Mean Field Game Model Research Articles

Second order local minimal-time mean field games

The paper considers a forward–backward system of parabolic PDEs arising in a Mean Field Game (MFG) model where every agent controls the drift of a trajectory subject to Brownian diffusion, trying to escape a given bounded domain $$\Omega $$ in minimal expected time. Agents are constrained by a bound on the drift depending on the density of other agents at their location. Existence for a finite time horizon T is proven via a fixed point argument, but the natural setting for this problem is in infinite time horizon. Estimates are needed to treat the limit $$T\rightarrow \infty $$ , and the asymptotic behavior of the solution obtained in this way is also studied. This passes through classical parabolic arguments and specific computations for MFGs. Both the Fokker–Planck equation on the density of agents and the Hamilton–Jacobi–Bellman equation on the value function display Dirichlet boundary conditions as a consequence of the fact that agents stop as soon as they reach $$\partial \Omega $$ . The initial datum for the density is given, and the long-time limit of the value function is characterized as the solution of a stationary problem.

Optimal Incentives to Mitigate Epidemics: A Stackelberg Mean Field Game Approach

Motivated by the models of epidemic control in large populations, we consider a Stackelberg mean field game model between a principal and a mean field of agents whose states evolve in a finite state space. The agents play a noncooperative game in which they control their rates of transition between states to minimize an individual cost. The principal influences the nature of the resulting Nash equilibrium through incentives to optimize its own objective. We analyze this game using a probabilistic approach. We then propose an application to an epidemic model of SIR type in which the agents control the intensities of their interactions, and the principal is a regulator acting with nonpharmaceutical interventions. To compute the solutions, we propose an innovative numerical approach based on Monte Carlo simulations and machine learning tools for stochastic optimization. We conclude with numerical experiments illustrating the impact of the agents' and the regulator's optimal decisions in two specific models: a basic SIR model with semiexplicit solutions and a more complex model with a larger state space.

Traveling waves for a nonlocal KPP equation and mean-field game models of knowledge diffusion

We analyze a mean-field game model proposed by economists Lucas and Moll [J. Political Econ. 122 (2014)] to describe economic systems where production is based on knowledge growth and diffusion. This model reduces to a PDE system where a backward Hamilton–Jacobi– Bellman equation is coupled with a forward KPP-type equation with nonlocal reaction term. We study the existence of traveling waves for this mean-field game system, obtaining the existence of both critical and supercritical waves. In particular, we prove a conjecture raised by economists on the existence of a critical balanced growth path for the described economy, supposed to be the expected stable growth in the long run. We also provide nonexistence results which clarify the role of parameters in the economic model. In order to prove these results, we build fixed point arguments on the sets of critical waves for the forced speed problem arising from the coupling in the KPP-type equation. To this purpose, we provide a full characterization of the whole family of traveling waves for a new class of KPP-type equations with nonlocal and nonhomogeneous reaction terms. This latter analysis has independent interest since it shows new phenomena induced by the nonlocal effects and a different picture of critical waves, compared to the classical literature on Fisher–KPP equations.

Cooperative Control of Physical Collision and Transmission Power for UAV Swarm: A Dual-Fields Enabled Approach

This article studies the collision avoidance and interference mitigation for unmanned aerial vehicle (UAV) swarm where many UAVs track a common target. The considered problem is formulated to jointly minimize the average interference and ensure the collision avoidance among UAVs. By exploiting the problem characteristics, our major contributions are summarized as follows. First, the singular case tolerance (SCT)-artificial potential field (APF) is proposed to overcome the failure of traditional APFs in collision avoidance, where the repulsive force gain coefficient among UAVs is dynamically controlled by the corresponding interferences. Second, the mean-field game (MFG) model is established to control communication power, where the instantaneous interferences among flying UAVs are represented by the mean-field approximation. Third, considering the tight coupling of trajectory and interference of UAVs, a cooperative control approach enabled by dual-fields is proposed to jointly adjust the trajectories and power of UAVs. Simulation results validate the significant performance improvement of the cooperative control approach enabled by dual-fields. Compared with separate APF and MFG, the proposed dual-field-cooperation approach can achieve about 117% throughput gain and 88% interference reduction when the UAV swarm is close to the target.

Mean Field Models to Regulate Carbon Emissions in Electricity Production.

The most serious threat to ecosystems is the global climate change fueled by the uncontrolled increase in carbon emissions. In this project, we use mean field control and mean field game models to analyze and inform the decisions of electricity producers on how much renewable sources of production ought to be used in the presence of a carbon tax. The trade-off between higher revenues from production and the negative externality of carbon emissions is quantified for each producer who needs to balance in real time reliance on reliable but polluting (fossil fuel) thermal power stations versus investing in and depending upon clean production from uncertain wind and solar technologies. We compare the impacts of these decisions in two different scenarios: (1) the producers are competitive and hopefully reach a Nash equilibrium; (2) they cooperate and reach a social optimum. In the model, the producers have both time dependent and independent controls. We first propose nonstandard forward–backward stochastic differential equation systems that characterize the Nash equilibrium and the social optimum. Then, we prove that both problems have a unique solution using these equations. We then illustrate with numerical experiments the producers’ behavior in each scenario. We further introduce and analyze the impact of a regulator in control of the carbon tax policy, and we study the resulting Stackelberg equilibrium with the field of producers.

