Existence of weak solutions to first-order stationary mean-field games with Dirichlet conditions

Ferreira, Rita; Gomes, Diogo A.; Tada, Teruo

doi:10.1090/proc/14475

Cited by 14 publications

(23 citation statements)

References 31 publications

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“…For these modeling scenarios it is of particular relevance the study of MFG with Dirichlet boundary conditions since they arise naturally in situations where agents can leave the domain, such as in evacuation problems, in the case of financial default or in an exhaustible resources framework [35]. We mention that, in the paper [30], the authors have proven the existence and uniqueness of solutions to a stationary MFG with Dirichlet boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

Mean Field Games of Controls with Dirichlet boundary conditions

Bongini¹,

Salvarani²

2021

Preprint

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In this paper we study a mean-field games system with Dirichlet boundary conditions in a closed domain and in a mean-field of control setting, that is in which the dynamics of each agent is affected not only by the average position of the rest of the agents but also by their average optimal choice. This setting allows the modeling of more realistic real-life scenarios in which agents not only will leave the domain at a certain point in time (like during the evacuation of pedestrians or in debt refinancing dynamics) but also act competitively to anticipate the strategies of the other agents. We shall establish the existence of Nash Equilibria for such class of mean-field of controls systems under certain regularity assumptions on the dynamics and the Lagrangian cost. Much of the paper is devoted to establishing several a priori estimates which are needed to circumvent the fact that the mass is not conserved (as we are in a Dirichlet boundary condition setting). In the conclusive sections, we provide examples of systems falling into our framework as well as numerical implementations.

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Section: Introductionmentioning

confidence: 99%

Mean Field Games of Controls with Dirichlet boundary conditions

Bongini¹,

Salvarani²

2021

Preprint

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“…However, apart from a small number of papers e.g. [16,24], almost all results focus on problems posed on the torus in order avoid dealing with boundary conditions. In [2] the Dirichlet problem was motivated as a stopping time problem, and it was analysed in [24].…”

Section: Introductionmentioning

confidence: 99%

“…[16,24], almost all results focus on problems posed on the torus in order avoid dealing with boundary conditions. In [2] the Dirichlet problem was motivated as a stopping time problem, and it was analysed in [24]. In this paper we consider Neumann boundary conditions, which relate to a no-flux boundary.…”

Section: Introductionmentioning

confidence: 99%

“…The methods used there and in many subsequent works (e.g. [24]) rely on monotonicity properties of the operators. In our work presented here monotonicity similarly plays a central role in proving existence and uniqueness -through both the use of the maximum principle to prove existence and uniqueness for strictly increasing functions h and through the ability to uniformly perturb an increasing function into a strictly increasing function through the addition of a logarithmic congestion term.…”

Section: Introductionmentioning

confidence: 99%

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Comparing the best-reply strategy and mean-field games: The stationary case

2020

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Mean-field games (MFGs) and the best-reply strategy (BRS) are two methods of describing competitive optimisation of systems of interacting agents. The latter can be interpreted as an approximation of the respective MFG system. In this paper, we present an analysis and comparison of the two approaches in the stationary case. We provide novel existence and uniqueness results for the stationary boundary value problems related to the MFG and BRS formulations, and we present an analytical and numerical comparison of the two paradigms in some specific modelling situations.

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“…According to [16] (also see [17]), Problem 2 admits weak solutions under suitable polynomial growth conditions of g, see Corollary 6.3 in [16]. Here, in Section 7, under a different set of hypothesis and using a limiting argument, we establish the existence of solutions for Problem 2.…”

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confidence: 96%

The selection problem for some first-order stationary Mean-field games

Gomes¹,

Mitake²,

Terai³

2020

Networks &Amp; Heterogeneous Media

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Here, we study the existence and the convergence of solutions for the vanishing discount MFG problem with a quadratic Hamiltonian. We give conditions under which the discounted problem has a unique classical solution and prove convergence of the vanishing-discount limit to a unique solution up to constants. Then, we establish refined asymptotics for the limit. When those conditions do not hold, the limit problem may not have a unique solution and its solutions may not be smooth, as we illustrate in an elementary example. Finally, we investigate the stability of regular weak solutions and address the selection problem. Using ideas from Aubry-Mather theory, we establish a selection criterion for the limit.

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Existence of weak solutions to first-order stationary mean-field games with Dirichlet conditions

Cited by 14 publications

References 31 publications

Mean Field Games of Controls with Dirichlet boundary conditions

Mean Field Games of Controls with Dirichlet boundary conditions

Comparing the best-reply strategy and mean-field games: The stationary case

The selection problem for some first-order stationary Mean-field games

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