Finite State Graphon Games with Applications to Epidemics

We consider a game for a continuum of non-identical players evolving on a finite state space. Their heterogeneous interactions are represented with a graphon, which can be viewed as the limit of a dense random graph. A player’s transition rates between the states depend on their control and the strength of interaction with the other players. We develop a rigorous mathematical framework for the game and analyze Nash equilibria. We provide a sufficient condition for a Nash equilibrium and prove existence of solutions to a continuum of fully coupled forward-backward ordinary differential equations characterizing Nash equilibria. Moreover, we propose a numerical approach based on machine learning methods and we present experimental results on different applications to compartmental models in epidemiology.

[2]  Salah Eddine Choutri,et al.  Mean-field risk sensitive control and zero-sum games for Markov chains , 2018, Bulletin des Sciences Mathématiques.

[3]  P. Lions,et al.  Jeux à champ moyen. I – Le cas stationnaire , 2006 .

[4]  Francesco Salvarani,et al.  Optimal individual strategies for influenza vaccines with imperfect efficacy and durability of protection. , 2018, Mathematical biosciences and engineering : MBE.

[5]  Federica Garin,et al.  Graphon-Based Sensitivity Analysis of SIS Epidemics , 2020, IEEE Control Systems Letters.

[6]  Mathieu Lauriere,et al.  Convergence Analysis of Machine Learning Algorithms for the Numerical Solution of Mean Field Control and Games I: The Ergodic Case , 2019, SIAM J. Numer. Anal..

[7]  Dylan Dronnier,et al.  Targeted Vaccination Strategies for an Infinite-dimensional SIS Model , 2021 .

[8]  Peter E. Caines,et al.  Spectral Representations of Graphons in Very Large Network Systems Control , 2019, 2019 IEEE 58th Conference on Decision and Control (CDC).

[9]  R. Carmona,et al.  Finite State Mean Field Games with Major and Minor Players , 2016 .

[10]  H. Brezis.  Functional Analysis, Sobolev Spaces and Partial Differential Equations , 2010 .

[11]  Hamidou Tembine,et al.  A Stochastic Maximum Principle for Markov Chains of Mean-Field Type , 2018, Games.

[12]  Xavier Warin,et al.  Incentives, lockdown, and testing: from Thucydides's analysis to the COVID-19 pandemic , 2020 .

[13]  Jaksa Cvitanic,et al.  Dynamic programming approach to principal–agent problems , 2015, Finance Stochastics.

[14]  Peter E. Caines,et al.  Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle , 2006, Commun. Inf. Syst..

[15]  A Remark on a Fixed-Point Theorem for Iterated Mappings , 1968 .

[16]  D. Dawson,et al.  Superprocesses at Saint-Flour , 2011 .

[17]  Hannah R. Meredith,et al.  The Incubation Period of Coronavirus Disease 2019 (COVID-19) From Publicly Reported Confirmed Cases: Estimation and Application , 2020, Annals of Internal Medicine.

[18]  M. Keeling,et al.  Networks and epidemic models , 2005, Journal of The Royal Society Interface.

[19]  M. Bálek,et al.  Large Networks and Graph Limits , 2022 .

[20]  Bruno Gaujal,et al.  A MEAN FIELD GAME ANALYSIS OF SIR DYNAMICS WITH VACCINATION , 2020, Probability in the Engineering and Informational Sciences.

[21]  H. Andersson,et al.  Limit theorems for multitype epidemics , 1995 .

[22]  Peiqi Wang,et al.  A Probabilistic Approach to Extended Finite State Mean Field Games , 2018, Math. Oper. Res..

[23]  Gabriel Turinici,et al.  Individual Vaccination as Nash Equilibrium in a SIR Model with Application to the 2009–2010 Influenza A (H1N1) Epidemic in France , 2015, Bulletin of Mathematical Biology.

[24]  Yongchao Zhang,et al.  Individual risk and Lebesgue extension without aggregate uncertainty , 2008, J. Econ. Theory.

[25]  Romuald Elie,et al.  Contact rate epidemic control of COVID-19: an equilibrium view , 2020, Mathematical Modelling of Natural Phenomena.

[26]  Peter E. Caines,et al.  Graphon Mean Field Games and the GMFG Equations: ε-Nash Equilibria , 2019, 2019 IEEE 58th Conference on Decision and Control (CDC).