A Mean Field Game Model for COVID-19 with Human Capital Accumulation

A Mean Field Game Model for COVID-19 with Human Capital Accumulation

Nonsmooth mean field games with state constraints

In this paper, we consider a mean field game model inspired by crowd motion where agents aim to reach a closed set, called target set, in minimal time. Congestion phenomena are modeled through a constraint on the velocity of an agent that depends on the average density of agents around their position. The model is considered in the presence of state constraints: roughly speaking, these constraints may model walls, columns, fences, hedges, or other kinds of obstacles at the boundary of the domain which agents cannot cross. After providing a more detailed description of the model, the paper recalls some previous results on the existence of equilibria for such games and presents the main difficulties that arise due to the presence of state constraints. Our main contribution is to show that equilibria of the game satisfy a system of coupled partial differential equations, known mean field game system, thanks to recent techniques to characterize optimal controls in the presence of state constraints. These techniques not only allow to deal with state constraints but also require very few regularity assumptions on the dynamics of the agents.

A Mean-Field Game Model for Optimal Trading at the Intraday Electricity Market

A Mean-Field Game Model for Optimal Trading at the Intraday Electricity Market

Mean Field Game Model for an Advertising Competition in a Duopoly

In this study, we analyze an advertising competition in a duopoly. We consider two different notions of equilibrium. We model the companies in the duopoly as major players, and the consumers as minor players. In our first game model, we identify Nash Equilibrium (NE) between all the players. Next we frame the model to lead to the search for Multi-Leader–Follower Nash Equilibrium (MLF-NE). This approach is reminiscent of Stackelberg games in the sense that the major players design their advertisement policies assuming that the minor players are rational and settle in a Nash Equilibrium among themselves. This rationality assumption reduces the competition between the major players to a two-player game. After solving these two models for the notions of equilibrium, we analyze the similarities and differences of the two different sets of equilibria.

A Cucker--Smale Inspired Deterministic Mean Field Game with Velocity Interactions

We introduce a mean field game (MFG) model for pedestrians moving in a given domain and choosing their trajectories so as to minimize a cost including a penalization on the difference between their...

A mean field game model for the evolution of cities

<p style='text-indent:20px;'>We propose a (toy) MFG model for the evolution of residents and firms densities, coupled both by labour market equilibrium conditions and competition for land use (congestion). This results in a system of two Hamilton-Jacobi-Bellman and two Fokker-Planck equations with a new form of coupling related to optimal transport. This MFG has a convex potential which enables us to find weak solutions by a variational approach. In the case of quadratic Hamiltonians, the problem can be reformulated in Lagrangian terms and solved numerically by an IPFP/Sinkhorn-like scheme as in [<xref ref-type="bibr" rid="b4">4</xref>]. We present numerical results based on this approach, these simulations exhibit different behaviours with either agglomeration or segregation dominating depending on the initial conditions and parameters.

Discrete-time mean field games with risk-averse agents

We propose and investigate a discrete-time mean field game model involving risk-averse agents, each of them controlling a linear dynamical system. The model under study is a coupled system of dynamic programming equations with a Kolmogorov equation. The agents’ risk aversion is modeled by composite risk measures. The existence of a solution to the coupled system is obtained with a fixed point approach. The corresponding feedback control allows to construct an approximate Nash equilibrium for a related dynamic game with finitely many players.

Controlling Propagation of Epidemics via Mean-Field Control

The coronavirus disease 2019 (COVID-19) pandemic is changing and impacting lives on a global scale. In this paper, we introduce a mean-field game model in controlling the propagation of epidemics o...

A MEAN FIELD GAME ANALYSIS OF SIR DYNAMICS WITH VACCINATION

We analyze a mean field game model of SIR dynamics (Susceptible, Infected, and Recovered) where players choose when to vaccinate. We show that this game admits a unique mean field equilibrium (MFE) that consists in vaccinating at a maximal rate until a given time and then not vaccinating. The vaccination strategy that minimizes the total cost has the same structure as the MFE. We prove that the vaccination period of the MFE is always smaller than the one minimizing the total cost. This implies that, to encourage optimal vaccination behavior, vaccination should always be subsidized. Finally, we provide numerical experiments to study the convergence of the equilibrium when the system is composed by a finite number of agents ($N$) to the MFE. These experiments show that the convergence rate of the cost is$1/N$and the convergence of the switching curve is monotone.

