\mathbb{E}[L_i],\\ \text{mbox}subjectto\\ \mathbb{P}(s(t+ϵ)=s^{\prime }|s)={\int }_t^{t+ϵ}{\tilde{q}}_{s{s}^{\prime }}d{t}^{\prime }+o(ϵ),s^{\prime }\ne s\\ dx=[\frac{b_1{l}_1(x)}{l_1^{\prime }(x)}+\frac{h(x){\sum }_j{b}_{2j}{u}_j}{l_1^{\prime }(x)}]dt+\sqrt{\frac{{\sigma }_1^2+{\sigma }_2^2{l}_1(x)}{l_1^{\prime \prime }(x)}}dB,\\ x(0)=x_0,s(0)=s_0\end{array}$$

$$-l^{\ast }(x)=\underset{u}{\inf }\{l(u)-xu\}.$$

$$\begin{array}[]{ll}u^{*}_{i}=\sum_{s\in\mathcal{S}}{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}_{\{s(t)=s\}}(l^{*}_{2})^{\prime}(-\frac{b_{2i}\alpha_{i}}{r_{ii}}h(x)),\\ \mathbb{E}[L_{i}(x,u)]=\mathbb{E}[\alpha_{i}(0,s_{0})l(x_{0})+\delta_{i}(0,s_{0})],\\ \end{array}$$

$$\begin{array}[]{ll}\dot{\alpha}_{i}+q_{i}+\alpha_{i}(b_{1}+\frac{\sigma_{2}^{2}}{2})+\sum_{s^{\prime}}(\alpha_{i}(t,s^{\prime})-\alpha_{i}(t,s))\tilde{q}_{ss^{\prime}}\\ -\eta_{ii}+\sum_{j\neq i}\eta_{ij}+\alpha_{i}b_{2j}\gamma_{j}=0,\\ \dot{\delta}_{i}+\frac{\sigma_{1}^{2}}{2}+\sum_{s^{\prime}}(\delta_{i}(t,s^{\prime})-\delta_{i}(t,s))\tilde{q}_{ss^{\prime}}=0\\ \end{array}$$

$$\begin{array}[]{ll}r_{ii}l^{*}_{2}(-\frac{b_{2i}\alpha_{i}}{r_{ii}}h(x))=\eta_{ii}l_{1}(x),\\ r_{ij}l_{2}(u^{*}_{j})=\eta_{ij}l_{1}(x),\\ h(x)(l_{2}^{*})^{\prime}[-\frac{\alpha_{j}b_{2j}}{r_{jj}}h(x)]=\gamma_{j}l_{1}(x),\\ \end{array}$$

$$l_2(y)=yh(y),l_1=\kappa l_2,h(x)=\frac{x^{2k-1}}{2k}.$$

Full text

**Keywords : ** Non-Linear, non-quadratic systems, mean-field-type games, risk-awareness, direct method.

1. 1 Introduction

2. 1.1 Direct Method for LQ-MFTG

3. 1.2 Direct Method beyond LQ-MFTG

4. 1.3 Conditional dynamics of mean-field type

5. 1.4 Direct Method

6. 2 Some Solvable Mean-Field-Free Games

7. 2.1 Logarithmic Scale

8. 2.2 Logarithm square

9. 2.3 Legendre-Fenchel

10. 2.4 Geometric Gauss-Volterra Game

11. 3 Some Solvable Mean-Field-Type Games

12. 3.1 Control-dependent switching MFTG

13. 3.2 Quadratic-Quadratic MFTG

14. 3.3 Quadratic State and Power Utility

15. 3.4 Non-Linear State and Log-Utility

16. 3.5 Cotangent Drift

17. 3.6 Hyperbolic coTangent Drift

18. 3.7 A Delayed and Trend-based MFTG

19. 3.8 Mean-Field of MFTG

20. 4 MFTG beyond Brownian motions and Poisson

21. 4.1 Noncooperative MFTG under Gauss-Volterra processes

22. 4.2 Fully Cooperative MFTG under Gauss-Volterra Noise

23. 4.3 Adversarial Mean-Field-Type Game under Gauss-Volterra Noise

24. 5 Numerical Examples

25. 6 Conclusion

1 Introduction

Mean-field-type game theory studies a class of games in which the payoffs and or state dynamics depend not only on the state-action pairs but also the distribution of them. In mean-field-type games, (i) a single decision-maker can have a strong impact on the mean-field terms, (ii) the expected payoffs are not necessarily linear with respect to the state distribution, (iii) the number of decision-makers (“true decision-makers”) is not necessarily infinite.

Games with non-linearly distribution-dependent quantity-of-interest [1, 2, 3] are very attractive in terms of applications because the non-linear dependence of the payoff functions in terms of state distribution allow us to capture risk measures which are functionals of variance, inverse quantiles, and or higher moments. During the past, a significant amount of research on mean-field-type games has been performed [4, 5, 6, 8, 9, 10]. In the time-dependent case, the analysis of mean-field-type games has several challenges. Previous works have devoted tremendous effort in terms of partial integro-differential system of equations (PIDEs), in infinite dimensions, of conditional Liouville, Boltzmann, Kolmogorov or McKean-Vlasov type. At the same time, an important set of numerical tools have been developed to address the master equilibrium system. However, the current state-of-the-art of numerical schemes is problem-specific and needs to be adjusted properly depending on the underlying problem. To date, the question of computation of the master system in the general setting remains open. This work provides explicit solutions of a class of master systems. These explicit solutions can be used to build reference trajectories and several numerical schemes developed to solve PIDEs can be tested beyond the linear-quadratic setting.

1.1 Direct Method for LQ-MFTG

In the current literature, only relatively few examples of explicitly solvable mean-field-type game problems are available. The most notable examples are (i) linear-quadratic mean-field-type games (LQ-MFTG) [6], (ii) linear-exponentiated quadratic mean-field-type games (LEQ-MFTG) [7] , (ii) adversarial linear-quadratic mean-field-type games (minmax LQ, minmax LEQ-MFTG) [6]. In LQ-MFTG the base state dynamics has two components: drift and noise.

• •

the drift is an affine function of the state, expected value of the state, control action and expected value of the control actions of all decision-makers. The coefficients are regime switching dependent.

• •

the noises are combination of diffusion, Gauss-Volterra, jump, regime-switching process where the noise coefficients are affine functions of the state, expected value of the state, control action and expected value of the control actions of all decision-makers. The coefficients are regime switching and jump dependent.

To the state dynamics, one can add a common noise which is a diffusion-Gauss-Volterra-jump-regime-switching process. The cost functions are polynomial of degree two and include the weighted conditional variances, co-variances between state and control actions of all decision-makers. In addition, the cost functional is not measured perfectly. Only a noisy cost is available.

This basic model of LQ mean-field-type games captures several interesting features such as heterogeneity, risk-awareness and empathy of the decision-makers.