[27]  Hamidou Tembine,et al.  Optimal Control and Zero-Sum Games for Markov Chains of Mean-Field Type , 2016, Mathematical Control & Related Fields.

[28]  Mathieu Laurière,et al.  Convergence Analysis of Machine Learning Algorithms for the Numerical Solution of Mean Field Control and Games: I - The Ergodic Case , 2019, The Annals of Applied Probability.

[29]  Romuald Elie,et al.  COVID-19 pandemic control: balancing detection policy and lockdown intervention under ICU sustainability* , 2020 .

[30]  Gabriel Turinici,et al.  Nash-MFG equilibrium in a SIR model with time dependent newborn vaccination , 2018 .

[31]  Samuel Cho.  Mean-Field Game Analysis of SIR Model with Social Distancing , 2020 .

[32]  Mean Field Games , 2021, Encyclopedia of Systems and Control.

[33]  Alexander Schied,et al.  Large Deviations for Hierarchical Systems of Interacting Jump Processes , 1998 .

[34]  Peter E. Caines,et al.  LQG Graphon Mean Field Games , 2020, ArXiv.

[35]  D. Gomes,et al.  Discrete Time, Finite State Space Mean Field Games , 2010 .

[36]  Erhan Bayraktar,et al.  Analysis of a Finite State Many Player Game Using Its Master Equation , 2017, SIAM J. Control. Optim..

[37]  Markus Fischer,et al.  Probabilistic Approach to Finite State Mean Field Games , 2017, Applied Mathematics & Optimization.

[38]  Yeneng Sun,et al.  The exact law of large numbers via Fubini extension and characterization of insurable risks , 2006, J. Econ. Theory.

[39]  B. Gaujal,et al.  Vaccination in a Large Population: Mean Field Equilibrium Versus Social Optimum , 2020, NetGCooP.

[40]  P. Lions,et al.  Jeux à champ moyen. II – Horizon fini et contrôle optimal , 2006 .

[41]  Gabriel Turinici,et al.  Equilibrium in an individual - societal SIR vaccination model in presence of discounting and finite vaccination capacity , 2016 .

[42]  Mathieu Lauriere,et al.  Optimal Incentives to Mitigate Epidemics: A Stackelberg Mean Field Game Approach , 2020, SIAM Journal of Control and Optimization.

[43]  Mathieu Lauriere,et al.  Stochastic Graphon Games: II. The Linear-Quadratic Case , 2021, Applied Mathematics & Optimization.

[44]  Xun Yu Zhou,et al.  Relationship Between Backward Stochastic Differential Equations and Stochastic Controls: A Linear-Quadratic Approach , 2000, SIAM J. Control. Optim..

[45]  Boualem Djehiche,et al.  The rate function for some measure-valued jump processes , 1995 .

[46]  Jakša Cvitanić,et al.  Contract Theory in Continuous-Time Models , 2012 .

[47]  Hamidou Tembine,et al.  Controlling Propagation of epidemics via mean-field games , 2020, 2006.01249.

[48]  A. Skorokhod,et al.  Studies in the theory of random processes , 1966 .

[49]  Mathieu Lauriere,et al.  Stochastic Graphon Games: I. The Static Case , 2019, Math. Oper. Res..

[50]  Peter E. Caines,et al.  Graphon Mean Field Games and the GMFG Equations , 2018, 2018 IEEE Conference on Decision and Control (CDC).

[51]  Yeneng Sun,et al.  A theory of hyperfinite processes: the complete removal of individual uncertainty via exact LLN1 , 1998 .

[52]  D. Gomes,et al.  Continuous Time Finite State Mean Field Games , 2012, 1203.3173.

[53]  François Delarue,et al.  MEAN FIELD GAMES: A TOY MODEL ON AN ERDOS-RENYI GRAPH , 2017 .

[54]  Asuman Ozdaglar,et al.  Graphon Games: A Statistical Framework for Network Games and Interventions , 2018, SSRN Electronic Journal.

[55]  V. Kolokoltsov.  Nonlinear Markov Games on a Finite State Space (Mean-field and Binary Interactions) , 2012 .

[56]  Jan van Neerven,et al.  Analysis in Banach Spaces , 2023, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics.

[57]  Hamidou Tembine.  COVID-19: Data-Driven Mean-Field-Type Game Perspective , 2020, Games.