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Selected topics in mean field games · 2022-econ-11

Dynamic, Stochastic and Differential Games · ICM 2022: Selected topics in mean field games

MSC 91B · Mathematical economics

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Problem statement

Let EE be a finite set, let

ΔE={m=(mi)i∈E∈[0,1]E:∑i∈Emi=1},ΔE∘={m∈ΔE:mi>0 for every i∈E}, \Delta_E=\left\{m=(m_i)_{i\in E}\in[0,1]^E:\sum_{i\in E}m_i=1\right\}, \qquad \Delta_E^\circ=\{m\in\Delta_E:m_i>0\text{ for every }i\in E\},

fix T>0T>0 and m0∈ΔE∘m_0\in\Delta_E^\circ, and let F=(Fi)i∈EF=(F^i)_{i\in E} and G=(Gi)i∈EG=(G^i)_{i\in E} be sufficiently smooth bounded functions on ΔE\Delta_E. Assume that the mean field game is potential: there are smooth functionals F,G:ΔE→R\mathcal F,\mathcal G:\Delta_E\to\mathbb R such that, for every m∈ΔEm\in\Delta_E and every tangent vector h∈REh\in\mathbb R^E satisfying ∑ihi=0\sum_i h_i=0,

DF(m)[h]=∑i∈EFi(m)hi,DG(m)[h]=∑i∈EGi(m)hi. D\mathcal F(m)[h]=\sum_{i\in E}F^i(m)h_i, \qquad D\mathcal G(m)[h]=\sum_{i\in E}G^i(m)h_i.

The associated deterministic mean field control problem is to minimize

J(β,p)=∫0T[12∑i,j∈Ei≠jpti∣βti,j∣2+F(pt)]dt+G(pT) \mathcal J(\beta,p)= \int_0^T\left[ \frac12\sum_{\substack{i,j\in E\\i\ne j}}p_t^i\lvert\beta_t^{i,j}\rvert^2 +\mathcal F(p_t) \right]dt+\mathcal G(p_T)

over bounded measurable rate matrices βt=(βti,j)i,j∈E\beta_t=(\beta_t^{i,j})_{i,j\in E} satisfying

βti,j≥0(i≠j),βti,i=−∑j≠iβti,j, \beta_t^{i,j}\ge 0\quad(i\ne j), \qquad \beta_t^{i,i}=-\sum_{j\ne i}\beta_t^{i,j},

and absolutely continuous trajectories p:[0,T]→ΔEp:[0,T]\to\Delta_E satisfying

p˙ti=∑j∈Eptjβtj,ifor almost every t∈[0,T],p0=m0. \dot p_t^i=\sum_{j\in E}p_t^j\beta_t^{j,i}\quad\text{for almost every }t\in[0,T], \qquad p_0=m_0.

For x∈Rx\in\mathbb R, write x+=max⁡{x,0}x_+=\max\{x,0\}. For each ε>0\varepsilon>0, let φε:[0,∞)→[0,∞)\varphi^\varepsilon:[0,\infty)\to[0,\infty) be a smooth boundary penalty that is positive in a neighborhood of 00, whose value at 00 is large enough to ensure the well-posedness stated below, and such that φε→0\varphi^\varepsilon\to0 uniformly on every compact subset of (0,∞)(0,\infty). On a filtered probability space, let (Wi,j)i,j∈E(W^{i,j})_{i,j\in E} be independent standard Brownian motions. For every ε>0\varepsilon>0, let (uε,mε,vε)(u^\varepsilon,m^\varepsilon,v^\varepsilon) be the unique adapted solution on [0,T][0,T], with m0ε=m0m_0^\varepsilon=m_0 and uTε,i=Gi(mTε)u_T^{\varepsilon,i}=G^i(m_T^\varepsilon), of