Linear-Quadratic Mixed Stackelberg–Nash Stochastic Differential Game with Major–Minor Agents

In this paper, we study a controlled linear-quadratic-Gaussian large population system combining three types of interactive agents mixed, which are respectively, major leader, minor leaders, and minor followers. In reality, they may represent three typical types of participants involved in market price formation: major supplier, minor suppliers and minor producers. The Stackelberg–Nash–Cournot (SNC) approximate equilibrium is derived from the combination of a major–minor mean-field game (MFG) and a leader–follower Stackelberg game. Although all agents are of forward states in that only initial conditions are specified in their dynamics, our SNC analysis provides an MFG framework that is naturally in a forward–backward state in that both initial and terminal conditions are specified. This result differs from those reported in the literature on standard MFG frameworks, mainly as a result of the adoption of a Stackelberg structure. Through variational analysis, the consistency condition system can be represented by some fully-coupled forward–backward-stochastic-differential-equations with a high-dimensional block structure in an open-loop case. To sufficiently address the related solvability, we also derive the feedback form of the SNC approximate. equilibrium strategy via some coupled Riccati equations. Our study includes various mean-field game models as its special cases.

A mean field game price model with noise

In this paper, we propose a mean-field game model for the price formation of a commodity whose production is subjected to random fluctuations. The model generalizes existing deterministic price formation models. Agents seek to minimize their average cost by choosing their trading rates with a price that is characterized by a balance between supply and demand. The supply and the price processes are assumed to follow stochastic differential equations. Here, we show that, for linear dynamics and quadratic costs, the optimal trading rates are determined in feedback form. Hence, the price arises as the solution to a stochastic differential equation, whose coefficients depend on the solution of a system of ordinary differential equations.

Remarks on Nash equilibria in mean field game models with a major player

For a mean field game model with a major and infinite minor players, we characterize a notion of Nash equilibrium via a system of so-called master equations, namely a system of nonlinear transport equations in the space of measures. Then, for games with a finite number N of minor players and a major player, we prove that the solution of the corresponding Nash system converges to the solution of the system of master equations as N tends to infinity.

Mean Field Games Theoretic for Mobile Privacy Security Enhancement in Edge Computing

Edge computing paradigm extends the cloud service to the edge of networks and reduces service latency to edge users. Meanwhile, the characteristics of edge computing, such as mobility and heterogeneity, arise new privacy security challenges. Current research on privacy security relies on information security techniques like encryption and anonymization. In this paper, we present an application of mean field game theory for privacy leakage with large scale edge devices in edge computing because the edge devices are resource constrained. The contributions are threefold: first, we construct an individual cost function with a mean field term and discuss the evolution of the state if the number of devices is large enough. Subsequently, we elaborate the Nash equilibrium of the mean field game models which are coupled through the Hamiltonian–Jacobi–Bellman (HJB) backward and Fokker–Planck–Kolmogorov (FPK) forward equations. In addition, an approximate method is introduced to analyze the stable solution of the state. Finally, numerical examples are provided to illustrate the stability of the state and the presented strategy.

On Mean Field Games models for exhaustible commodities trade

We investigate a mean field game model for the production of exhaustible resources. In this model, firms produce comparable goods, strategically set their production rate in order to maximise profit, and leave the market as soon as they deplete their capacities. We examine the related Mean Field Game system and prove well-posedness for initial measure data by deriving suitable a priori estimates. Then, we show that feedback strategies which are computed from the Mean Field Game system provideε-Nash equilibria to the correspondingN-Player Cournot game, for large values ofN. This is done by showing tightness of the empirical process in the Skorokhod M 1 topology, which is defined for distribution-valued processes.

Mean-field games and swarms dynamics in Gaussian and non-Gaussian environments

The collective behaviour of stochastic multi-agents swarms driven by Gaussian and non-Gaussian environments is analytically discussed in a mean-field approach. We first exogenously implement long range mutual interactions rules with strengths that are weighted by the real-time distance separating each agent with the swarm barycentre. Depending on the form of this barycentric modulation, a transition between two drastically different collective behaviours can be unveiled. A behavioural bifurcation threshold due to the tradeoff between the desynchronisation effects of the stochastic environment and the synchronising interactions is analytically calculated. For strong enough interactions, the emergence of a soliton propagating wave is established. Alternatively, weaker interactions cannot overcome the environmental noise and evanescent diffusive waves result. In a second and complementary approach, we show that the emergent solitons can alternatively be interpreted as being the optimal equilibrium of mean-field games (MFG) models with ad-hoc running cost functions which are here exactly determined. These MFG's soliton equilibria are therefore endogenously generated. Hence for the classes of models here proposed, an explicit correspondence between exogenous and endogenous interaction rules leading to similar collective effects is explicitly constructed. For non-Gaussian environments our results offer a new class of exactly solvable mean-field games dynamics.