To solve LQ-MFTG problems one can use the direct method proposed in Figure 3. This solution approach does not require solving the Bellman-Kolmogorov equations or backward-forward stochastic differential equations of Pontryagin’s type. The proposed direct method can be easily implemented by beginners and engineers who are new to the emerging field of mean-field-type game theory.

For this broader class of LQ-MFTG problem one can derive a semi-explicit solution under sufficient conditions. The existence of solution to the master system corresponding to the LQ-MFTG problem can be converted into an existence of solution to a system of ordinary differential equations driven by common noises. In some particular cases, these systems are stochastic Riccati systems and extensions of Riccati to include some fractional order terms.

1.2 Direct Method beyond LQ-MFTG

The direct method is not limited to the linear-quadratic case. The direct method can be extended to a class of LEQ-MFTG, minmax LQ-MFTG and minmax LEQ-MFTG. In this article, we present several examples to illustrate how the direct method addresses non-linear and/or non-quadratic mean-field-type games. The examples below go beyond LQ-MFTG, LEQ-MFTG and minmax LQ problems.

The contributions of this article can be summarized as follows. We provide semi-explicit solution for classes of mean-field-type game problems presented in Table 1. Several noises are examined: Brownian motion $B$, regime switching $s$, jump process $N$, and Gauss-Volterra process $B_{gv}$. The Gauss-Volterra noise processes are obtained from the integral of a Brownian motion with a suitable kernel function. In addition, several type of common noises are considered: $s,B_{o},N_{o},B_{o,gv}.$. We limit ourselves to the class of state-and-conditional mean-field type feedback strategies. The analysis for more general class of strategies is beyond the scope of this article.

To the best of the authors’ knowledge this is the first work to provide semi-explicit solutions of mean-field-type games beyond LQ and under Gauss-Volterra processes.

Structure

The rest of the article is structured as follows. Section 2 presents semi-explicit solutions to some non-linear non-quadratic stochastic differential games. In Section 3 we formulate and solve various mean-field-type games with non-quadratic quantity-of-interest and provide semi-explicit solutions using a direct method. Section 4 presents semi-explicit solutions to some non-quadratic mean-field-type games driven by Gauss-Volterra processes. Numerical examples are presented in Section 5. The last section summarizes the work.

Preliminary

We introduce the following notations (see Table 2). Let $[0,T],\ T>0$ be a fixed time horizon and $(\Omega,\mathcal{F},\mathbb{F}^{B,N,B_{gv},s,B_{o},B_{o,gv},N_{o}},\mathbb{P})$ be a given filtered probability space. The filtration $\mathbb{F}=\{{\mathcal{F}}^{B,N,B_{gv},s,B_{o},B_{o,gv},N_{o}}_{t},\ 0\leq t\leq T\}$ is the natural filtration of the union of the family $\{B,N,B_{gv},s,B_{o},B_{o,gv},N_{o}\}$ augmented by $\mathbb{P}-$null sets of ${\mathcal{F}}.$ In practice, $B$ is used to capture smaller disturbance, $N$ is used for larger jumps of the system, $B_{gv}$ is used for Gauss-Volterra processes (including sub- or super diffusion). Let $k\geq 1.$ ${L}^{k}([0,T]\times\mathcal{S};\mathbb{R})$ is the set of measurable functions $f:\ [0,T]\times\mathcal{S}\rightarrow\mathbb{R}$ such that $\int_{0}^{T}|f(t,s)|^{k}dt<\infty$. $\mathcal{L}^{k}_{\mathbb{F}}([0,T]\times\mathcal{S};\mathbb{R})$ is the set of $\mathbb{F}$-adapted $\mathbb{R}$-valued processes $X(\cdot)$ such that $\mathbb{E}[\int_{0}^{T}|X(t)|^{k}dt]<\infty.$ The stochastic quantity $\bar{x}(t)=\mathbb{E}[X(t)|\ \mathcal{F}^{s,B_{o},B_{o,gv},N_{o}}_{t}]$ denotes the conditional expectation of the random variable $X(t)$ with respect to the filtration $\mathcal{F}^{s,B_{o},B_{o,gv},N_{o}}_{t}.$ Note that $\bar{x}$ is a random process. Below, by abuse of notation we use $s(t),x(t)$ for the values $s(t_{-}),x(t_{-})$ inside the jump processes $N,N_{o}$ or the regime-switching process $s$. The set of decision-makers is denoted by $\mathcal{I}=\{1,\ldots,I\}.$ An admissible control strategy $u_{i}$ of the decision-maker $i$ is an $\mathbb{F}$-adapted. We denote the set of all admissible controls by $\mathcal{U}_{i}$: Decision-maker $i$ chooses a control strategy $u_{i}\in\mathcal{U}_{i}$ to optimize its performance functional. The information structure of the problem under perfect state observation and under common noise observation $(s,B_{o},B_{o,gv},N_{o}).$

1.3 Conditional dynamics of mean-field type

Consider the following state dynamics of conditional McKean-Vlasov type with time delays, trend, diffusion, jump, regime switching, Gauss-Volterra and common noises.

[TABLE]

where

• •

$\mathcal{I}=\{1,\dots,I\}$ is the set of decision-makers.

• •

$x_{i}=x_{i}(t)$ is the basic state at time $t$ of the decision-maker $i$

• •

$\tau_{ik}>0$ represents a time delay,

• •

$y_{i}=(x_{i}(t-\tau_{ik}))_{1\leq k\leq K},$ is a $K-$dimensional delayed state vector,

• •

$z_{i}(t)=(\int_{t-\tau_{i}}^{t}\lambda_{i}(dt^{\prime})\phi_{il}(t,t^{\prime})x_{i}(t^{\prime}))_{l\leq I}$ is the integral state vector of the recent past state over $[t-\tau_{i},t].$ The trend of the state of decision-maker $i$ is its latest moving averages. $z_{i}(t)$ represents the trend of the state of $i$. The process $\phi_{il}(t,t^{\prime})$ is an $\mathcal{F}_{t^{\prime}}-$adapted locally bounded process, $\lambda_{i}$ is a positive and $\sigma-$finite measure on $[-\tau_{i},T]$.

• •

$m_{1}$ is the distribution states of all the other decision-makers,

• •

$m_{2}$ the distribution of actions of all other decision-makers,

• •

$x_{i0}$ is a initial deterministic function of state of $i$ defined on $[-\tau_{i},0].$

• •

$B$ be a Brownian motion on $\mathcal{T}=[0,T]$ with suitable dimension. $B_{o}$ be a Brownian motion observed by all decision-makers.