dutε,i=[12∑j∈E(utε,i−utε,j)+2−Fi(mtε)−ε∑j∈Emtε,imtε,j(vtε,i,i,j−vtε,i,j,i)−∑j∈Eφε(mtε,j)(utε,j−utε,i)]dt+∑j,k∈Evtε,i,j,k dWtj,k,dmtε,i=∑j∈E[mtε,j(φε(mtε,i)+(utε,j−utε,i)+)−mtε,i(φε(mtε,j)+(utε,i−utε,j)+)]dt+ε∑j∈Emtε,imtε,j d(Wti,j−Wtj,i),i∈E. \begin{aligned} d u_t^{\varepsilon,i} ={}&\Bigg[ \frac12\sum_{j\in E}(u_t^{\varepsilon,i}-u_t^{\varepsilon,j})_+^2 -F^i(m_t^\varepsilon) -\sqrt\varepsilon\sum_{j\in E}\sqrt{m_t^{\varepsilon,i}m_t^{\varepsilon,j}} \bigl(v_t^{\varepsilon,i,i,j}-v_t^{\varepsilon,i,j,i}\bigr)\\ &\hspace{2cm}-\sum_{j\in E}\varphi^\varepsilon(m_t^{\varepsilon,j}) (u_t^{\varepsilon,j}-u_t^{\varepsilon,i}) \Bigg]dt +\sum_{j,k\in E}v_t^{\varepsilon,i,j,k}\,dW_t^{j,k},\\ d m_t^{\varepsilon,i} ={}&\sum_{j\in E}\Big[ m_t^{\varepsilon,j}\bigl(\varphi^\varepsilon(m_t^{\varepsilon,i}) +(u_t^{\varepsilon,j}-u_t^{\varepsilon,i})_+\bigr) -m_t^{\varepsilon,i}\bigl(\varphi^\varepsilon(m_t^{\varepsilon,j}) +(u_t^{\varepsilon,i}-u_t^{\varepsilon,j})_+\bigr) \Big]dt\\ &+\sqrt\varepsilon\sum_{j\in E}\sqrt{m_t^{\varepsilon,i}m_t^{\varepsilon,j}} \,d(W_t^{i,j}-W_t^{j,i}), \qquad i\in E. \end{aligned}

Determine whether every weak subsequential limit, as ε↓0\varepsilon\downarrow0, of the laws of mεm^\varepsilon on C([0,T];ΔE)C([0,T];\Delta_E) is supported on trajectories pp arising from minimizers (β,p)(\beta,p) of the deterministic mean field control problem above.

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Question 1

Fix T>12T>\tfrac12. For each N≥2N\ge2, consider the continuous-time NN-player game in which every player ii has a state XtN,i∈{−1,1}X_t^{N,i}\in\{-1,1\} and chooses, in Markov feedback form, a nonnegative rate αN,i(t,x)\alpha^{N,i}(t,x) at which its state flips from xix_i to −xi-x_i, where x=(x1,…,xN)∈{−1,1}Nx=(x_1,\ldots,x_N)\in\{-1,1\}^N. Given a strategy profile αN\alpha^N, player ii minimizes

JiN(αN)=E[∫0T12∣αN,i(t,XtN)∣2 dt−XTN,imTN,i],mtN,i=1N−1∑j≠iXtN,j. J_i^N(\alpha^N) =\mathbb E\left[ \int_0^T \frac12\bigl|\alpha^{N,i}(t,X_t^N)\bigr|^2\,dt -X_T^{N,i}m_T^{N,i} \right], \qquad m_t^{N,i}=\frac1{N-1}\sum_{j\ne i}X_t^{N,j}.

Assume that X0N,1,…,X0N,NX_0^{N,1},\ldots,X_0^{N,N} are independent and uniformly distributed on {−1,1}\{-1,1\}, and let XN,∗X^{N,*} be the state process under the unique symmetric Markov feedback Nash equilibrium. Define its empirical mean by

mtN=1N∑i=1NXtN,∗,i,0≤t≤T. m_t^N=\frac1N\sum_{i=1}^N X_t^{N,*,i},\qquad 0\le t\le T.

