Regularity for Mean-Field Games Systems with Initial-Initial Boundary Conditions: The Subquadratic Case

Gomes, Diogo A.; Pimentel, Edgard A.

doi:10.1007/978-3-319-16118-1_15

Cited by 10 publications

(7 citation statements)

References 29 publications

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“…In this case, the existence and uniqueness of smooth solutions to (1.1) is well understood for stationary problems [18,19,20,33], weakly coupled MFG systems [11], the obstacle MFG problem [12] and extended MFGs [13]. In the time-dependent setting, similar results are obtained in [14,15,24] for standard MFGs and in [16,23] for forward-forward problems. The theory of weak solutions is also well developed for first-order and second-order problems (see [4,5,7] and [6,8,34,35], respectively).…”

Section: Introductionmentioning

confidence: 60%

One-Dimensional Stationary Mean-Field Games with Local Coupling

2017

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A standard assumption in mean-field game (MFG) theory is that the coupling between the Hamilton-Jacobi equation and the transport equation is monotonically non-decreasing in the density of the population. In many cases, this assumption implies the existence and uniqueness of solutions. Here, we drop that assumption and construct explicit solutions for one-dimensional MFGs. These solutions exhibit phenomena not present in monotonically increasing MFGs: lowregularity, non-uniqueness, and the formation of regions with no agents.

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Section: Introductionmentioning

confidence: 60%

One-Dimensional Stationary Mean-Field Games with Local Coupling

2017

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“…Autore batzuek ( [5,3] esate baterako) onartzen dute kostua funtzio konbexua dela, eta, beraz, EH(π) funtzioak Kakutaniren teoremaren baldintzak betetzen dituela. Zoritxarrez, gure hipotesiak dio kostua funtzio jarraitua dela; beraz, Kakutaniren teoremaren baldintzak betetzen direla ikusteko, beste modu batean definitu behar dugu EH(π) funtzioa.…”

Section: Joko Matematikoen Hurbilketa: Kasu Diskretuaunclassified

Joko matematikoen hurbilketa: kasu diskretua

Doncel

2018

EKAIA

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Laburpena: Joko-teoriaren helburua eragile arrazioanalen portaera estrategikoak aztertzea da. Eragile kopurua oso handia bada, batez besteko eremuko jokoak joko lehiakor finituen hurbilketa onak dira eta, hori dela eta, biziki ikertuak izan dira azken urteotan. Hala ere, literaturan, artikulu gutxitan onartzen da jokalarien akzio-espazioa diskretua dela. Eredu horiek dira, hain zuzen ere, gure ikerketan aztertzen ditugunak. Jokalariak simetrikoak direla onartuko dugu, baita jokalarien kostu-funtzioa eta dinamikak jarraituak direla ere. Artikulu honetan erakutsiko dugu badela beti oreka bat batez besteko eremuko joko horietan.Hitz gakoak: Joko-teoria, batez besteko hurbilketa, kasu diskretua.Abstract: Game-theory is the set of mathematical tools to study the rational behavior of interacting agents or players. When the number of players is large, mean-field games have been proposed as an approximation of non-cooperative finite games. Contrarily to most of the literature in this area, we analyse mean-field games when the number of possible actions that players can take is discrete. We consider that players are symmetric and that the cost function and the dynamics of the players are continuous. In this article we show that a solution of these type of mean-field games always exists.

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“…Thanks to the regularizing properties of the Laplacian, both elliptic and parabolic MFGs are now well-understood. For example, the existence of solutions to second-order time-dependent MFGs without congestion was examined in [4], [8], [12], [13], [14], [15], and [16]. Time-dependent cases with congestion were investigated in [17], [26], and [27].…”

Section: Introductionmentioning

confidence: 99%

Existence of weak solutions to time-dependent mean-field games

Ferreira

Gomes

Tada

2020

Preprint

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Here, we establish the existence of weak solutions to a wide class of time-dependent monotone mean-field games (MFGs). These MFGs are given as a system of degenerate parabolic equations with initial and terminal conditions. To construct these solutions, we consider a high-order elliptic regularization in space-time. Then, using Schaefer's fixed-point theorem, we obtain the existence and uniqueness for this regularized problem. Using Minty's method, we prove the existence of a weak solution to the original MFG. Finally, the paper ends with a discussion on congestion problems and density constrained MFGs.

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Regularity for Mean-Field Games Systems with Initial-Initial Boundary Conditions: The Subquadratic Case

Cited by 10 publications

References 29 publications

One-Dimensional Stationary Mean-Field Games with Local Coupling

One-Dimensional Stationary Mean-Field Games with Local Coupling

Joko matematikoen hurbilketa: kasu diskretua

Existence of weak solutions to time-dependent mean-field games

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