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On Quasi-Stationary Mean Field Games of Controls

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In Mean Field Games of Controls, the dynamics of the single agent is influenced not only by the distribution of the agents, as in the classical theory, but also by the distribution of their optimal strategies. In this paper, we study quasi-stationary Mean Field Games of Controls, in which the strategy-choice mechanism of the agent is different from the classical case: the generic agent cannot predict the evolution of the population, but chooses its strategy only on the basis of the information available at the given instant of time, without anticipating. We prove existence and uniqueness for the solution of the corresponding quasi-stationary Mean Field Games system under different sets of hypotheses and we provide some examples of models which fall within these hypotheses.

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The authors were partially supported by INdAM-GNAMPA Project, codice CUP\(_{E55F22000270001}\). The second author was partially supported also by the Fondazione CaRiPaRo Project “Nonlinear Partial Differential Equations: Asymptotic Problems and Mean-Field Games”. The authors warmly thank the anonymous referees for useful suggestions and comments to improve the paper.

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Correspondence to Claudio Marchi.

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A Appendix: Proofs of Some Results

A Appendix: Proofs of Some Results

1.1 A.1 Proof of Lemma 2.5


Set \(b^i(x,a)=b(x,a;\nu _i)\) and \(\ell ^i(x,a)=\ell (x,a;\nu _i)\), \(i=1,2\). We first claim that

$$\begin{aligned} \rho \Vert u^\rho _1-u^\rho _2\Vert _{L^\infty }\le C \max _{x,a}|b^1(x,a)-b^2(x,a)|+\max _{x,a}|\ell ^1(x,a)-\ell ^2(x,a)|, \end{aligned}$$

where \(C=C_1(1+K+L)\) as in (2.4). Indeed, to prove the claim, it is sufficient to observe that

$$\begin{aligned} v^\rho _{\pm }(x)= u^\rho _2(x)\pm \rho ^{-1}\left( C\max _{x,a}|b^1(x,a)-b^2(x,a)|+\max _{x,a}|\ell ^1(x,a)-\ell ^2(x,a)|\right) \end{aligned}$$

are a subsolution and a supersolution of the equation satisfied by \(u^\rho _1\). Then estimate (A.1) follows by the comparison principle. The rest of the proof follows the corresponding argument in [15, Thm. 2.2]. \(\square \)

1.2 A.2 Proof of Lemma 3.1


Given m and p as in the statement, let \(\Psi :\mathcal{P}(\mathbb {T}^d\times A)\rightarrow \mathcal{P}(\mathbb {T}^d\times A)\) be the map defined by \(\Psi (\mu )=\left( \mathop {Id},\alpha ^*(\cdot ,p(\cdot );\mu )\right) \sharp m\). Given \(\mu _1,\mu _2 \in \mathcal{P}(\mathbb {T}^d\times A)\), by (3.1), we have

$$\begin{aligned}&{{\,\mathrm{\textbf{d}_1}\,}}(\Psi (\mu _1),\Psi (\mu _2))\\ {}&\quad = \sup _\phi \left\{ \int _{\mathbb {T}^d\times A} \phi (x,a)d(\Psi (\mu _1) -\Psi (\mu _2))\right\} \\&\quad =\sup _\phi \left\{ \int _{\mathbb {T}^d}\Big (\phi (x,\alpha ^*(x,p(x) ; \mu _1 ))- \phi (x,\alpha ^*(x,p(x); \mu _2))\Big )m(x)dx\right\} \\&\quad \le \Vert \alpha ^*(\cdot ,p(\cdot ) ; \mu _1 ) -\alpha ^*(\cdot ,p (\cdot ); \mu _2 )\Vert _{L^\infty }\le \lambda _0{{\,\mathrm{\textbf{d}_1}\,}}(\mu _1,\mu _2), \end{aligned}$$

where the \(\sup \) here and in the following formulas are taken with respect to 1-Lipschitz functions on \(\mathbb {T}^d\times A\). Hence, by (H4), \(\Psi \) is a contraction and therefore there exists a unique fixed point to (3.2).

(i). By (3.1), we have

$$\begin{aligned} {{\,\mathrm{\textbf{d}_1}\,}}(\mu _n,\mu )&=\sup _\phi \left\{ \int _{\mathbb {T}^d}\left[ \phi (x,\alpha ^*(x,p_n(x) ; \mu _n))- \phi (x,\alpha ^*(x,p(x); \mu ))\right] m(x)dx\right\} \\&\le \int _{\mathbb {T}^d}\lambda _1|p_n(x)-p(x)|m(x)dx+\lambda _0{{\,\mathrm{\textbf{d}_1}\,}}(\mu _n,\mu ). \end{aligned}$$

We deduce

$$\begin{aligned} {{\,\mathrm{\textbf{d}_1}\,}}(\mu _n,\mu )\le (1-\lambda _0)^{-1}\lambda _1\int _{\mathbb {T}^d}|p_n(x)-p(x)|m(x)dx \end{aligned}$$

and the statement is an easy consequence of the Cauchy-Schwarz inequality applied to (A.2).

