Risk-Sensitive Mean Field Games via the Stochastic Maximum Principle

Abstract

In this paper, we consider risk-sensitive mean field games via the risk-sensitive maximum principle. The problem is analyzed through two sequential steps: (i) risk-sensitive optimal control for a fixed probability measure, and (ii) the associated fixed-point problem. For step (i), we use the risk-sensitive maximum principle to obtain the optimal solution, which is characterized in terms of the associated forward–backward stochastic differential equation (FBSDE). In step (ii), we solve for the probability law induced by the state process with the optimal control in step (i). In particular, we show the existence of the fixed point of the probability law of the state process determined by step (i) via Schauder’s fixed-point theorem. After analyzing steps (i) and (ii), we prove that the set of N optimal distributed controls obtained from steps (i) and (ii) constitutes an approximate Nash equilibrium or $\epsilon$-Nash equilibrium for the N player risk-sensitive game, where $\epsilon \rightarrow 0$ as $N \rightarrow \infty$ at the rate of $O(\frac{1}{N^{1/(n+4)}})$. Finally, we discuss extensions to heterogeneous (non-symmetric) risk-sensitive mean field games.

Appendices

Appendix A: The Risk-Sensitive Maximum Principle

This appendix proves the maximum principle for the risk-sensitive optimal control problem. We note that the risk-sensitive maximum principle in [42] considered the one-dimensional Brownian motion with the maximization of the Hamiltonian. Here, we extend this to the general r-dimensional Brownian motion with the minimization of the Hamiltonian (but we still call it the “maximum principle”).⁶

Consider the SDE

$$\begin{aligned} \mathrm{d}x(t)&= f(t,x,u)\mathrm{d}t + \sigma (t) \mathrm{d}B(t),~ x(0)=x_0, \end{aligned}$$

and the risk-sensitive cost function

$$\begin{aligned} J(u)&= \gamma \log {\mathbb {E}} \left[ \exp \left\{ \frac{1}{\gamma } \int _0^T l(t,x(t),u(t))\mathrm{d}t + \frac{1}{\gamma }m(x(T))\right\} \right] , \end{aligned}$$

(A.1)

where B is the r-dimensional standard Brownian motion. Let $\{\mathcal {F} \}_{t \ge 0}$ be the filtration generated by B.

We first state the risk-sensitive maximum principle, which is similar to [42, Theorem 3.1]. See also [65, Theorem 3.2, Chapter 3] for the risk-neutral stochastic maximum principle.

Theorem A1

Let $(x,\bar{u})$ be an optimal pair for the risk-sensitive optimal control problem in (A.1). Then there exists a unique pair $(p,q) \in \mathcal {L}_{\mathcal {F}}^2(0,T;{\mathbb {R}}^n) \times \mathcal {L}_{\mathcal {F}}^2(0,T;{\mathbb {R}}^{n \times r})$ such that it is the solution of the following BSDE:

$$\begin{aligned} \mathrm{d}p(t)&= -\left[ f_x^\top (t,x,\bar{u}) p(t) + l_x(t,x,\bar{u}) + \frac{1}{\gamma } q(t) \sigma ^\top (t) p(t) \right] \mathrm{d}t + q(t) \mathrm{d}B(t)\nonumber \\ p(T)&= m_x(x(T)), \end{aligned}$$

(A.2)

Also, the following optimality condition holds:

$$\begin{aligned} H(t,x,\bar{u},p,q) = \min _{u \in U} H(t,x,u,p,q), \end{aligned}$$

(A.3)

where the Hamiltonian H is given by

$$\begin{aligned} H(t,x,u,p,q) = p^\top f + l + {{\,\mathrm{Tr}\,}}\left( q^\top \sigma \right) + \frac{1}{\gamma }p^\top \sigma \sigma ^\top p. \end{aligned}$$

(A.4)

$\square$

Remark A1

Note that the term $\frac{1}{\gamma } q(t) \sigma ^\top (t) p(t)$ in the BSDE p in (A.2) is different from that of the one-dimensional Brownian motion case in [42]. $\square$

We now prove Theorem A1.

