A unifying framework for submodular mean field games

Dianetti, Jodi; Ferrari, Giorgio; Fischer, Markus; Nendel, Max

doi:10.48550/arxiv.2201.07850

Cited by 4 publications

(4 citation statements)

References 43 publications

(74 reference statements)

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“…The well-posedness of both strong and weak solutions of MFG master equations in the continuum state space setting has been explored under various sets of monotonicity assumptions [1,8,11,15,16,18,31,32,34,35,47]. The approach in our setting, which involves appealing to Tarski's fixed point theorem for increasing functions on lattices [54], has also been taken in the continuum state space setting, where maximal and minimal solutions were found under related assumptions; see for instance [25][26][27]48]. The partial order used in [48] comes from the notion of stochastic dominance for probability measures.…”

Section: Summary Of Main Resultsmentioning

confidence: 99%

Interpolation results for pathwise Hamilton-Jacobi equations

Lions¹,

Seeger²,

Souganidis³

2022

Indiana Univ. Math. J.

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We study first-and second-order linear transport equations, as well as ODE and SDE flows, with velocity fields satisfying a one-sided Lipschitz condition. Depending on the time direction, the flows are either compressive or expansive. In the compressive regime, we characterize the stable continuous distributional solutions of both the first and second-order nonconservative transport equations as the unique viscosity solution. Our results in the expansive regime complement the theory of Bouchut, James, and Mancini [23], and we provide a complete theory for both the conservative and nonconservative equations in Lebesgue spaces, as well as proving the existence, uniqueness, and stability of the regular Lagrangian ODE flow. We also provide analogous results in this context for second order equations and SDEs with degenerate noise coefficients that are constant in the spatial variable.

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Section: Summary Of Main Resultsmentioning

confidence: 99%

Interpolation results for pathwise Hamilton-Jacobi equations

Lions¹,

Seeger²,

Souganidis³

2022

Indiana Univ. Math. J.

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“…While there is a substantial literature on N -player games and, more so on mean-field games (MFGs) with singular controls (see e.g. [9,10,11,18,20,22,26]), MFGs with semimartingale strategies have not yet been considered in the literature to the best of our knowledge.…”

Section: Discussionmentioning

confidence: 99%

A Mean-Field Control Problem of Optimal Portfolio Liquidation with Semimartingale Strategies

Fu¹,

Horst²,

Xia³

2022

Preprint

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We consider a mean-field control problem with càdlàg semimartingale strategies arising in portfolio liquidation models with transient market impact and self-exciting order flow. We show that the value function depends on the state process only through its law, and that it is of linear-quadratic form and that its coefficients satisfy a coupled system of non-standard Riccati-type equations. The Riccati equations are obtained heuristically by passing to the continuous-time limit from a sequence of discrete-time models. A sophisticated transformation shows that the system can be brought into standard Riccati form from which we deduce the existence of a global solution. Our analysis shows that the optimal strategy jumps only at the beginning and the end of the trading period.

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“…This LPFP algorithm has already been used for applications to water management and electricity markets in [13] and [3], respectively, but no theoretical convergence results have been provided. In the recent paper [23], the authors study the class of submodular MFGs. In particular, in the case of optimal stopping, they prove the existence of equilibria using Tarski's fixed point theorem and provide an approximation of the minimal equilibria (with respect to a specific order structure) by starting with the minimal measure flow and iterating the minimal best response.…”

Section: Introductionmentioning

confidence: 99%

Linear programming fictitious play algorithm for mean field games with optimal stopping and absorption

Dumitrescu¹,

Leutscher²,

Tankov³

2023

ESAIM: M2AN

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We develop the fictitious play algorithm in the context of the linear programming approach for mean field games of optimal stopping and mean field games with regular control and absorption. This algorithm allows to approximate the mean field game population dynamics without computing the value function by solving linear programming problems associated with the distributions of the players still in the game and their stopping times/controls. We show the convergence of the algorithm using the topology of convergence in measure in the space of subprobability measures, which is needed to deal with the lack of continuity of the flows of measures. Numerical examples are provided to illustrate the convergence of the algorithm.

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A unifying framework for submodular mean field games

Cited by 4 publications

References 43 publications

Interpolation results for pathwise Hamilton-Jacobi equations

Interpolation results for pathwise Hamilton-Jacobi equations

A Mean-Field Control Problem of Optimal Portfolio Liquidation with Semimartingale Strategies

Linear programming fictitious play algorithm for mean field games with optimal stopping and absorption

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