• •

$B_{gv}$ be a Gauss-Volterra process on $\mathcal{T}$ with suitable dimension and with integrable kernel $K.$ $B_{o,gv}$ be a Gauss-Volterra process observed by all decision-makers

• •

${N}(dt,d\theta,s)$ be a jump process with suitable dimension on $\mathcal{T}$ with compensated jump $\tilde{N}(dt,d\theta,s)={N}(dt,d\theta,s)-\nu(d\theta)dt$, $\nu$ is a Radon measure over $\Theta.$ ${N}_{o}$ is a common jump process observed by all decision-makers.

• •

$s(t)$ is a regime switching process defined over the finite set $\mathcal{S}=\{1,2,\ldots,S\}$ with switching rate $\tilde{q}$ satisfying $\tilde{q}_{ss^{\prime}}>0,s\neq s^{\prime}$ and $\tilde{q}_{ss}:=-\sum_{s^{\prime}\neq s}\tilde{q}_{ss^{\prime}}.$ We use ${\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}_{\{s(t)=s\}}$ to denote the indicator function on the condition $\{s(t)=s\}.$

• •

$u=(u_{i})_{i\in\mathcal{I}}$ is the control strategy profile of all decision-makers. An admissible control strategy $u_{i}$ of the decision-maker $i$ is an $\mathbb{F}$-adapted process.

• •

The processes $B,B_{gv},N,B_{o},B_{o,gv},N_{o},s$, are defined in a given filtered probability space $(\Omega,\mathbb{F},\mathcal{F}^{B,N,B_{gv},s,B_{o},B_{o,gv},N_{o}},\mathbb{P})$ $(\mathbb{F}=\{\mathcal{F}^{B,N,B_{gv},s,B_{o},B_{o,gv},N_{o}}_{t}\}_{t\in\mathcal{T}}).$ The processes $B_{o},B_{o,gv},N_{o},s$ are common noises assumed to be observable by all decision-makers. All the processes are assumed to be mutually independent.

• •

The coefficient functionals $b,\sigma,\sigma_{gv},\mu,\sigma_{o},\sigma_{o,gv},\mu_{o}$ are of compatible dimensions with $x.$

• •

The quantity $\bar{\omega}(t)=\mathbb{E}[\omega(t)|\ \mathcal{F}^{B_{o},B_{o,gv},N_{o},s}_{t}]$ denotes the conditional expectation of the random variable $\omega(t)$ with respect to the filtration $\mathcal{F}^{B_{o},B_{o,gv},N_{o},s}_{t}.$ Note that $\bar{\omega}$ is a random process. We take $\omega\in\{u,x,y,z\}.$ By abuse of notation we use $s(t),x(t)$ for the values $s(t_{-}),x(t_{-})$ inside the jump processes $N,N_{o}$ or the regime-switching process $s$.

Let $f(t,x,s)$ be a twice continuously differentiable function in $x$ and continuously differentiable in time $t$ for each regime $s\in\mathcal{S}.$ Using [34] and [33, Theorem 4.1], the stochastic integration formula, which is an extended Itô’s formula, yields

[TABLE]

Notice that (2) applies to a one-dimensional state as well as to a vector, matrix, tensor, lattice or another object in a Hilbert space. For vectors in an Euclidean space, the inner product is $\langle a,b\rangle=\sum_{l=1}^{d}a_{l}b_{l},$ for matrices, $\langle a,b\rangle=trace[a^{*}b]=trace[b^{*}a],$ where $a^{*}$ is the transpose of $a.$

1.4 Direct Method

Consider $I$ decision-makers under perfect state observation $x$ and common noise observation $(B_{o},B_{o,gv},N_{o},s).$ Given $I$ cost functionals $L_{i}(x,y,z,u,s)$ associated with (1), we use (2) in the direct method described as follows. The direct method consists of five elementary steps (see Figure 3).

• •

The first step starts by setting the mean-field terms of the problem.

• •

The second step consists of the identification of a partial guess functional where the coefficient functionals are random and regime switching dependent. For each decision-maker $i$, one needs to identify a guess functional $f_{i}(t,x,y,z,u,s).$

• •

In the third step we compute the difference $L_{i}-f_{i}(t,x,y,z,u,s)$ using the stochastic integration formula (2).

• •

In the fourth step, we use completion of terms in one-shot optimization for both control actions and conditional mean-field of the control actions of all decision-makers. Terms completion make $\mathbb{E}[L_{i}-f_{i}(t,x,y,z,u,s)]\geq 0$ by matching coefficients. The latter inequality becomes equality iff the optimal control strategies are used.

• •

The fifth and last step uses an algebraic basis of linearly independent processes to identify the coefficients. The identification leads to a (possibly stochastic) differential system of equations, providing a semi-explicit representation of the solution. The matched coefficients provide simpler differential systems that are uncoupled with the state.

2 Some Solvable Mean-Field-Free Games

We start with mean-field-free settings where logarithm, logarithm square, Legendre-Fenchel duality, and power payoffs are presented. The cost functions are not necessarily quadratic and the state dynamics is not necessarily linear.

2.1 Logarithmic Scale

Consider a set of decision makers $\mathcal{I}=\{1,\dots,I\}$ interacting in the following non-linear non-quadratic mean-field-free game:

[TABLE]

and with a given initial condition $x(0)\triangleq x_{0}>>e,s(0)=s_{0}\in\mathcal{S},$ $k\geq 1$ is an integer, and $\tilde{q}_{ss^{\prime}}>0,s\neq s^{\prime}$ and $\tilde{q}_{ss}:=-\sum_{s^{\prime}\neq s}\tilde{q}_{ss^{\prime}}.$ $\blacksquare$

Proposition 1

Assume that $x_{0}>>e,r_{i}(.)>\delta>0,q_{i}(.)\geq 0,\ \mu(\theta)\geq 0,$ $\int_{\Theta}[\ln(1+\mu(\theta))-\mu(\theta)]\nu(d\theta)<\infty.$ The non-linear non-quadratic mean-field-free Nash equilibrium and the corresponding equilibrium cost are given by:

[TABLE]

where $\alpha_{i}$ and $\delta_{i}$ satisfies the following differential equations:

[TABLE]

where $\alpha_{i}(T,s)=q_{i}(T,s)$, and $\delta_{i}(T,s)=0$ $\blacksquare$.

.

