scholarly journals Long time behaviour and turnpike solutions in mildly non-monotone mean field games

2021 ◽  
Author(s):  
Marco Cirant ◽  
Alessio Porretta
Keyword(s):  
Infinite Horizon ◽  
Mean Field ◽  
Mean Field Games ◽  
Long Time Behaviour ◽  
Quadratic Hamiltonians ◽  
Long Time ◽  
Linearized System ◽  
The Individual ◽  
The Cost ◽  
Well Posed

We consider mean field game systems in time-horizon (0,T), where the individual cost depends locally on the density distribution of the agents, and the Hamiltonian is locally uniformly convex. We show that, even if the coupling cost functions are mildly non-monotone, then the system is still well posed due to the effect of individual noise. The rate of anti-monotonicity (the aggregation rate of the cost function) which can be afforded depends on the intensity of the diffusion and on global bounds of solutions. We give applications to either globally Lipschitz Hamiltonians or quadratic Hamiltonians and couplings having mild growth.  Under similar conditions,  we give a complete description of the ergodic and long time properties of the system. In particular we prove: (i) the turnpike property of solutions in the finite (long) horizon (0,T), (ii) the convergence of the system from (0,T) towards  (0,\infty), (iii) the vanishing discount limit of the infinite horizon problem and the long time convergence towards the ergodic stationary solution. We extend previous results which were known only for the case of monotone and smoothing couplings; our approach is self-contained and does not need the use of the  linearized system or of the master equation.

scholarly journals Long-Time Behavior of First-Order Mean Field Games on Euclidean Space

2019 ◽  
Vol 10 (2) ◽  
pp. 361-390
Author(s):  
Piermarco Cannarsa ◽  
Wei Cheng ◽  
Cristian Mendico ◽  
Kaizhi Wang
Keyword(s):  
Euclidean Space ◽  
Mean Field ◽  
Mean Field Games ◽  
Time Behavior ◽  
Long Time Behavior ◽  
First Order ◽  
Long Time

A finite difference analogue of the “mean field” equilibrium problem

2020 ◽  
pp. 31-44
Author(s):  
Виктория Сергеевна Корниенко ◽  
Владимир Викторович Шайдуров ◽  
Евгения Дмитриевна Карепова
Keyword(s):  
Control Function ◽  
Mean Field ◽  
Mean Field Games ◽  
Differential Problem ◽  
Control Functions ◽  
Economic Interaction ◽  
The Mean ◽  
Hamilton Jacobi Bellman ◽  
The Stability ◽  
The Cost

Представлен конечно-разностный аналог дифференциальной задачи, сформулированной в терминах теории “игр среднего поля” (mean field games). Задачи оптимизации такого типа формулируются как связанные системы параболических дифференциальных уравнений в частных производных типа Фоккера - Планка и Гамильтона - Якоби - Беллмана. Предложенный конечно-разностный аналог обладает основными свойствами оптимизационной дифференциальной задачи непосредственно на дискретном уровне. В итоге он может служить как приближение, сходящееся к исходной дифференциальной задаче при стремлении шагов дискретизации к нулю, так и как самостоятельная оптимизационная задача с конечным числом участников. Для предложенного аналога построен алгоритм монотонной минимизации функционала стоимости, проиллюстрированный на модельной экономической задаче In most forecasting problems, overstating or understating forecast leads to various losses. Traditionally, in the theory of “mean field games”, the functional responsible for the costs of implementing the interaction of the continuum of agents between each other is supposed to be dependent on the squared function of control of the system. Since additional external factors can influence the player’s strategy, the control function of a dynamic system is more complex. Therefore, the purpose of this article is to develop a computational algorithm applicable for more general set of control functions. As a research method, a computational experiment and proof of the stability of the constructed computational scheme are used in this study. As a result, the numerical algorithm was applied on the problem of economic interaction in the presence of alternative resources. We consider the model, in which a continuum of consumer agents consists of households deciding on heating, having a choice between the cost of installing and maintaining the thermal insulation or the additional cost of electricity. In the framework of the problem, the convergence of the method is numerically demonstrated. Conclusions. The article considers a model of the strategic interaction of continuum of agents, the interaction of which is determined by a coupled differential equations, namely, the Fokker - Planck and the Hamilton - Jacobi - Bellman one. To approximate the differential problem, difference schemes with a semi-Lagrangian approximation are used, which give a direct rule for minimizing the cost functional

scholarly journals Linear quadratic mean field game with control input constraint