Proof of Theorem A1

First, it is easy to see that the risk-sensitive optimal control problem in (A.1) can be converted into the following Mayer form:

$$\begin{aligned} J(u)&= \gamma \log {\mathbb {E}} \left[ \exp \{ \frac{1}{\gamma } (m(x(T)) + y(T))\} \right] \end{aligned}$$

(A.5)

$$\begin{aligned} \mathrm{d}x(t)&= f(t,x,u)\mathrm{d}t + \sigma (t) \mathrm{d}B(t),~ x(0)=x_0 \end{aligned}$$

(A.6)

$$\begin{aligned} \mathrm{d}y(t)&= l(t,x,u)\mathrm{d}t,~ y(0) = 0, \end{aligned}$$

(A.7)

Note that with this reformulation, we can apply the risk-neutral maximum principle in [65, Theorem 3.2, Chapter 3].

From [65, Theorem 3.2, Chapter 3], the Hamiltonian for the optimal control problem in (A.5) is given by

$$\begin{aligned} \bar{H}(t,x,u,p,q)&= \left\langle p,\begin{pmatrix} f \\ l \end{pmatrix} \right\rangle + {{\,\mathrm{Tr}\,}}(q^\top \begin{pmatrix} \sigma \\ 0 \end{pmatrix}), \end{aligned}$$

(A.8)

where p is the adjoint process satisfying

$$\begin{aligned} \mathrm{d}p(t)&= - \begin{pmatrix} f_x(t,x,\bar{u}) &{} 0 \\ l_x^\top (t,x,\bar{u}) &{} 0 \end{pmatrix}^\top p(t) + q(t) \mathrm{d}B(t) \nonumber \\ p(T)&= \frac{1}{\gamma }\exp \left\{ \frac{1}{\gamma } (m(x(T)) + y(T))\right\} \begin{pmatrix} m_x(x(T)) \\ 1 \end{pmatrix}. \end{aligned}$$

(A.9)

Note that p is an $(n+1)$-dimensional adjoint process with $p^\top =(p_1^\top ,p_2)^\top$, where $p_1$ is an n-dimensional stochastic process associated with the constraint (A.6). Moreover, q is an $(n+1)\times r$ dimensional matrix stochastic process.

We define the associated value function for (A.5):

$$\begin{aligned} v(t) = \inf _{u} {\mathbb {E}} \left[ \exp \left\{ \frac{1}{\gamma } (m(x(T)) + y(T))\right\} \right] , \end{aligned}$$

where $v(t) >0$ and $v(T) = \exp \{ \frac{1}{\gamma } (m(x(T)) + y(T))\}$. Due to the relationship between the maximum principle and dynamic programming, the associated value function logarithmic transformation (see [29, Chapter VI], [5, 42]) leads to

$$\begin{aligned} p(t) = v_{(x,y)}(t),~~V = \gamma \log v, \end{aligned}$$

(A.10)

where p(T) satisfies the terminal condition in (A.9). The gradient of V can be written as

$$\begin{aligned} \tilde{p}(t) = \gamma \frac{p(t)}{v(t)}, \end{aligned}$$

(A.11)

where $\tilde{p} = (\bar{p}^\top , \tilde{p}_2)^\top \in {\mathbb {R}}^{n+1}$, in which $\bar{p}$ is an n-dimensional backward stochastic process.

We now obtain the expression of $\tilde{p}$ in (A.11). Under the non-degeneracy assumption (stated in A4) in Sect. 2), v is the smooth value function of the optimal control problem in (A.5) as mentioned in [65, Chapter 4] and [42]. Also, we can see that there is no running cost. Then in view of the proof in [65, Theorem 4.1, Chapter 5] and [42], and by using the Itô formula, we have

$$\begin{aligned} \mathrm{d}v(t)&= p_1^\top (t) \sigma (t) \mathrm{d}B(t). \end{aligned}$$

(A.12)

By using the Itô formula again with (A.12), and from (A.11), we have

$$\begin{aligned} \mathrm{d}\frac{1}{v(t)}&= - \frac{1}{\gamma v(t)} \bar{p}^\top (t) \sigma (t) \mathrm{d}B(t) + \frac{1}{\gamma ^2 v(t)} \bar{p}^\top (t) \sigma (t) \sigma ^\top (t) \bar{p}(t) \mathrm{d}t. \end{aligned}$$

(A.13)

By using the Itô formula for (A.11) with (A.13) and (A.9), we have

$$\begin{aligned} \mathrm{d}\tilde{p}(t)&= \mathrm{d} \left( \gamma \frac{p(t)}{v(t)}\right) \nonumber \\&= - \begin{pmatrix} f_x(t,x,\bar{u}) &{} 0 \\ l_x^\top (t,x,\bar{u}) &{} 0 \end{pmatrix}^\top \tilde{p}(t)\mathrm{d}t - \frac{1}{\gamma } \tilde{q}(t) \sigma ^\top (t) \bar{p}(t) \mathrm{d}t + \tilde{q}(t) \mathrm{d}B(t), \end{aligned}$$