Proof. Consider the following guess functional:

[TABLE]

By applying Itô’s formula for jump-diffusion-regime switching processes, the gap between the cost and the guess functional $\mathbb{E}[L_{i}(x,u)-f_{i}(0,x_{0})]$ can be computed and it is given by

[TABLE]

Noting that

[TABLE]

with equality iff $u_{i}=u_{i}^{*}:=[\frac{1}{2k}\frac{\alpha_{i}b_{2i}}{r_{i}}]^{\frac{1}{2k-1}},$ the announced result follows. $\blacksquare$

Notice that the differential system (4) has a unique solution: the system in $\alpha$ is linear and the system in $\delta$ is obtained by integration. $\ln(x)$ is well-defined because the state $x$ stays positive in $[0,T]$ almost surely if one starts at $x_{0}>>e.$

Remark 1

For $k=1$ the system reduces to the following ordinary differential equations:

[TABLE]

2.2 Logarithm square

Consider the following non-linear non-quadratic mean-field-free game:

[TABLE]

$\blacksquare$

Proposition 2

Assume that $x_{0}>>e,\ q_{i}(t,s)\geq 0,r_{i}(t,s)>\delta>0,$ and $\int_{\theta\in\Theta}[\ln(1+\mu(\theta))-\mu(\theta)]\nu(d\theta)<\infty.$ The non-linear non-quadratic mean-field-free Nash equilibrium and corresponding optimal cost are given by:

[TABLE]

where $\alpha_{i}$ satisfies the following differential equation:

[TABLE]

$\blacksquare$

Proof. Consider the following guess functional:

[TABLE]

Applying the Itô’s formula yields

[TABLE]

Thus, the gap $\mathbb{E}[L_{i}(x,u)-f_{i}(0,x_{0})]$ is given by

[TABLE]

By performing square completion one obtains

[TABLE]

then,

[TABLE]

Finally, the announced result is obtained by minimizing the terms. $\blacksquare$

2.3 Legendre-Fenchel

We consider a convex running loss functions $l_{1},l_{2}.$

[TABLE]

where $\tilde{q}_{ss^{\prime}}>0,s\neq s^{\prime}$ and $\tilde{q}_{ss}:=-\sum_{s^{\prime}\neq s}\tilde{q}_{ss^{\prime}}.$ Recall that the Legendre-Fenchel transform of $l,$ is given by

[TABLE]

Proposition 3

Assume that l_{1},l_{2},l^{\prime\prime}_{2},r_{ij},q_{i}\ are positive. Then, the game problem (8) has a solution:

[TABLE]

with

[TABLE]

where

[TABLE]

These conditions are fulfilled by choosing for example

[TABLE]

$\blacksquare$

Proof Step 1: we observe that the structure of the problem is mainly driven by the evolution of the function $l_{1}.$

Step 2: Inspired the nature of the problem, we propose a guess functional in the form of $l_{1}$ with deterministic coefficients $\alpha_{i},\delta_{i}.$ Let $f_{i}(t,x,s)=\alpha_{i}(t,s)l_{1}(x)+\delta_{i}(t,s).$

Step 3: We apply stochastic integration formula for diffusion-regime switching to obtain the difference between the cost and the guess functional as

[TABLE]

Step 4: Observing that

[TABLE]

with equality iff $u_{i}=u_{i}^{*},$ the one-shot optimization provides $u_{j}^{*}=(l_{2}^{*})^{\prime}[-\frac{\alpha_{j}b_{2j}}{r_{jj}}h(x)],$

Step 5: By identification of processes, the announced result follows. This completes the proof. $\blacksquare$

2.4 Geometric Gauss-Volterra Game

The Gauss-Volterra processes are singular integrals of a standard Brownian motion and include (i) fractional Brownian motions, (ii) Liouville fractional Brownian motions, and (iii) multi-fractional Brownian motions. The difficulty of finding semi-explicit solution is significantly increased if the noise process and thereby the state process is driven by non-Markov processes or non-martingales. Let $B_{gv}$ be a Gauss-Volterra process with zero mean and covariance

[TABLE]

The kernel $K$ is assumed to have causality, continuity and integrability properties as in [27]. The variance of the process $\int_{0}^{t}\sigma_{gv}(t^{\prime},s)dB_{gv}(t^{\prime})$ is given by

[TABLE]

Consider the following geometric Gauss-Volterra game with unobserved processes $B_{o},B_{o,gv},N_{o}$ which are assumed to be independent.

[TABLE]

where $b=b_{1}\sqrt{x}+\sum_{j\in\mathcal{I}}{b_{2j}}u_{j},$ and $\sigma$, $\sigma_{o}$, $\sigma_{gv}$, $\sigma_{o,gv}$, $\mu$, and $\mu_{o}$ are real valued and regime-switching dependent functions $s(t)$ with $\tilde{q}_{ss^{\prime}}>0,\ s\neq s^{\prime}$ and $\tilde{q}_{ss}:=-\sum_{s^{\prime}\neq s}\tilde{q}_{ss^{\prime}}.$

Proposition 4

Assume that $q_{i}\geq 0,\ r_{i}>\delta>0,\ x_{0}>0,k\geq 1.$ The mean-field-free equilibrium for the Geometric Gauss-Volterra Game in (19) is given by

[TABLE]

where $\alpha_{i}$ satisfies the following differential equation:

[TABLE]

$\blacksquare$

Proof. This proof is developed following a direct method.

Step 1: Observe that the problem is a mean-field free problem driven by $\sqrt{x}$.

Step 2: Based on the structure of the problem we propose the following guess functional:

[TABLE]

Step 3: We apply stochastic integration formula for jump-diffusion-regime-switching Gauss-Volterra and common noises to compute the difference between the costs and the guess functionals, i.e.,

[TABLE]

Step 4: we perform terms completion:

[TABLE]

Step 5: We perform process identification after having replaced back the optimal control inputs in the gap $\mathbb{E}[L_{i}(x,u)-f_{i}(0,s_{0})]$, i.e.,

[TABLE]

Finally, the announced result is obtained by minimizing terms, which completes the proof. $\blacksquare$

Notice that under the conditions: $q_{i}\geq 0,r_{i}>\delta>0,i\in\mathcal{I},k\geq 1$ the differential system (20) has a positive solution.

When $s,B_{o},B_{o,gv},N_{o}$ are noises observed by all decision-makers (observed common noises), the ordinary differential system in $\alpha$ becomes a stochastic differential system driven by the union of events with $B_{o},B_{o,gv},N_{o}.$

[TABLE]

with the terminal condition $q_{i}(T,s)\geq 0$ being $\mathcal{F}^{B_{o},B_{o,gv},N_{o}}$-measurable random coefficient.

3 Some Solvable Mean-Field-Type Games

3.1 Control-dependent switching MFTG

In most continuous time MFTG models with regime switching considered in the literature it is assumed that the switching rate $\tilde{q}_{ss^{\prime}}$ is control-independent. In this subsection, we provide an example with control-dependent switching rate $\tilde{q}_{ss^{\prime}}(u)$ in which the MFTG problem can be solved semi-explicitly.

[TABLE]

where $b_{kjss^{\prime}}>0$ for $s^{\prime}\neq s$ and $\mathbb{E}{b}_{1jss^{\prime}}=0.$

Proposition 5

Assume that $r_{i},\bar{r}_{i}>0.$ The equilibrium strategy is

[TABLE]

and the equilibrium cost is $V_{i}(t,s),$ which satisfies the following ordinary differential system:

[TABLE]

$\blacksquare$

Proof.