2018 ◽  
Vol 24 (2) ◽  
pp. 901-919 ◽  
Author(s):  
Ying Hu ◽  
Jianhui Huang ◽  
Xun Li
Keyword(s):  
Backward Stochastic Differential Equation ◽  
Consistency Condition ◽  
Mean Field ◽  
Control Process ◽  
Mean Field Games ◽  
Monotonicity Condition ◽  
Projection Operators ◽  
Control Input ◽  
Linear Quadratic ◽  
The Individual

In this paper, we study a class of linear-quadratic (LQ) mean-field games in which the individual control process is constrained in a closed convex subset Γ of full space ℝm. The decentralized strategies and consistency condition are represented by a class of mean-field forward-backward stochastic differential equation (MF-FBSDE) with projection operators on Γ. The wellposedness of consistency condition system is obtained using the monotonicity condition method. The related ϵ-Nash equilibrium property is also verified.

scholarly journals A Quadratic Mean Field Games Model for the Langevin Equation

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 68
Author(s):  
Fabio Camilli
Keyword(s):  
Langevin Equation ◽  
Existence Result ◽  
Mean Field ◽  
Mean Field Games ◽  
Existence Of A Solution ◽  
Quadratic Mean ◽  
Change Of Variables ◽  
The Mean ◽  
The Cost ◽  
Fokker Planck Equations

We consider a Mean Field Games model where the dynamics of the agents is given by a controlled Langevin equation and the cost is quadratic. An appropriate change of variables transforms the Mean Field Games system into a system of two coupled kinetic Fokker–Planck equations. We prove an existence result for the latter system, obtaining consequently existence of a solution for the Mean Field Games system.

scholarly journals A class of infinite horizon mean field games on networks

2019 ◽  
Vol 14 (3) ◽  
pp. 537-566
Author(s):  
Yves Achdou ◽  
◽  
Manh-Khang Dao ◽  
Olivier Ley ◽  
Nicoletta Tchou ◽  
...  
Keyword(s):  
Infinite Horizon ◽  
Mean Field ◽  
Mean Field Games

scholarly journals Long time average of mean field games

2012 ◽  
Vol 7 (2) ◽  
pp. 279-301 ◽  
Author(s):  
Pierre Cardaliaguet ◽  
Jean-Michel Lasry ◽  
Pierre-Louis Lions ◽  
Alessio Porretta
Keyword(s):  
Mean Field ◽  
Mean Field Games ◽  
Long Time ◽  
Time Average

scholarly journals Contact rate epidemic control of COVID-19: an equilibrium view

2020 ◽  
Vol 15 ◽  
pp. 35 ◽  
Author(s):  
Romuald Elie ◽  
Emma Hubert ◽  
Gabriel Turinici
Keyword(s):  
Compartmental Model ◽  
Transmission Rate ◽  
Mean Field ◽  
Optimal Solution ◽  
Population Level ◽  
Contact Rate ◽  
The Individual ◽  
The Cost ◽  
Effective Transmission ◽  
Latent Phase

We consider the control of the COVID-19 pandemic through a standard SIR compartmental model. This control is induced by the aggregation of individuals’ decisions to limit their social interactions: when the epidemic is ongoing, an individual can diminish his/her contact rate in order to avoid getting infected, but this effort comes at a social cost. If each individual lowers his/her contact rate, the epidemic vanishes faster, but the effort cost may be high. A Mean Field Nash equilibrium at the population level is formed, resulting in a lower effective transmission rate of the virus. We prove theoretically that equilibrium exists and compute it numerically. However, this equilibrium selects a sub-optimal solution in comparison to the societal optimum (a centralized decision respected fully by all individuals), meaning that the cost of anarchy is strictly positive. We provide numerical examples and a sensitivity analysis, as well as an extension to a SEIR compartmental model to account for the relatively long latent phase of the COVID-19 disease. In all the scenario considered, the divergence between the individual and societal strategies happens both before the peak of the epidemic, due to individuals’ fears, and after, when a significant propagation is still underway.

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