(A.14)

where

$$\begin{aligned} \tilde{p}(T) = \begin{pmatrix} m_x(x(T)) \\ 1 \end{pmatrix}, \end{aligned}$$

and $\tilde{q}$ is an $(n+1) \times r$ dimensional stochastic process:

$$\begin{aligned} \tilde{q}(t)&= \frac{\gamma q(t)}{v(t)} - \frac{1}{\gamma } \tilde{p}(t) \bar{p}^\top (t) \sigma (t). \end{aligned}$$

In view of the value function transformation in (A.10), it is easy to see that $\tilde{p}_2(t) = 1$ for all $t \in [0,T]$, and $\tilde{q}(t)$ satisfies

$$\begin{aligned} \tilde{q}(t)&= \frac{\gamma q(t)}{v(t)} - \frac{1}{\gamma } \tilde{p}(t) \bar{p}^\top (t) \sigma (t) =: \begin{pmatrix} \bar{q}(t) \\ \tilde{q}_2(t) \end{pmatrix}, \end{aligned}$$

(A.15)

where $\tilde{q}(t)$ is an $(n \times r)$-dimensional stochastic process, whereas $\tilde{q}_2(t)$ is an $(1 \times r)$-dimensional stochastic process with $\tilde{q}_2(t) = 0$ a.s. for all $t \in [0,T]$.

We note that $\tilde{p} = (\bar{p}^\top ,\tilde{p}_2)^\top$ and $\bar{p}(T) = m_x(x(T))$. Then expanding (A.14) and together with (A.15), we can easily show that $\bar{p}$ satisfies the backward SDE in (A.2). Moreover, by substituting the relationships of (A.11) and (A.15) into the Hamiltonian in (A.8), one can arrive at the Hamiltonian of the risk-sensitive optimal control problem in (A.4) with the optimality condition given in (A.3). Since our derivation can be reversed, this completes the proof of the risk-sensitive maximum principle for the r-dimensional Brownian motion. The proof of Theorem A1 is done. $\square$

Appendix B: Proof of Theorem 1

First, we note that from the risk-sensitive maximum principle in Theorem A1 in “Appendix A” section, with the optimal control $\bar{u}$, there exists a unique solution of the FBSDE in (7). Then under (A2)–(A5), by applying Four Step Scheme introduced in [25] (see also [53] and [44, Chapter 4] for Four Step Scheme under the strong regularity assumptions), the BSDE for p can be expressed in terms of x as follows:

$$\begin{aligned} \theta (t,x(t)) = p(t),~ \theta (T,x) = p(T) \end{aligned}$$

(B.1)

almost surely for $t \in [0,T]$ [25, Corollary 1.5]. In fact, in view of Four Step Scheme and Itô formula, one can show that $\theta (t,x)$ is a classical solution of the particular quasi-linear parabolic partial differential equation with the terminal condition $\theta (T,x) = p(T) = m_x(x(T),\mu (T))$ [25, ($\hbox {E}^\prime$)], [44, Chapter 4] and [53]. Also, from [25, Corollary 1.5], we have

$$\begin{aligned} |\theta (t,x_1) - \theta (t,x_2) | \le c |x_1 - x_2 |,~ \forall x_1,x_2 \in {\mathbb {R}}^n, \end{aligned}$$

(B.2)

for some constant $c \ge 0$. Hence, with (B.1), the SDE for x in (7) can be written as follows:

$$\begin{aligned} \mathrm{d}x(t)&=f(t,x,\mu ,w(t,x,\mu ,\theta (t,x)))\mathrm{d}t + \sigma (t) \mathrm{d}B(t), \end{aligned}$$

(B.3)

where $x(0) = x_0$. Note that (B.3) is now decoupled with the BSDE p in (7).

We now use Schauder’s fixed-point theorem to complete the proof. Its statement is given as follows: LetXbe a nonempty closed and bounded convex subset of a normed spaceS. LetTbe a continuous mapping ofXinto a compact subset$K \subset X$. ThenThas a fixed point. [58, Theorem 4.1.1].