Step 1: We observe that the structure of the problem does not have a drift and is driven by regime switching.

Step 2: Based on step 1, we propose guess in the following form: $V_{i}(t,s).$

Step 3: we use the stochastic integration formula for regime-switching to compute the difference between the cost and the guess functional as:

[TABLE]

Step 4: Assuming that $[r_{i}+\sum_{s^{\prime}\in\mathcal{S}}V_{i}(t,s^{\prime})b_{2iss^{\prime}}]>0,$ and $[\bar{r}_{i}+\sum_{s^{\prime}\in\mathcal{S}}V_{i}(t,s^{\prime})\bar{b}_{2iss^{\prime}}]>0$ the terms completion lead to a one-shot optimization of a strictly convex and coercive function.

Step 5: the minimization and the identification of the processes provides the announced result.

3.2 Quadratic-Quadratic MFTG

This example examines a class of Quadratic-Quadratic Mean-Field-Type Game (QQ-MFTG) problem. The state is non-linear in $u.$ A semi-explicit solution is derived.

[TABLE]

where the coefficients are regime-switching dependent with switching rate matrix $\tilde{Q}=(\tilde{q}_{ss^{\prime}},(s,s^{\prime})\in\mathcal{S}^{2}).$

Proposition 6

Assume that $r_{i}>0,q_{i}\geq 0,\ r_{i}>\delta>0,\bar{r}_{i}>0,\bar{q}_{i}\geq 0,\bar{r}_{i}>\delta>0,$ $\mathbb{E}[\epsilon_{1i}]=0,~{}\forall i\in\mathcal{I}.$ The QQ-MFTG problem (26) has unique solution and it is given by

[TABLE]

Note that the semi-explicit solution is in fact an explicit solution. Let $\vec{\alpha}_{i}(t)=[\alpha_{i}(t,s)]_{s\in\mathcal{S}}$, and $\vec{q}_{i}(T)=[q_{i}(T,s)]_{s\in\mathcal{S}}$, then $\vec{\alpha}_{i}(t)$ is explicitly given by

[TABLE]

*in particular $\vec{\alpha}_{i}(0)=\vec{q}_{i}(T)\exp[{-\int_{0}^{T}\tilde{Q}\mathrm{d}t^{\prime}}]$. $\blacksquare$

Proof. Let us consider the following guess functional: $V_{i}=\alpha_{i}x$. Then,

[TABLE]

Itô’s formula yields

[TABLE]

Thus, the difference $\mathbb{E}[L_{i}-V_{i}(0)]$ is given by

[TABLE]

Performing square completion yields

[TABLE]

[TABLE]

Minimizing terms it yields

[TABLE]

completing the proof. $\blacksquare$

Remark 2

Notice that using the result presented in Proposition 6, the following Quadratic-Exponential-Quadratic Mean-Field-Type Game (QEQ-MFTG) problem:

[TABLE]

can be solved explicitly. $\blacksquare$

3.3 Quadratic State and Power Utility

This subsection examines a class of mean-field-type games with power payoffs and a non-linear state. The model is inspired from the modern portfolio optimization under shared asset platform by several decision-makers. The state $x(t)$ is the total amount of money. Decision-maker $i$ can decide to consume certain amount $u_{1i}$ and re-allocate the remaining between less-risky assets $(1-u_{2i})\kappa_{2}x$ and more risky assets $u_{2i}\kappa_{2}x+\bar{u}_{2i}{x}[{\sigma}dB+\int{\mu}d\tilde{N}].$ The coefficients $\kappa_{1},\kappa_{2}$ depend on time $t$ and on the switching regime $s(t)$ which takes values in $\mathcal{S}.$ The set $S$ is non-empty and finite. We have modified the model to include mean-field terms, a function of the expected value of the state and a function of the expected value of the control action.

[TABLE]

where the coefficients are regime-switching dependent, and $k_{i}\geq 1,\bar{\rho}_{i}\in(0,1).$ The coefficients $q_{i},\bar{q}_{i},r_{i},\bar{r}_{i}$ are positive. The state dynamics (33) is not linear in $(x,u).$

Following the same method as in the problem (33), a semi-explicit solution can be derived.

Note that a similar method can be used to derive semi-explicit solution to the following game problem in which decision-makers minimize with $k_{i}(t,s)>1,\ \bar{\rho}_{i}(t,s)>1.$

[TABLE]

This can be easily extended to include multi-type power utilities in the following form:

[TABLE]

In this case, the guess functional will be

[TABLE]

with an ordinary differential system for $p_{ik},$ and $\bar{p}_{ik}.$

3.4 Non-Linear State and Log-Utility

We consider the following logarithmic Cobb-Douglas utility.

[TABLE]

where the coefficients are regime-switching dependent.

Note that the state dynamics (36) is not linear in $(x,u).$

Following the same method as above, the problem (36) can be solved explicitly.

3.5 Cotangent Drift

This subsection we examine mean-field-type games with cotangent drift. This class of games is inspired from [11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. We have modified the model to include mean-field terms. Using trigonometric relationships a semi-explicit equilibrium solution is derived.

[TABLE]

where the coefficients are regime-switching dependent and $cot(\theta)=\frac{1}{\tan(\theta)}=-cot(-\theta).$

Proposition 7

Assume that $q_{i}>0,\bar{q}_{i}>0.$ The mean-field-type game problem with cotangent drift (37) has a unique equilibrium solution which is given by

[TABLE]

whenever the following system

[TABLE]

has a unique solution with positive $\alpha_{i},\bar{\alpha}_{i}$ which do not blow up within $[0,T].$ $\blacksquare$

Proof:

We prove the statement using a direct method. Step 1: We observe that the mean-field-type problem is driven by functionals of $x-\bar{x}$ and $\bar{x}$ which are conditionally orthogonal processes.

Step 2: Given the structure of the problem, we propose the following guess functional:

[TABLE]

be a guess functional.

Step 3: we apply Brownian with regime switching to obtain the difference between the cost functional and the guess functional as:

[TABLE]

Step 4: Noting the terms completion leads to a strictly concave one-shot optimization with coercive function $\cos^{2}(\frac{\bar{x}}{4})[\bar{u}_{i}+\frac{\bar{\alpha}_{i}}{4}\bar{b}_{2i}tan(\frac{\bar{x}}{4})]^{2}$ whenever $\cos^{2}(\frac{\bar{x}}{4})>0.$

Step 5: By identification of processes one obtains the announced result. This completes the proof. $\blacksquare$

The mean-field term $\bar{x}$ solves

[TABLE]

which has a unique solution for $\bar{x}_{0}\in(0,\pi).$

3.6 Hyperbolic coTangent Drift

Problem (37) can be modified to handle the hyperbolic cotangent drift case as specified below. The functions $\cos,\sin,\tan,\cot$ are replaced by $\cosh,\sinh,\tanh,\coth$ respectively.

