Mean field games models of segregation

Achdou, Yves; Bardi, Martino; Cirant, Marco

doi:10.1142/s0218202517400036

Cited by 61 publications

(94 citation statements)

References 47 publications

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“…Mean-field type games is a tool to extend the mean-field approach to distinguishable sub-crowds and influential individuals, and was applied in [5]. Other ways to break the anonymity within the mean-field approach are multi-population MFGs [28,21,1] and major agent models [32, and references therein].…”

Section: Pedestrian Crowd Modelingmentioning

confidence: 99%

Modeling tagged pedestrian motion: A mean-field type game approach

Aurell

Djehiche

2019

Transportation Research Part B: Methodological

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This paper suggests a model for the motion of tagged pedestrians: pedestrians moving towards a specified targeted destination, which they are forced to reach. It aims to be a decision-making tool for the positioning of fire fighters, security personnel and other services in a pedestrian environment. Taking interaction with the surrounding crowd into account leads to a differential nonzero-sum game model where the tagged pedestrians compete with the surrounding crowd of ordinary pedestrians. When deciding how to act, pedestrians consider crowd distribution-dependent effects, like congestion and crowd aversion. Including such effects in the parameters of the game, makes it a mean-field type game. The equilibrium control is characterized, and special cases are discussed. Behavior in the model is studied by numerical simulations.MSC 2010: 49N90, 49J55, 60H10, 60K30, 91A80, 93E20 Financial support from the Swedish Research Council (2016-04086) is gratefully acknowledged. We thank the anonymous reviewers for comments and suggestions that greatly helped to improve the presentation of the results, and we thank E. Cristiani for directing us to [23].Aurell, Djehiche/Tagged pedestrian model 2 the model include cancer cell dynamics and smart medicine in the human body, and malware propagation in a network, among other.The central planner's decision making is based on knowledge of the pedestrian distribution. As noted in [45], the pedestrian behavior in dense crowds is empirically random to some extent, likely due to the large number of external inputs. In a noisy environment, the central planner anticipates the behavior of the crowd and predicts the tagged's path to the target. As is standard in the mean-field approach, interaction with tagged and ordinary pedestrians is modeled as reactions to the state distribution of a representative tagged and ordinary pedestrian, respectively. This leads us to formulate a mean-field type game based model, which in certain scenarios reduces to an optimal control based model. Related work Optimal control and games of mean-field typeRational pedestrian behavior is, in this paper, characterized by either a game equilibrium, or an optimal control. The tool used to find the equilibrium/optimal behavior is Pontryagin's stochastic maximum principle (SMP). For stochastic control problems, SMP yields, when available, necessary conditions that must be satisfied by any optimal solution. The necessary conditions become sufficient under additional convexity conditions. Early results show that an optimal control along with the corresponding optimal state trajectory must solve the so-called Hamiltonian system, which is a two-point (forward-backward) boundary value problem, together with a maximum condition of the so-called Hamiltonian function (see [51] for a detailed account). In stochastic differential games, both zero-sum and nonzero-sum, SMP is one of the main tools for obtaining conditions for an equilibrium, essentially inherited from the theory of stochastic optimal control. For recent examples ...

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Section: Pedestrian Crowd Modelingmentioning

confidence: 99%

Modeling tagged pedestrian motion: A mean-field type game approach

Aurell

Djehiche

2019

Transportation Research Part B: Methodological

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“…In the second example, we consider a particular instance of a two population MFG system modelling segregation (see e.g. [2,10]). As discussed in [9], standard MFGs can be seen as a particular (F P K) equation with M = 1, where the drift term b 1 satisfies that for each (…”

Section: Simulationsmentioning

confidence: 99%

“…where * denotes the space convolution and W 11 (x) = W 21 (x) = W 22 (x) := |x| 2 2 , W 12 (x) := −|x| 2 2 . With these choices, the drift terms…”

Section: Simulationsmentioning

confidence: 99%

“…are smooth approximations of (·) − and (·) + , respectively, and m 1 (t) * φ δ (·), m 2 (t) * φ δ (·) are defined as the convolutions of m 1 (t) and m 2 (t) with R d x → φ δ (x) = √ 2πδ exp (−|x| 2 /(2δ 2 )) ∈ R. When ν > 0 andm 0 ( = 1, 2) are sufficiently regular, the existence of classical solutions to (MFG) can be proved by standard methods (see [2,Theorem 12], where the proof is provided when the space domain in (MFG) is bounded and Neumann boundary conditions are imposed on its boundary).…”

Section: 2mentioning

confidence: 99%

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A Fully-Discrete Scheme for Systems of Nonlinear Fokker-Planck-Kolmogorov Equations

Carlini

Silva

2018

Springer INdAM Series

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We consider a system of Fokker-Planck-Kolmogorov (FPK) equations, where the dependence of the coefficients is nonlinear and nonlocal in time with respect to the unknowns. We extend the numerical scheme proposed and studied in [9] for a single FPK equation of this type. We analyse the convergence of the scheme and we study its applicability in two examples. The first one concerns a population model involving two interacting species and the second one concerns two populations Mean Field Games.

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“…Mean field games with multiple populations arise naturally in most of the practical applications, and were already considered in the first original work of [22]. Lachapelle & Wolfram [25] modeled a congestion problem of pedestrian crowds, and Achdou, Bardi & Cirant [1] studied the issue of urban settlements and residential choice using the mean-field game representation. Feleqi [17] and Cirant [14] dealt with ergodic mean field games of multiple populations under different boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

Probabilistic Approach to Mean Field Games and Mean Field Type Control Problems with Multiple Populations

Fujii

2019

SSRN Journal

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In this work, we systematically investigate mean field games and mean field type control problems with multiple populations using a coupled system of forward-backward stochastic differential equations of McKean-Vlasov type stemming from Pontryagin's stochastic maximum principle. Although the same cost functions as well as the coefficient functions of the state dynamics are shared among the agents within each population, they can be different population by population. We study the mean field limit for the three different situations; (i) every agent is non-cooperative; (ii) the agents within each population are cooperative; and (iii) the agents in some populations are cooperative but those in the other populations are not. We provide several sets of sufficient conditions for the existence of a mean field equilibrium for each of these cases. Furthermore, under appropriate conditions, we show that the mean field solution to each of these problems actually provides an approximate Nash equilibrium for the corresponding game with a large but finite number of agents.Keywords : mean field game, mean field type control, FBSDE of McKean-Vlasov typewhere (X i,µ i ,µ j ) i∈{1,2},j =i are the solutions to the optimal control problems in (ii).Remark 3.1. It is just for convenience to use the common dimension d (as well as k for A i ) for both populations. Optimization for given flows of probability measuresThe main assumptions in this section are as follows: Assumption 3.1. (MFG-a) L, K ≥ 0 and λ > 0 are some constants. For 1 ≤ i ≤ 2, the measurable functionsand g i : R d × P 2 (R d ) 2 → R satisfy the following conditions: (A1) The functions b i and σ i are affine in (x, α) in the sense that, for any (t, x, µ, ν, α) ∈ [0, T ] ×

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Mean field games models of segregation

Cited by 61 publications

References 47 publications

Modeling tagged pedestrian motion: A mean-field type game approach

Modeling tagged pedestrian motion: A mean-field type game approach

A Fully-Discrete Scheme for Systems of Nonlinear Fokker-Planck-Kolmogorov Equations

Probabilistic Approach to Mean Field Games and Mean Field Type Control Problems with Multiple Populations

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