We first note that the 1-Wasserstein metric on $\mathcal {P}_1(\mathcal {C}([0,T];{\mathbb {R}}^n))$ is equivalent to the Kantorovich–Rubinstein distance [16, Theorem 5.5]

$$\begin{aligned} W_1(\mu ^*(t),\mu ^\prime (t))&= \sup \left\{ \int _{\mathcal {C}([0,T];{\mathbb {R}}^n)} f(x) \mathrm{d}\mu ^*(t,x) - \int _{\mathcal {C}([0,T];{\mathbb {R}}^n)} f(x) \mathrm{d}\mu ^\prime (t,x) \right\} , \end{aligned}$$

where $\mu ^*,\mu ^\prime \in \mathcal {P}_1(\mathcal {C}([0,T];{\mathbb {R}}^n))$, and the supremum is taken over the set of all 1-Lipschitz continuous maps f. Indeed, it can be seen that the 1-Wasserstein metric is induced by the Kantorovich–Rubinstein norm [57], which, together with the fact that $\mathcal {C}([0,T];{\mathbb {R}}^n)$ is a normed space with the norm $|\cdot |_{\infty } := \sup _{0 \le t \le T} |\cdot |$ [43], implies that $\mathcal {P}_1(\mathcal {C}([0,T];{\mathbb {R}}^n))$ is a normed space [15].

We define the following set for $c > 0$:

$$\begin{aligned} \mathcal {E}=\{\mu \in \mathcal {P}_4(\mathcal {C}\left( [0,T];{\mathbb {R}}^n\right) )~:~M_4(\mu ) \le c\}, \end{aligned}$$

where we have the inclusion $\mathcal {E} \subset \mathcal {P}_2(\mathcal {C}([0,T];{\mathbb {R}}^n)) \subset \mathcal {P}_1(\mathcal {C}([0,T];{\mathbb {R}}^n))$. Then, it is easy to check that $\mathcal {E}$ is bounded and convex and is closed with respect to the 1-Wasserstein metric. The latter follows from the fact that for any convergent sequence of measures $\mu _k \in \mathcal {E}$, $k \ge 1$, to $\mu$ with the 1-Wasserstein metric, we have $W_4(\mu _k,\mu ) \le W_1(\mu _k,\mu )$ for $k \ge 1$ [16, Section 5], which implies $\mu \in \mathcal {E}$. In the proof below, a constant c can vary from line to line.

Now, note that $\bar{u} \in {\mathcal {U}}$ is the optimal control that minimizes the Hamiltonian in (8). Then with (A2), (A5) and (B.2), the standard estimate of the SDEs in [65, Theorem 6.3, Chapter 1] implies that there exists a constant c, depending on $x_0$, $\beta$ and T, such that ${\mathbb {E}}[\sup _{0 \le t \le T} |x(t)|^4] \le c$. Hence, by considering the mapping $\varPsi$ on $\mathcal {E}$, and noticing that $\mathcal {E} \subset \mathcal {P}_1(\mathcal {C}([0,T];{\mathbb {R}}^n))$, we have $\varPsi : \mathcal {E} \rightarrow \mathcal {E}$, i.e., $\varPsi \mu \in \mathcal {E}$, for any $\mu \in \mathcal {E}$.

To prove compactness of $\varPsi (\mathcal {E})$, we show tightness of a sequence of measures $\varPsi \mu _k$ with respect to the 1-Wasserstein metric, where $\mu _k \in \mathcal {E}$, $k \ge 1$. Note that the initial condition of the SDE is not random and $\sigma$ is uniformly bounded in $t \in [0,T]$ from (A4). Then, from [65, Theorem 6.3, Chapter 1], for any $\delta > 0$ and $s \in [t,t+\delta ]$, ${\mathbb {E}}[|x(s) - x(t)|^2] \le c$, where c depends on the initial condition of the SDE and $\delta$. Hence, in view of [14, Theorem 7.3] and [14, Corollary, page 83], $\{\varPsi \mu _k\}$ is tight, which implies that $\varPsi (\mathcal {E})$ is relatively compact with respect to the 1-Wasserstein metric [14, Theorem 5.1].