[TABLE]

where the coefficients are regime-switching dependent and $\coth(\theta)=\frac{e^{\theta}+e^{\theta}}{e^{\theta}-e^{-\theta}}=-\coth(-\theta).$

Proposition 8

The equilibrium strategies and the equilibrium costs are given by

[TABLE]

whenever the following system:

[TABLE]

has unique solution with positive $(\alpha_{i},\bar{\alpha}_{i})_{i}$ which do not blow up within $[0,T].$ $\blacksquare$

The system in (44) shares some similarities with the system in (39) of Problem (37). However, these two systems are different. In particular, the sign of the terms $(2+\sigma^{2})\frac{\alpha_{i}}{8},$ and $\frac{\bar{\alpha}_{i}}{4}$ have changed.

Proof.

We prove the statement on the hyperbolic game using a direct method. Let

[TABLE]

be a guess functional combining hyperbolic functions.

[TABLE]

By identification one obtains the announced result. This completes the proof. $\blacksquare$

The mean-field term $\bar{x}$ solves

[TABLE]

which has a unique global solution within $[0,T].$

3.7 A Delayed and Trend-based MFTG

We present a cooperative MFTG with basic state dynamics $x(t)$, regime switching $s(t),$ a trend $y(t):=\int_{-\tau}^{0}e^{\lambda t^{\prime}}x(t+t^{\prime})dt^{\prime}$ on the time window $[t-\tau,t],$ the delayed state $z(t)=x(t-\tau).$ This class of examples plays an important role in real-world applications as the effects of actions are not instantaneous in general [28, 29, 30, 31]. It may take a certain time delay. This leads to delayed and trend-based stochastic differential equations of mean-field type.

[TABLE]

where $var(X)$ denotes the variance of the random variable $X,$ and $\bar{X}(t)=\mathbb{E}[X(t)|\ \mathcal{F}^{s,B_{o}}]$ is the conditional expectation with respect to the common noises $s,B_{o}.$

Lemma 1

The conditional expected trend $\bar{y}$ satisfies the following stochastic differential equation:

[TABLE]

$\blacksquare$

**Proof: **

[TABLE]

Taking the conditional expected values one obtains

[TABLE]

This completes the proof. $\blacksquare$

Proposition 9

The equilibrium strategies and the equilibrium payoff of the delayed MFTG (47) are given by

[TABLE]

whenever the following system:

[TABLE]

has a unique solution $\alpha,\beta,\beta_{B_{o}}$ $\blacksquare$

Note that the system in $\alpha$ has a positive solution if $q\geq 0,q_{T}\geq 0,\ r_{1}>0.$ With single regime $\mathcal{S}=\{s_{0}\}$ the $\beta$ equation yields

[TABLE]

This is completely solvable with an explicit solution given by

[TABLE]

where

[TABLE]

**Proof:  **

Let $f(x)=-\alpha\ var(x)+\beta\frac{(\bar{x}+\bar{\eta}\bar{y})^{\rho}}{\rho},$ be a guess functional.

[TABLE]

with the following careful matching $\eta=\bar{b}_{13}e^{\lambda\tau},\ \bar{b}_{12}=\bar{b}_{13}e^{\lambda\tau}(b_{11}+\lambda+\bar{b}_{13}e^{\lambda\tau}).$ The joint optimization over $(u_{1},u_{2})$ together with the mean-field terms $(\bar{u}_{1},\bar{u}_{2})$ gives the announced result provided that $\rho<1.$

3.8 Mean-Field of MFTG

This subsection we examine a class of mean-field of mean-field-type games.

In view of the delayed mean-field-type game (47), we have modified $\bar{r}_{1}$ to be $\bar{r}_{1}(m)$ where $m$ is the conditional total consumption of the large population. Then, $m$ is obtained as

[TABLE]

where $\mu^{m}(t,d\bar{x},d\bar{y})$ is the conditional distribution of all players’ states and trends in the large population under $m,$ which reduces to the fixed-point problem $m=(\bar{x}(m)+\bar{\eta}\bar{y}(m))(\frac{\beta(m)}{\bar{r}_{1}(m)})^{\frac{1}{\rho-1}}.$

Now consider the following modified Cournot-Ross game with $I$ producers and a large population of potential consumers. The mean-field-type version of the game under common noise is analyzed in [21].

[TABLE]

Let $D(m)$ be the demand generated by a large population of consumers. Given a demand $D(m),$ each macro-player $i$ has a certain utility of mean-field type. The payoff function in the Cournot game $\bar{x}\bar{u}_{i}+\bar{r}_{i}\frac{\bar{u}_{i}^{2}}{2}$ is modified to be $\bar{x}^{2k-1}\bar{u}_{i}-\bar{r}_{i}\frac{\bar{u}_{i}^{2k}}{2k}$ and some extra mean-field dependent terms.

By means of a direct method one can fully characterize the mean-field equilibrium of (52). It is given by following set of equations:

[TABLE]

where $\alpha_{i},\bar{\alpha}_{i}$ solve a system of ordinary differential equations.

4 MFTG beyond Brownian motions and Poisson

In this section class of mean-field-type games with a state dependent Gauss-Volterra noise is formulated and solved with a polynomial and mean-field dependent payoff for an arbitrary number of players and a finite time horizon. The control strategies are linear state and mean-field feedbacks. A mean-field-type Nash equilibrium is verified for the game and the optimal strategies are obtained using a direct method that does not require solving nonlinear partial integro-differential equations or forward-backward stochastic differential equations. The example below is inspired from [22, 23, 24, 25, 26, 27]. We add mean-field terms to these previous works. This will allow us to solve variance or higher moment reduction problems.

4.1 Noncooperative MFTG under Gauss-Volterra processes

This section examines a class of noncooperative mean-field-type games with non-quadratic cost and state driven by Gauss-Volterra processes.

[TABLE]

where $k_{i}\geq 1,\bar{k}_{i}\geq 1$ are natural numbers, the coefficients are time and switching dependent,

Remark 3

The cost functional is clearly non-quadratic for $k_{i}>1$ or $\bar{k}_{i}>1.$ For $k_{i}=1$ the equilibrium of the variance reduction game (54) under Gauss-Volterra processes is obtained.