The associated mean field game system is

z˙t=12zt∣zt∣,m˙t=−mt∣zt∣+zt,zT=2mT,m0=0. \dot z_t=\frac12 z_t|z_t|, \qquad \dot m_t=-m_t|z_t|+z_t, \qquad z_T=2m_T, \qquad m_0=0.

For T>12T>\tfrac12, besides the zero solution this system has two nonzero solutions (z+,m+)(z^+,m^+) and (z−,m−)(z^-,m^-) satisfying mt+>0m_t^+>0 and mt−=−mt+<0m_t^-=-m_t^+<0 for every t∈(0,T]t\in(0,T]. Prove that, on the Skorokhod space D([0,T];[−1,1])D([0,T];[-1,1]),

Law⁡(mN)→N→∞weakly12δm++12δm−. \operatorname{Law}(m^N)\xrightarrow[N\to\infty]{\mathrm{weakly}} \frac12\delta_{m^+}+\frac12\delta_{m^-}.

Source reference: Alekos Cecchin, Paolo Dai Pra, Markus Fischer, and Guglielmo Pelino, “On the Convergence Problem in Mean Field Games: A Two State Model without Uniqueness,” SIAM Journal on Control and Optimization 57 (2019), no. 4, 2443–2466, open problem in Section 4; DOI: 10.1137/18M1222454; arXiv: 1810.05492.

Progress & source assessment

Source-reported assessment from the ICM catalogue snapshot. This is not an independent review by Frontier Science.

The exact problem is unresolved. The available selection theorem applies only after an ε\varepsilon-dependent modification of the running costs; it does not prove the claimed support property for the unmodified common-noise system in the statement.

  • “Finite state mean field games with Wright–Fisher common noise,” Erhan Bayraktar, Alekos Cecchin, Asaf Cohen, and François Delarue, Journal de Mathématiques Pures et Appliquées 147 (2021), 98–162, DOI 10.1016/j.matpur.2021.01.003. The paper proves smooth solvability of the Kimura-type master equation and uniqueness of the associated common-noise finite-state mean field game when the inward boundary forcing is sufficiently strong. This supplies the well-posed noisy system used in the question, but it does not analyze its zero-noise selection limit.

  • “Selection by vanishing common noise for potential finite state mean field games,” Alekos Cecchin and François Delarue, Communications in Partial Differential Equations 47 (2022), no. 1, 89–168, DOI 10.1080/03605302.2021.1955256, arXiv:2005.12153. Theorem 2.4 (refined as Theorem 5.1) proves that weak limits are supported on minimizers of the deterministic mean field control problem for a specially constructed viscous potential game. Theorem 2.2 adds a correction ϑε,φ\vartheta^{\varepsilon,\varphi} to the running costs (equivalently, the survey formulation uses FεF^\varepsilon instead of FF). The paper explicitly states that it cannot address the limit of the primary, non-potential noisy game. Thus its theorem is substantive partial progress but does not cover the system in the statement, where FF is unchanged.

  • “Selected topics in mean field games,” Pierre Cardaliaguet and François Delarue, Proceedings of the International Congress of Mathematicians 2022, vol. 5 (2022), 3660–3703, DOI 10.4171/ICM2022/17. Section 4.3 explicitly identifies the unmodified vanishing-common-noise question as open. Its Theorem 4.5 records the corrected-cost result above: the selected limits minimize the original deterministic control problem, while the corrected games are only approximate equilibria for the original common-noise games.

Later literature found through direct title, phrase, author, and citation searches develops weak master-equation solutions, other common-noise models, and other selection mechanisms, but no citable result was found that removes the ε\varepsilon-dependent cost correction or gives a counterexample. The remaining gap is precisely to control the zero-noise limit of the unique unmodified system stated in the problem and prove or disprove that all its limiting laws are supported on deterministic minimizers.

References
  1. [1]ICM Conjectures · 2022-econ-11 link
  2. [2]ICM 2022: Selected topics in mean field games link

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