It remains to show that $\varPsi$ is continuous on $\mathcal {E}$ with respect to the 1-Wasserstein metric. That is, for every $\epsilon > 0$, there exists $\eta > 0$ such that with $\mu ^*,\mu ^\prime \in \mathcal {E}$, $W_1(\mu ^*,\mu ^\prime ) < \eta$ implies $W_1(\varPsi \mu ^*,\varPsi \mu ^\prime )< \epsilon$. Note that $\mu ^*$ is not a fixed point of $\varPsi$. Let $x^*$ and $x^\prime$ be generated by two SDEs corresponding to $\mu ^*$ and $\mu ^\prime$, respectively. From the definition of $W_1$, for any $\mu ^*,\mu ^\prime \in \mathcal {E} \subset \mathcal {P}_2(\mathcal {C}([0,T];{\mathbb {R}}^n)) \subset \mathcal {P}_1(\mathcal {C}([0,T];{\mathbb {R}}^n))$,

$$\begin{aligned} W_1(\varPsi \mu ^*,\varPsi \mu ^\prime ) \le {\mathbb {E}} \left[ \sup _{0 \le t \le T} |x^*(t) - x^\prime (t)| \right] . \end{aligned}$$

(B.4)

From Gronwall’s lemma, (A2), (A5) and (B.2) and by following the proof in [19, Proposition 3.8], there exists a constant $c > 0$ such that ${\mathbb {E}} [ \sup _{0 \le t \le T} |x^*(t) - x^\prime (t)|^2 ] \le c (\int _0^T W_2^2(\mu ^*(t),\mu ^\prime (t)) \mathrm{d}t )^{1/2}$. Then by using Jensen’s inequality and the fact that $W_2(\mu ^*(t),\mu ^\prime (t)) \le W_1(\mu ^*(t),\mu ^\prime (t))$ [16, Section 5], we have

$$\begin{aligned} {\mathbb {E}} \left[ \sup _{0 \le t \le T} |x^*(t) - x^\prime (t) | \right] \le c \left( \int _0^T W_1^2(\mu ^*(t),\mu ^\prime (t)) \mathrm{d}t \right) ^{1/4}. \end{aligned}$$

This, together with (B.4), implies continuity of $\varPsi$ on $\mathcal {E}$ with respect to the 1-Wasserstein metric. This completes the proof of the theorem. $\square$

Appendix C: Proof of Theorem 2

To prove Theorem 2, we first need the following lemma:

Lemma C1

There exists a constant $c>0$, dependent on n, $M_{5+n}< \infty$ and T, such that

$$\begin{aligned} {\mathbb {E}}\left[ W_2^2\left( \nu _N^*(t),\mu ^*(t)\right) \right] \le \frac{c}{N^{2/(n+4)}},~ \forall t \in [0,T]. \end{aligned}$$

Moreover, $W_2(\nu _N^*(t),\mu ^*(t)) \rightarrow 0$ as $N \rightarrow \infty$ almost surely for all $t \in [0,T]$.

A proof of this lemma can be found in [57, Theorem 10.2.1] and [59, Proposition 5.1], or [31, 38]. In fact, the proof relies on Gronwall’s lemma with the Lipschitz property, the strong law of large numbers of the empirical distribution, and exchangeability of the stochastic processes $x_i^*$. The second part of Lemma C1 follows from [57, page 323].

We now proceed with the proof of Theorem 2.

Proof of Theorem 2

Since the players are symmetric, that is, the players are invariant under arbitrary permutations, we only need to consider the case when $i=1$. In the proof below, the constant c can vary from line to line.

We note that $x_1^*$ defined in (10) is the SDE for player $i=1$ with the optimal distributed control $u_1^*$ given in (11). As mentioned, $x_1^*$ is decoupled with other players since f does not depend on the mean field from (A6), which implies that it is statistically independent from other players. Furthermore, we note that $x_1$ is the SDE of player $i=1$ with an arbitrary control $u_1 \in {\mathcal {U}}_{\mathcal {F}}^1$. Then it should be clear from the definitions of $x_1$ and $x_1^*$ that $x_1$ is identical to $x_1^*$ when $u_1 = u_1^*$.

In the proof below, note that the empirical distribution $\nu _N^*$ in (12) is obtained when the N players are under the optimal distributed control in (11). Then in view of Lemma C1, we have for $t \in [0,T]$,

$$\begin{aligned} {\mathbb {E}} \left[ W_2^2(\nu _N^*(t),\mu ^*(t)) \right] = O\left( \frac{1}{N^{2/(n+4)}} \right) . \end{aligned}$$

We also note that

$$\begin{aligned} \nu _N(t) = \frac{1}{N}\delta _{x_1(t)} + \frac{1}{N}\sum _{i=2}^N \delta _{x_i^*(t)}, \end{aligned}$$

which is the empirical distribution when $x_1$ is under an arbitrary control $u_1$, while other players are with the optimal distributed control in (11).