Proposition 10

Assume $q_{i}>0,\bar{q}_{i}>0,r_{i},\bar{r}_{i}>\delta$ and $\int_{\theta}[(1+\mu)^{2k_{i}}-1-2k_{i}\mu]\nu(d\theta)<+\infty.$ The mean-field Nash equilibrium of the mean-field type game (54) under Gauss-Volterra process is given by

[TABLE]

whenever the following system of ordinary differential equations admit a positive solution which does not blowup within $[0,T].$

[TABLE]

$\blacksquare$

**Proof: **

Consider the guess functional $f_{i}=\alpha_{i}\frac{(x-\bar{x})^{2k_{i}}}{2k_{i}}+\bar{\alpha}_{i}\frac{\bar{x}^{2\bar{k}_{i}}}{2\bar{k}_{i}}$

[TABLE]

We complete the following term:

[TABLE]

A similar completion is done for the terms in $\bar{u}_{i}.$ Thus,

[TABLE]

Noting for $k_{i}>\frac{1}{2},r_{i}>0$ one has:

[TABLE]

with equalities in (60) iff

[TABLE]

By identification, one obtains the announced result. $\blacksquare$

The derived system of equations (56) are inhomogeneous differential system where it is known that existence and uniqueness, nonexistence or nonuniqueness may occur.

Existence

Some results on sufficient conditions for the existence of trajectories satisfying the associated set of non-linear differential equations (56) are outlined. Below we present Carathéodory conditions for existence of a solution.

Here, the non-linear differential system (56) can be written as

[TABLE]

Assume that

• •

For each fixed time $t\in[0,T],$ $h$ is continuous in $\alpha.$

• •

For each fixed $\alpha,$ $h$ is measurable in $t.$

• •

Given a nonempty compact $C\subset\mathbb{R}^{n|\mathcal{S}|}$ and interval $[T-\epsilon,T],$ there is an integrable positive function $\hat{h}$ on the time interval $[T-\epsilon,T]$ such that |h(t,\alpha)|\leq\hat{h}(t),\ for all $(t,\alpha)\in[0,T]\times C.$

The interval of definition of the solution depends on the terminal value $q(T,s).$ For each terminal condition $q(T,.)\in\mathbb{R}^{n|\mathcal{S}|},$ there is an interval $(T-\epsilon,T),\ \epsilon>0$ where this non-linear differential system (61) has at least one solution. We refer to [32, Theorem 1.1, Chapter 2] for a detailed proof.

Uniqueness

It is a well-known result that not every non-linear differential system has a unique solution. Therefore, the uniqueness issue is dealt with in a separate result. We provide two sufficient conditions for having at most one solution:

• •

If $h$ is continuously differentiable in $\alpha$ and $|h_{\alpha}(t,\alpha)|\leq\hat{h}(t),$ on $(t,\alpha)\in[0,T]\times C,$ then there is at most one solution on $[T-\epsilon,T].$

• •

If $|h(t,\alpha_{1})-h(t,\alpha_{2})|\leq\hat{h}(t)|\alpha_{1}-\alpha_{2}|,$ on $(t,\alpha)\in[0,T]\times C,$ then there is at most one solution on $[T-\epsilon,T].$

It is important the notice that the function $h$ is not necessarily globally Lipschitz in $\alpha.$ For example, for $k\geq 1,$ $\alpha_{i}^{\frac{2k}{2k-1}}$ is not necessarily globally Lipschitz. Therefore we need estimates of $\alpha.$ We rely on the original dynamic optimization problem to derive lower and upper bounds on $\alpha_{i}.$ Since $q(T,.),q(t,.)\succ 0,r_{i}>\delta>0$ by assumption, lower bound for $\alpha_{i}$ is zero. This can also be obtained directly from the problem formulation as the cost is positive. By summing up (56) over $i\in\mathcal{I},$ an upper bound is obtained as $\sum_{i}\alpha_{i}$ is bounded subject to integrability condition of the coefficients.

Note, however, that the stationary system may have multiple solutions, depending on the parameters.

Admissibility of the coefficient solution

As $\epsilon$ in the Carathéodory existence result depends on $q(T,.),$ the maximal interval in which the solution is defined may depend on $q(T,.).$ Thus, we need to examine the singularity of $\alpha$ in (56). In order for the control strategies to be admissible, we seek sufficient conditions for non-blow-up (no escape) within $[0,T$ If $q_{i}>0,r_{i}>\delta>0$ and all coefficients continuous, then there is no escape within $[T-\epsilon,T].$ If in addition the coefficient functions $b_{1},\sigma^{2},\sigma^{2}_{cogv},\int_{\theta}[(1+\mu)^{2k_{i}}-1-2k_{i}\mu]\nu(d\theta),$ $r_{i}(\frac{b_{2i}}{r_{i}})^{\frac{2k_{i}}{2k_{i}-1}}$ and $(\frac{{b}_{2i}}{{r}_{i}})^{\frac{1}{2{k}_{j}-1}}$ are all integrable within $\mathcal{T}$, then there is no escape of $\alpha$ within the entire $[0,T]$ as the estimates of $\sum_{i\in\mathcal{I}}\alpha_{i}$ is finite in $\mathcal{T}.$

Similar reasoning works for $\bar{\alpha}$ when $\bar{q}_{i}>0,\bar{r}_{i}>\delta>0$ and the coefficient functions $\bar{b}_{1},$ $\bar{r}_{i}(\frac{\bar{b}_{2i}}{\bar{r}_{i}})^{\frac{2\bar{k}_{i}}{2\bar{k}_{i}-1}},$ $(\frac{\bar{b}_{2i}}{r_{i}})^{\frac{1}{2\bar{k}_{i}-1}}$ are integrable within $\mathcal{T}.$

At equilibrium, the mean-field term $\bar{x}$ in Proposition 10 solves

[TABLE]

which admits a unique solution within $[0,T]$ subject to the integrability of the regime switching dependent coefficient [\bar{b}_{1}+\sum_{j}\bar{b}_{2j}\Bigg{(}-\frac{\bar{b}_{2j}\bar{\alpha}_{j}}{\bar{r}_{j}}\Bigg{)}^{\frac{1}{2\bar{k}_{j}-1}}] over $[0,T].$

4.2 Fully Cooperative MFTG under Gauss-Volterra Noise

In this subsection we choose $k_{i}=k,\ \bar{k}_{i}=\bar{k}$ and assume that the $I$ decision-makers are fully cooperative. They jointly decide and solve the following problem:

[TABLE]

Proposition 11

The global optimum of the fully cooperative mean-field type game (63) under Gauss-Volterra process is given by

[TABLE]

whenever the following system of ordinary differential equations admit a positive solution which does not blowup within $[0,T].$

[TABLE]

