Convergence of large population games to mean field games with interaction through the controls

Laurière, Mathieu; Tangpi, Ludovic

doi:10.48550/arxiv.2004.08351

Cited by 15 publications

(35 citation statements)

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“…A third and final contribution is the implementation, through an iterative procedure based on the particle method, of mean-field games system with Dirichlet boundary conditions in a mean-field of control setting. This method draws its theoretical justification with the results of convergence of large population games to mean field games (see, for example, [44]. Up to the best of our knowledge, this is one of the first uses of particle methods to numerically solve mean-field games system with Dirichlet boundary conditions in a mean-field of control setting, in which an iterative procedure is implemented for reducing the control space.…”

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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“…The previous assumptions are standard in the probabilistic approach of mean field game and control problems. The separability condition is more specific to the extended mean field game and control problems (see Carmona and Lacker [15], Cardaliaguet and Lehalle [10], Laurière and Tangpi [46], Possamaï and Tangpi [46], Djete [24]). It is mainly used for technical reasons.…”

Section: Setup and Main Resultsmentioning

confidence: 99%

“…For the extended MFG, while allowing the volatility to be controlled, a similar study to [38] is provided by Djete [24] thanks to the notion of measure-valued MFG equilibrium. With stronger assumptions but by providing convergence rates, Laurière and Tangpi [46] study these convergence problems in the situation without common noise using notably a notion of backward propagation of chaos.…”

Section: Introductionmentioning

confidence: 99%

“…In the situation of MFG of controls without common noise, the only article to our knowledge dealing with convergence issues for the closed-loop controls is Possamaï and Tangpi [49]. They use the result of backward propagation of chaos of [46] to deal with this issue among many others. Nevertheless, it seems that no many cases of non-unique MFG equilibria are considered because of their assumptions of existence and uniqueness for a McKean-Vlasov BSDE.…”

Section: Introductionmentioning

confidence: 99%

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Large population games with interactions through controls and common noise: convergence results and equivalence between $open$--$loop$ and $closed$--$loop$ controls

Djete¹

2021

Preprint

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In the presence of a common noise, we study the convergence problems in mean field game (MFG) and mean field control (MFC) problem where the cost function and the state dynamics depend upon the joint conditional distribution of the controlled state and the control process. In the first part, we consider the MFG setting. We start by recalling the notions of measure-valued MFG equilibria and of approximate closed-loop Nash equilibria associated to the corresponding N -player game. Then, we show that all convergent sequences of approximate closed-loop Nash equilibria, when N → ∞, converge to measure-valued MFG equilibria. And conversely, any measure-valued MFG equilibrium is the limit of a sequence of approximate closed-loop Nash equilibria. In other words, measure-valued MFG equilibria are the accumulation points of the approximate closed-loop Nash equilibria. Previous work has shown that measure-valued MFG equilibria are the accumulation points of the approximate open-loop Nash equilibria. Therefore, we obtain that the limits of approximate closed-loop Nash equilibria and approximate open-loop Nash equilibria are the same. In the second part, we deal with the MFC setting. After recalling the closed-loop and open-loop formulations of the MFC problem, we prove that they are equivalent. We also provide some convergence results related to approximate closed-loop Pareto equilibria.

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“…Some related work has appeared recently on convergence of the finite-player Nash equilibrium to the corresponding mean-field equilibrium in various settings. Laurière and Tangpi (2020) proved such convergence results for open-loop equilibria of games with idiosyncratic noise for each of the players. Neuman and Voß (2021) studied the corresponding problem for execution games with common noise.…”

Section: Introductionmentioning

confidence: 84%

Closed-Loop Nash Competition for Liquidity

Micheli¹,

Muhle‐Karbe²,

Neuman³

2021

Preprint

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We study a multi-player stochastic differential game, where agents interact through their joint price impact on an asset that they trade to exploit a common trading signal. In this context, we prove that a closed-loop Nash equilibrium exists if the price impact parameter is small enough. Compared to the corresponding open-loop Nash equilibrium, both the agents' optimal trading rates and their performance move towards the central-planner solution, in that excessive trading due to lack of coordination is reduced. However, the size of this effect is modest for plausible parameter values.

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Convergence of large population games to mean field games with interaction through the controls

Cited by 15 publications

References 0 publications

Mean Field Games of Controls with Dirichlet boundary conditions

Mean Field Games of Controls with Dirichlet boundary conditions

Large population games with interactions through controls and common noise: convergence results and equivalence between $open$--$loop$ and $closed$--$loop$ controls

Closed-Loop Nash Competition for Liquidity

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