Due to boundedness of f and $\sigma$ in t, one can show that by using Itô isometry, there exists a constant $c > 0$ (dependent on $\beta$ and T) such that

$$\begin{aligned} {\mathbb {E}} \left[ \sup _{0 \le t \le T} |x_1(t)|^2 \right] \le c + c {\mathbb {E}} \left[ \int _0^T |u_1(t)|^2 \mathrm{d}t \right] . \end{aligned}$$

(C.1)

Since $u_i^*$ satisfies ${\mathbb {E}}[\int _0^T |u_i^*(t)|^2 \mathrm{d}t] < \infty$, $2 \le i \le N$ and f and $\sigma$ are bounded, from (A2), (A5), (B.2) and [65, Theorem 6.3, Chapter 1], we can show the estimate ${\mathbb {E}} [ \sup _{0 \le t \le T} |x_i^*(t)|^2 ] \le c$ for $2 \le i \le N$. This, together with (C.1) and Itô isometry leads to the following inequality:

$$\begin{aligned}&\frac{1}{N}\left( {\mathbb {E}} \left[ \sup _{0 \le t \le T} |x_1(t)|^2 \right] + \sum _{i=2}^N {\mathbb {E}} \left[ \sup _{0 \le t \le T} |x_i^*(t)|^2 \right] \right) \le c + \frac{c}{N}{\mathbb {E}} \left[ \int _0^T |u_1(t)|^2 \mathrm{d}t \right] , \end{aligned}$$

(C.2)

which is also bounded since we have $u_1 \in {\mathcal {U}}_{\mathcal {F}}^1$ with ${\mathbb {E}}[\int _0^T |u_1(t)|^2 \mathrm{d}t ] < \infty$.

Consider the following inequality:

$$\begin{aligned}&{\mathbb {E}} \left[ W_2^2(\nu _N(t),\mu ^*(t)) \right] \end{aligned}$$

(C.3)

$$\begin{aligned}&\quad \le c {\mathbb {E}} \left[ W_2^2 \left( \nu _N(t),\frac{1}{N-1} \sum _{i=2}^N \delta _{x_i^*(t)} \right) \right] \end{aligned}$$

(C.4)

$$\begin{aligned}&\qquad + c {\mathbb {E}} \left[ W_2^2 \left( \frac{1}{N-1} \sum _{i=2}^N \delta _{x_i^*(t)},\mu ^*(t) \right) \right] . \end{aligned}$$

(C.5)

We now show boundedness of ${\mathbb {E}} [ W_2^2(\nu _N(t),\mu ^*) ]$ in (C.3) with respect to N for $t \in [0,T]$.

First, from Lemma C1, we have

$$\begin{aligned} {\mathbb {E}} \left[ W_2^2 \left( \frac{1}{N-1} \sum _{i=2}^N \delta _{x_i^*(t)},\mu ^*(t) \right) \right] \le \frac{c}{N^{2/(n+4)}}, \end{aligned}$$

(C.6)

which is the bound for (C.5).

Also, for (C.4), by the definition of $W_2$, we have

$$\begin{aligned}&{\mathbb {E}} \left[ W_2^2 \left( \nu _N(t),\frac{1}{N-1} \sum _{i=2}^N \delta _{x_i^*(t)} \right) \right] \le \frac{c}{N(N-1)} \sum _{i=2}^n {\mathbb {E}} \left[ |x_1(t) - x_i^*(t)|^2 \right] \le \frac{c}{N}, \end{aligned}$$

(C.7)

where the last inequality follows from (C.1) and (C.2). Hence, for (C.3), in view of (C.6), and (C.7), we have

$$\begin{aligned} {\mathbb {E}} \left[ W_2^2(\nu _N(t),\mu ^*(t)) \right] \le \frac{c}{N^{2/(n+4)}},~ t \in [0,T]. \end{aligned}$$

(C.8)

By applying Jensen’s inequality, we have (see also [29, Chapter VI])

$$\begin{aligned} J_1^N\left( u^{N*}\right)&\ge {\mathbb {E}} \left[ \int _0^T l(t,x_1^*(t),\nu _N^*(t),u_1^*(t))\mathrm{d}t + m(x_1^*(T),\nu _N^*(T)) \right] =: L_1^N\left( u^{N*}\right) \\ \bar{J}_1\left( u_1^*,\mu ^*\right)&\ge {\mathbb {E}} \left[ \int _0^T l(t,x_1^*(t),\mu ^*(t),u_1^*(t))\mathrm{d}t + m(x_1^*(T),\mu ^*(T)) \right] =: \bar{L}_1\left( u_1^*,\mu ^*\right) . \end{aligned}$$