$\blacksquare$

**Proof: ** The proof follows similar steps as in Proposition 10.

Remark 4

A sufficient condition for existence and uniqueness of the global optimum of mean-field type is obtained for $r_{i}>\delta>0,\ \sum_{i}q_{i}>0,\int_{\theta}[(1+\mu)^{2k}-1-2k\mu]\nu(d\theta)<+\infty$ and all coefficients continuous. Then there is no escape within $[T-\epsilon,T].$ If in addition the coefficient functions $b_{1},\sigma^{2},\sigma^{2}_{cogv},\int_{\theta}[(1+\mu)^{2k}-1-2k\mu]\nu(d\theta),$ $r_{i}(\frac{b_{2i}}{r_{i}})^{\frac{2k}{2k-1}}$ and $(\frac{{b}_{2i}}{{r}_{i}})^{\frac{1}{2{k}-1}}$ are all integrable within $[0,T]$, then is no escape of $\alpha$ within the entire interval $[0,T].$ Similar reasoning works for $\bar{\alpha}$ when $\bar{r}_{i}>\delta>0,i\in\mathcal{I},\ \sum_{j}\bar{q}_{j}>0,$ and the coefficient functions $\bar{b}_{1},$ $\bar{r}_{i}(\frac{\bar{b}_{2i}}{\bar{r}_{i}})^{\frac{2\bar{k}_{i}}{2\bar{k}-1}},$ $(\frac{\bar{b}_{2i}}{r_{i}})^{\frac{1}{2\bar{k}-1}}$ are integrable within $[0,T].$

At the global optimum, the mean-field term $\bar{x}$ in Proposition 11 solves

[TABLE]

*which admits a unique solution within $[0,T]$ subject to the integrability of the regime switching dependent coefficient \bar{b}_{1}+\sum_{j}\bar{b}_{2j}\Bigg{(}-\frac{\bar{b}_{2j}\bar{\alpha}_{0}}{\bar{r}_{j}}\Bigg{)}^{\frac{1}{2\bar{k}-1}} over $[0,T].$ *

Remark 5

Notice that the differential equation

[TABLE]

has a unique solution within $[0,T].$ Moreover, the unique solution is positive.

4.3 Adversarial Mean-Field-Type Game under Gauss-Volterra Noise

In this subsection we choose $k_{i}=k\geq 1,\ \bar{k}_{i}=\bar{k}\geq 1$ and assume that the $n$ decision-makers are divided into two teams $\mathcal{I}_{+}=\{i\in\mathcal{I}\ |\ r_{i}>\delta>0,\bar{r}_{i}>\delta>0\},\mathcal{I}_{-}=\{i\in\mathcal{I}|\ r_{i}<-\delta<0,\bar{r}_{i}<-\delta<0\}$ and $\mathcal{I}=\mathcal{I}_{+}\cup\mathcal{I}_{-}.$ The decision-makers in team $\mathcal{I}_{+}$ minimize the functional $\mathbb{E}L_{ad}(x,u)$ over $(u_{i})_{i\in\mathcal{I}_{+}}.$ The decision-makers in team $\mathcal{I}_{-}$ maximize $\mathbb{E}L_{ad}(x,u)$ over $(u_{j})_{j\in\mathcal{I}_{-}}$ This leads to a minmax game problem:

[TABLE]

A mean-field-type risk-neutral saddle point is a strategy profile $(u^{*}_{j},\ j\in I_{+}),$ of the team of defenders and $(u^{*}_{j},\ j\in I_{-})$ of the team of attackers such that

[TABLE]

Proposition 12

Assume $\mathcal{I}_{+}\cup\mathcal{I}_{-}=\mathcal{I}$ and $q>0,\bar{q}>0.$ Then, the minmax solution of the adversarial mean-field type game (68) under Gauss-Volterra process is given by

[TABLE]

whenever the following system of ordinary differential equations admit a positive solution which does not blowup within the horizon $[0,T].$

[TABLE]

In this case, the minmax solution is also a maxmin solution, hence $(u_{i})_{i\in\mathcal{I}_{+}},(u_{j})_{j\in\mathcal{I}_{-}}$ is a saddle point. Thus, the adversarial mean-field-type game has a value $\mathbb{E}L_{ad}(x,(u_{i})_{i\in\mathcal{I}_{+}},(u_{j})_{j\in\mathcal{I}_{-}}).$

The proof follows similar steps as above by exploiting the strict convex-concave and coercivity properties of the cost functional.

Remark 6

A sufficient condition for existence and uniqueness of the minmax point of mean-field type is obtained:

[TABLE]

and the coefficient functions $b_{1},\sigma^{2},\sigma^{2}_{cogv},\int_{\theta}[(1+\mu)^{2k}-1-2k\mu]\nu(d\theta),$ $r_{i}(\frac{b_{2i}}{r_{i}})^{\frac{2k}{2k-1}}$ and $(\frac{{b}_{2i}}{{r}_{i}})^{\frac{1}{2{k}-1}}$ are all integrable within $[0,T]$, then is no escape of $\alpha_{ad}$ within the entire interval $[0,T].$

Similar reasoning works for $\bar{\alpha}_{ad}$ when

[TABLE]

and the coefficient functions $\bar{b}_{1},$ $\bar{r}_{i}(\frac{\bar{b}_{2i}}{\bar{r}_{i}})^{\frac{2\bar{k}_{i}}{2\bar{k}-1}},$ $(\frac{\bar{b}_{2i}}{r_{i}})^{\frac{1}{2\bar{k}-1}}$ are integrable within $[0,T].$

5 Numerical Examples

In this section, we present some numerical illustrations of Problem (54) by choosing Gauss-Volterra process with the following kernel

[TABLE]

and $H\in(0,1),$ $c_{H}=\sqrt{\frac{2H\Gamma(\frac{3}{2}-H)}{\Gamma(\frac{1}{2}+H)\Gamma(2-2H)}},$ is a normalizing constant, where $\Gamma$ is the gamma function

[TABLE]

The Gauss-Volterra process with kernel $K_{H}$ is a fractional Brownian motion with the Hurst parameter $H.$ The parameters of the numerical setting are displayed in Table 4.

It is important to notice that under this setting the problem (54) is not Markov and the cost is not quadratic. From (10) we know that the mean-field Nash equilibrium of the mean-field type game (54) under Gauss-Volterra process is given by

[TABLE]

Figure 5 plots (a) a sample path of the optimal state trajectory starting from $x_{0}=50,$ (b) the optimal strategies of all decision-makers $i\leq 2018$ and 2019, and (c) sample noises. As expected the state is moving toward zero when the optimal strategies are employed.

6 Conclusion

In this article, we have shown that a mean-field equilibrium can be determined in a semi-explicit way for a broader class of non-linear, non-quadratic game problems with non-linearly distribution-dependent payoffs where the state dynamics is driven by conditional expected values of states, controls, Brownian motions, Gauss-Volterra processes, jump and regime-switching. The method does not require the sophisticated non-elementary extension to backward-forward systems. It does not need PIDEs. It does not need SMPs. It is basic and applies the stochastic integration formula. The use of this simple method may open the accessibility of the tool to a broader audience including beginners and engineers to this emerging field of mean-field-type game theory. Another direct application of the results presented this article is that the explicit solution provides a reference trajectory to the numerical schemes of the corresponding master system beyond the LQ setting.

Acknowledgements

Authors gratefully acknowledge support from U.S. Air Force Office of Scientific Research under grants number FA9550-17-1-0259 , FA9550-12-1-0384 and NSF grant DMS 1411412.