A Numerical Scheme for a Mean Field Game in Some Queueing Systems Based on Markov Chain Approximation Method

Bayraktar, Erhan; Budhiraja, Amarjit; Cohen, Asaf

doi:10.1137/17m1154357

Cited by 16 publications

(12 citation statements)

References 30 publications

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“…However, the analysis is done under heavy-traffic so that the number of states increases with n and the limiting problem has a diffusion noise. A numerical scheme associated with this model is studied in [1]. Another paper that studies finite state MFGs in continuous time is given in [12].…”

Section: Introductionmentioning

confidence: 99%

“…The results above require regularity of the master equation, which we prove in the last section by constructing it from the coupled system of forward-backward partial differential equations that were described in [18]. 1 The rest of the paper is organized as follows. In Section 2, we analyze the n-player game: we introduce the game, the master equation associated with it, and provide the convergence results.…”

Section: Introductionmentioning

confidence: 99%

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Analysis of a Finite State Many Player Game Using Its Master Equation

Bayraktar

Cohen

2017

SSRN Journal

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We consider an n-player symmetric stochastic game with weak interactions between the players. Time is continuous and the horizon and the number of states are finite. We show that the value function of each of the players can be approximated by the solution of a partial differential equation called the master equation. Moreover, we analyze the fluctuations of the empirical measure of the states of the players in the game and show that it is governed by a solution to a stochastic differential equation. Finally, we prove the regularity of the master equation, which is required for the above results. , web: https://sites.google.com/site/asafcohentau/ 1 During the review of this paper it was brought to our attention that Cecchin and Pelino [10] also independently analyzed the finite-state MFG using the master equation approach. The n-player game in that paper is formulated using the formulation in [9] while we follow the formulation given in [18]. We prove our main result, the fluctuations, using a probabilistic approach which relies on coupling, whereas [10] uses an analytical approach relying on the convergence of the generators. It also happens that our assumptions for the convergence results (in Section 2) are slightly weaker. In obtaining sufficient conditions, Section 3.4 of this paper, both papers adapt the approach of [5] to discrete state space.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Analysis of a Finite State Many Player Game Using Its Master Equation

Bayraktar

Cohen

2017

SSRN Journal

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“…Section 6). These models have received recent interest since they naturally arise as suitable limits of interacting queuing systems, see [5] and [4]. As in [5], we employ a weak (distributional) approach, and, by enforcing additional mild technical requirements on the data of the problem, an application of Tanaka's formula for continuous semimartingales allows to embed the considered MFG into the class of abstract submodular MFGs for which Theorem 2.6 holds.…”

Section: 2mentioning

confidence: 99%

“…In this section, we consider a MFG model with reflecting boundary conditions, in which the state process of the representative player is forced to remain in a certain interval of the state space. These types of models were recently introduced in [5] (see also [4]), motivated by applications to queueing systems consisting of many strategic servers that are weakly interacting. Also, a particular setting in the same class of models is studied in [34], motivated by a model for the production of exhaustible resources.…”

Section: Submodular Mean Field Games With Reflecting Boundary Conditionsmentioning

confidence: 99%

A unifying framework for submodular mean field games

Dianetti¹,

Ferrari²,

Fischer³

et al. 2022

Preprint

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We provide an abstract framework for submodular mean field games and identify verifiable sufficient conditions that allow to prove existence and approximation of strong mean field equilibria in models where data may not be continuous with respect to the measure parameter and common noise is allowed. The setting is general enough to encompass qualitatively different problems, such as mean field games for discrete time finite space Markov chains, singularly controlled and reflected diffusions, and mean field games of optimal timing. Our analysis hinges on Tarski's fixed point theorem, along with technical results on lattices of flows of probability and sub-probability measures.

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“…A Markov chain based approximation approach was proposed in [3] for numerically solving MFGs with reflecting barriers and showed its convergence. Then in [16] it was shown that, under the notion of weak (distributional) NE, N -player stochastic games with singular controls of finite variation can be approximated by that of bounded velocity, if the set of NEs for the latter is relatively compact under an appropriate topology.…”

Section: Introductionmentioning

confidence: 99%

Approximation of N-player stochastic games with singular controls by mean field games

Cao¹,

Guo²,

Lee³

2022

Preprint

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This paper establishes that a class of N -player stochastic games with singular controls, either of bounded velocity or of finite variation, can both be approximated by mean field games (MFGs) with singular controls of bounded velocity. More specifically, it shows (i) the optimal control to an MFG with singular controls of a bounded velocity θ is shown to be an ǫ N -NE to an N -player game with singular controls of the bounded velocity, with ǫ N = O( 1 √ N ), and (ii) the optimal control to this MFG is an (ǫ N + ǫ θ )-NE to an N -player game with singular controls of finite variation, where ǫ θ is an error term that depends on θ. This work generalizes the classical result on approximation N -player games by MFGs, by allowing for discontinuous controls.

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A Numerical Scheme for a Mean Field Game in Some Queueing Systems Based on Markov Chain Approximation Method

Cited by 16 publications

References 30 publications

Analysis of a Finite State Many Player Game Using Its Master Equation

Analysis of a Finite State Many Player Game Using Its Master Equation

A unifying framework for submodular mean field games

Approximation of N-player stochastic games with singular controls by mean field games

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