Note that $L_1^N$ and $\bar{L}_1$ are risk-neutral cost functions. Then, there exists a constant c, dependent on T and the Lipschitz constant $\beta$ in (A2), such that

$$\begin{aligned} \left| J_1^N\left( u^{N*}\right) - \bar{J}_1\left( u_1^*,\mu ^*\right) \right|&\le c \left| L_1^N\left( u^{N*}\right) - \bar{L}_1\left( u_1^*,\mu ^*\right) \right| . \end{aligned}$$

(C.9)

Therefore, by using Cauchy–Schwarz inequality, the Lipschitz properties of l and m in (A2), and the fact that $u_1^* \in {\mathcal {U}}_{\mathcal {F}^1}^1$, we can show that

$$\begin{aligned}&\left| J_1^N\left( u^{N*}\right) - \bar{J}_1\left( u_1^*,\mu ^*\right) \right| \\&\quad \le c {\mathbb {E}} \left[ W_2^2(\mu ^*(T),\nu _N^*(T)) \right] ^{1/2} + c \int _0^T {\mathbb {E}} \left[ W_2^2(\mu ^*(t),\nu _N^*(t)) \right] ^{1/2} \mathrm{d}t\le \frac{c}{N^{1/(n+4)}}, \end{aligned}$$

where the first inequality follows from (C.9) and (A2), and the second inequality is due to Lemma C1. In the first inequality, we have used the fact that ${\mathbb {E}} \left[ W_2(\mu ^*(t),\nu _N^*(t)) \right] \le {\mathbb {E}} \left[ W_2^2(\mu ^*(t),\nu _N^*(t)) \right] ^{1/2}$ due to Jensen’s inequality.

In view of the above inequality, we have

$$\begin{aligned}&\left| J_1^N\left( u^{N*}\right) - \bar{J}_1\left( u_1^*,\mu ^*\right) \right| = O\left( \frac{1}{N^{1/(n+4)}} \right) , \end{aligned}$$

(C.10)

which shows that for any i, $1 \le i \le N$,

$$\begin{aligned} J_i^N\left( u^{N*}\right) - \frac{c}{N^{1/(n+4)}} \le \bar{J}_i\left( u_i^*,\mu ^*\right) . \end{aligned}$$

This implies that for sufficiently large N, the cost difference between $J_i^N(u^{N*})$ and $\bar{J}(u_i^*,\mu ^*)$ is negligible as a consequence of Lemma C1. This result can also be explained by the law of large numbers of the empirical distribution of $x_i^*$, $1 \le i \le N$, due to Lemma C1.

Furthermore, with a similar reasoning as in (C.9) and (C.10), and due to the empirical estimate obtained in (C.8), we have $J_1^N(u_1,u_2^*,\ldots ,u_N^*) \ge \bar{J}_1(u_1,\mu ^*) - \frac{c}{N^{1/(n+4)}} \ge \bar{J}_1(u_1^*,\mu ^*) - \frac{c}{N^{1/(n+4)}}$. Note that the second inequality follows from step (i) and (15), since $\bar{J}_1(u_1^*,\mu ^*) \le \bar{J}_1(u_1,\mu ^*)$ for $u_1 \in {\mathcal {U}}_{\mathcal {F}}^1$. This, together with (C.10), implies that the set of the optimal distributed controls, $u^{N*}=\{u_1^*,\ldots ,u_N^*\}$, where $u_i^*$ is given in (11), constitutes an $\epsilon _N$-Nash equilibrium. Also, we have $\epsilon _N \rightarrow 0$ as $N \rightarrow \infty$ with the convergence rate of $O(1/N^{1/(n+4)})$. This completes the proof of the theorem. $\square$

Appendix D: Lemma for Sect. 5

We have the following lemma for Sect. 5, which is a modified version of Lemma C1 in Appendix C.

Lemma D1

Suppose that the conditions in Theorem 3 hold. Then the following estimate holds: for $t \in [0,T]$,

$$\begin{aligned} {\mathbb {E}}\left[ W_2^2\left( \nu _N^*(t),\mu ^*(t)\right) \right] = O\left( \frac{1}{N^{2/(n+4)}} + \sup _{ k \in \mathcal {K}} | \pi _k^N - \pi _k|^2 \right) . \end{aligned}$$

Moreover, $W_2(\nu _N^*(t),\mu ^*(t)) \rightarrow 0$ as $N \rightarrow \infty$ almost surely for all $t \in [0,T]$.