(ii). Consider \(m_n\) and m, \(\mu _n\) and \(\mu \) as in the statement. We have

$$\begin{aligned}&{{\,\mathrm{\textbf{d}_1}\,}}(\mu _n,\mu )\\ {}&\quad = \sup _\phi \left\{ \int _{\mathbb {T}^d\times A}\phi (x,a)d(\mu _n-\mu )\right\} \\ {}&\quad =\sup _\phi \left\{ \int _{\mathbb {T}^d}\left[ \phi (x,\alpha ^*(x,p(x) ; \mu _n))m_n(x)- \phi (x,\alpha ^*(x,p(x); \mu ))m(x)\right] dx\right\} \\&\quad \le \sup _\phi \left\{ \int _{\mathbb {T}^d}\left[ \phi (x,\alpha ^*(x,p(x) ; \mu _n))- \phi (x,\alpha ^*(x,p(x); \mu ))\right] m_n(x) dx\right\} \\&\qquad +\sup _\phi \left\{ \int _{\mathbb {T}^d}\phi (x,\alpha ^*(x,p(x); \mu ))(m_n(x)-m(x))dx\right\} . \end{aligned}$$

We denote by \(I_1\) and \(I_2\) the two terms in the last side. Assumption (3.1) ensures

$$\begin{aligned} I_1\le \lambda _0{{\,\mathrm{\textbf{d}_1}\,}}(\mu _n,\mu ). \end{aligned}$$

On the other hand, by (3.1), the function \(\alpha ^*(\cdot ,p(\cdot );\mu )\) is Lipschitz continuous with Lipschitz constant \(1+\lambda _1(1+ L_p)\). We deduce

$$\begin{aligned} I_2= & {} (1+\lambda _1+\lambda _1L_p)\sup _\phi \left\{ \int _{\mathbb {T}^d}\frac{\phi (x,\alpha ^*(x,p(x); \mu ))}{1+(\lambda _1+\lambda _1L_p)}(m_n(x)-m(x))dx\right\} \\\le & {} [1+\lambda _1(1+L_p)]{{\,\mathrm{\textbf{d}_1}\,}}(m_n,m), \end{aligned}$$

where the last relation is due to the fact that the integrand is a 1-Lipschitz continuous function in x. Replacing the last two relations in the previous one we obtain the statement. \(\square \)

1.3 A.3 Proof of Lemma 4.2


We shall borrow some arguments of [5, Lemma 5.4]; we proceed by contradiction assuming that there exists \(\varepsilon >0\) such that for every \(n\in \mathbb {N}\setminus \{0\}\) there exist \(\mu _n\in C([0,T],\mathcal{P}(\mathbb {T}^d\times A))\), \(\rho _n\in (0,1)\) and \(t_n\in [0,T-h_n]\), with \(h_n\in (0,1/n)\) and

$$\begin{aligned} \Vert Dv_n-Dw_n\Vert _{L^\infty }\ge \varepsilon \end{aligned}$$

where \(v_n\) and \(w_n\) are the solutions to (4.3) with \(\rho \) replaced by \(\rho _n\) and with \((t,\mu )\) replaced by \((t_n,\mu _n)\) and respectively by \((t_n+h_n,\mu _n)\). Possibly passing to a subsequence, we may assume that the sequence \(\{\rho _n\}\) converges to some value \(\rho \in [0,1]\). We split the rest of the proof according to the fact that \(\rho =0\) or \(\rho \ne 0\).

Case \(\rho \ne 0\). Estimates (2.3) and (2.4) and \(\rho >0\) ensure:

$$\begin{aligned} \Vert v_n\Vert _{C^{2,\theta }(\mathbb {T}^d)},\Vert w_n\Vert _{C^{2,\theta }(\mathbb {T}^d)}\le \bar{K}. \end{aligned}$$

Hence, possibly passing to subsequences (that we still denote \(v_n\) and \(w_n\)), we may assume that \(v_n\) and \(w_n\) converge to some function \(\varphi _v\) and respectively \(\varphi _w\) in the topology of \(C^{1}(\mathbb {T}^d)\).

Since \(h_n\rightarrow 0\) as \(n\rightarrow \infty \), possibly passing to a subsequence and without any loss of generality, we assume that both \(\{t_n\}_n\) and \(\{t_n+h_n\}_n\) converge to a common value \(\bar{t}\). By hypothesis (H5), there exists \({\bar{H}}(x,p,t)\) such that, as \(n\rightarrow \infty \), \(H(\cdot ,\cdot ;\cdot , \mu _n)\) uniformly converges to \({\bar{H}}\) in \(\mathbb {T}^d\times B(0,\bar{K})\times [0,T]\). In particular, by Ascoli–Arzela theorem, we deduce that \({\bar{H}}\) is continuous on \(\mathbb {T}^d\times B(0,\bar{K})\times [0,T]\) and, exploiting \((H1')\), also that it satisfies

$$\begin{aligned} |{\bar{H}}(x_1,p_1,t)-{\bar{H}}(x_2,p_2,t)|\le L \bar{K}|x_1-x_2|+K|p_1-p_2| \end{aligned}$$

for any \(x_1,x_2\in \mathbb {T}^d\), \(p_1,p_2\in B_{\bar{K}}\) and \(t\in [0,T]\). We denote by \({\bar{u}}\) the unique bounded solution to

$$\begin{aligned} - \Delta {\bar{u}} + {\bar{H}}(x,D{\bar{u}},\bar{t})+\rho {\bar{u}} =0 \quad \text{ in } \mathbb {T}^d. \end{aligned}$$