Proof

First, observe that the empirical distribution of $x_i$, $1 \le i \le N$, $\nu _N^*$, with the individual optimal controls and the associated fixed point, that is, $\nu _N^*$, has the following relationship:

$$\begin{aligned} \nu _N^*(t)&= \frac{1}{N}\sum _{i=1}^N \delta _{x_i(t)} = \frac{1}{N} \sum _{k=1}^K \sum _{i \in \mathcal {N}_k} \delta _{x_i(t)} = \frac{1}{N} \sum _{k=1}^K N_k \bar{\nu }_k^{N,*}(t) = \sum _{k=1}^K \pi _k^N \bar{\nu }_k^{N,*}(t), \end{aligned}$$

where $\bar{\nu }_k^{N,*}(t) = \frac{1}{N_k} \sum _{i \in \mathcal {N}_k} \delta _{x_i(t)}$. Note that $\bar{\nu }_k^{N,*}(t) \rightarrow \mu _k$ almost surely as $N_k \rightarrow \infty$ for each $k \in \mathcal {K}$.

Since $W_2$ is a distance, we have

$$\begin{aligned} {\mathbb {E}} \left[ W_2^2(\nu _N^*(t),\mu ^*(t)) \right]&\le c {\mathbb {E}} \left[ W_2^2 \left( \sum _{k=1}^K \pi _k^N \bar{\nu }_k^{N,*}(t),\sum _{k=1}^K \pi _k^N \mu _k^*(t) \right) \right] \end{aligned}$$

(D.1)

$$\begin{aligned}&\quad + c {\mathbb {E}} \left[ W_2^2 \left( \sum _{k=1}^K \pi _k^N \mu _k^*(t),\sum _{k=1}^K \pi _k \mu _k^*(t) \right) \right] . \end{aligned}$$

(D.2)

For (D.2), we first show that there exists a constant c, independent of N, such that

$$\begin{aligned}&{\mathbb {E}} \left[ W_2^2 \left( \sum _{k=1}^K \pi _k^N \mu _k^*(t),\sum _{k=1}^K \pi _k \mu _k^*(t) \right) \right] \le c \sup _{ k \in \mathcal {K}} | \pi _k^N - \pi _k|^2. \end{aligned}$$

(D.3)

By (18), we have $\sum _{k=1}^K \pi _k^N \mu _k^*(t) \rightarrow \sum _{k=1}^K \pi _k \mu _k^*(t)$ as $N \rightarrow \infty$, which is equivalent to saying that $W_2 (\sum _{k=1}^K \pi _k^N \mu _k^*(t),\sum _{k=1}^K \pi _k \mu _k^*(t) ) \rightarrow 0$ as $N \rightarrow \infty$ [57, Chapter 10.2]. This implies (D.3).

We now consider (D.1). It satisfies the following inequality:

$$\begin{aligned}&{\mathbb {E}} \left[ W_2^2 \left( \sum _{k=1}^K \pi _k^N \bar{\nu }_k^{N,*}(t),\sum _{k=1}^K \pi _k^N \mu _k^*(t) \right) \right] \le {\mathbb {E}} \left[ W_2^2 \left( \sum _{k=1}^K \pi _k^N \bar{\nu }_k^{N,*}(t),\sum _{k=1}^K \pi _k \bar{\nu }_k^{N,*}(t) \right) \right] \nonumber \\&\qquad + {\mathbb {E}} \left[ W_2^2 \left( \sum _{k=1}^K \pi _k \bar{\nu }_k^{N,*}(t),\sum _{k=1}^K \pi _k \mu _k^*(t) \Bigr ) \right) \right] + {\mathbb {E}} \left[ W_2^2 \left( \sum _{k=1}^K \pi _k \mu _k^*(t),\sum _{k=1}^K \pi _k^N \mu _k^*(t) \Bigr ) \right) \right] . \end{aligned}$$

(D.4)

In view of the assumption in (18) and (D.3), the first and last terms in (D.4) are bounded above by $c \sup _{ k \in \mathcal {K}} | \pi _k^N - \pi _k|^2$. For the second term in (D.4), from Lemma C1 in “Appendix C” section, we have

$$\begin{aligned}&{\mathbb {E}} \left[ W_2^2 \left( \sum _{k=1}^K \pi _k \bar{\nu }_k^{N,*}(t),\sum _{k=1}^K \pi _k \mu _k^*(t) \Bigr ) \right) \right] = O\left( \frac{1}{N^{2/(n+4)}} \right) ,~ \forall t \in [0,T]. \end{aligned}$$

This yields the desired result, thus completing the proof. $\square$