By (H5) and the continuity of \({\bar{H}}\), for some sequence \(o_n(1)\) with \(\lim _{n}o_n(1)=0\), there holds

$$\begin{aligned}{} & {} |H(x,p;t_n,\mu _n)-{\bar{H}}(x,p,{\bar{t}})|\le |H(x,p;t_n,\mu _n)-{\bar{H}}(x,p, t_n)|\nonumber \\{} & {} \quad +|{\bar{H}}(x,p,t_n)-{\bar{H}}(x,p,{\bar{t}})|\le o_n(1) \end{aligned}$$

for every \((x,p)\in \mathbb {T}^d\times B_{\bar{K}}\). By the Comparison Principle (using the positivity of \(\rho \), the bound (A.4), and the last inequality), we deduce that \(v_n\pm o_n(1)\) are super- and subsolution to (A.5). Letting \(n\rightarrow \infty \), by uniqueness of the solution to (A.5), we obtain \(\varphi _v={\bar{u}}\). Repeating the same arguments for \(w_n\), we obtain \(\varphi _w={\bar{u}}=\varphi _v\). In conclusion, as \(n\rightarrow \infty \), we have

$$\begin{aligned} \Vert Dv_n(\cdot )-Dw_n(\cdot )\Vert _{L^\infty }\le \Vert Dv_n(\cdot )-D{\bar{u}}(\cdot )\Vert _{L^\infty }+\Vert Dw_n(\cdot )-D{\bar{u}}(\cdot )\Vert _{L^\infty } \rightarrow 0 \end{aligned}$$

which contradicts our claim (A.3).

Case \(\rho =0\). We introduce the functions \(v^*_n(\cdot ):=v_n(\cdot )-v_n(0)\) and \(w^*_n(\cdot ):=w_n(\cdot )-w_n(0)\). Again by estimates (2.3) and (2.4), we infer

$$\begin{aligned} \Vert v^*_n\Vert _{C^{2,\theta }(\mathbb {T}^d)},\Vert w^*_n\Vert _{C^{2,\theta }(\mathbb {T}^d)}\le \bar{K}\qquad \text {and}\qquad |\rho _nv_n(0)|, |\rho _nw_n(0)|\le K. \end{aligned}$$

By Ascoli–Arzela theorem, possibly passing to a subsequence, there exist a function \(V\in C^{2,\theta }(\mathbb {T}^d)\) and a constant \(\lambda _v\) such that

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert v^*_n-V\Vert _{C^{2}(\mathbb {T}^d)}=0\qquad \text {and } \lim _{n\rightarrow \infty }\rho _nv_n(0)=\lambda _v. \end{aligned}$$

By the same arguments as before, relation (A.6) still holds true. Hence, by stability result, we deduce that the function V solves

$$\begin{aligned} \lambda _v-\Delta V+{\bar{H}}(x,DV,\bar{t})=0,\qquad V(0)=0. \end{aligned}$$

By similar arguments, there exist a function \(W\in C^{2,\theta }(\mathbb {T}^d)\) and a constant \(\lambda _w\) such that

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert w^*_n-W\Vert _{C^{2}(\mathbb {T}^d)}=0\qquad \text {and } \lim _{n\rightarrow \infty }\rho _nw_n(0)=\lambda _w \end{aligned}$$

and consequently

$$\begin{aligned} \lambda _w-\Delta W+{\bar{H}}(x,DW,\bar{t})=0,\qquad W(0)=0. \end{aligned}$$

By (A.7) and (A.8), the couples \((\lambda _v,V)\) and \((\lambda _w,W)\) are both solutions to the ergodic problem

$$\begin{aligned} \lambda -\Delta u+{\bar{H}}(x,Du,\bar{t})=0,\qquad u(0)=0. \end{aligned}$$

By the same arguments as those used in the proof of [1, Thm. 4.1], this ergodic problem admits exactly one solution \((\lambda ,u)\in \mathbb {R}\times C(\mathbb {T}^d)\); hence, we have

$$\begin{aligned} \lambda _v=\lambda _w\qquad \text {and }V =W. \end{aligned}$$

Finally, as \(n\rightarrow \infty \), we conclude

$$\begin{aligned} \Vert Dv_n-Dw_n\Vert _{L^\infty }= & {} \Vert Dv^*_n-Dw^*_n\Vert _{L^\infty }\le \Vert Dv^*_n-DV\Vert _{L^\infty }\\ {}{} & {} +\Vert Dw^*_n-DV\Vert _{L^\infty }\rightarrow 0 \end{aligned}$$

which contradicts our claim (A.3). \(\square \)

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Camilli, F., Marchi, C. On Quasi-Stationary Mean Field Games of Controls. Appl Math Optim 87, 47 (2023).

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