Extended Deterministic Mean-Field Games

Gomes, Diogo A.; Voskanyan, Vardan

doi:10.1137/130944503

Cited by 70 publications

(48 citation statements)

References 18 publications

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“…This regularity is related with the time regularity of the optimal controls. In the first order case, it is not an issue because the optimal trajectories satisfy a Pontryagin maximum principle and thus are (at least) uniformly C 1 (see [Gomes et al, 2014, Gomes andVoskanyan, 2016]). In the second order setting, the Pontryagin maximum principle is not so simple to manipulate and a similar regularity for the controls would be much more heavy to express.…”

Section: The Hamiltonianmentioning

confidence: 99%

“…Similar-more general-MFG systems were introduced in [Gomes et al, 2014] under the terminology of extended MFG models. In [Gomes et al, 2014, Gomes andVoskanyan, 2016], existence of solutions is proved for deterministic MFG under suitable structure assumptions. A chapter in the monograph [Carmona and Delarue, 2017] is also devoted to this class of MFG, with a probabilistic point of view.…”

Section: Introductionmentioning

confidence: 99%

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Mean field game of controls and an application to trade crowding

2017

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In this paper we formulate the now classical problem of optimal liquidation (or optimal trading) inside a Mean Field Game (MFG). This is a noticeable change since usually mathematical frameworks focus on one large trader facing a "background noise" (or "mean field"). In standard frameworks, the interactions between the large trader and the price are a temporary and a permanent market impact terms, the latter influencing the public price.In this paper the trader faces the uncertainty of fair price changes too but not only. He also has to deal with price changes generated by other similar market participants, impacting the prices permanently too, and acting strategically.Our MFG formulation of this problem belongs to the class of "extended MFG", we hence provide generic results to address these "MFG of controls", before solving the one generated by the cost function of optimal trading. We provide a closed form formula of its solution, and address the case of "heterogenous preferences" (when each participant has a different risk aversion). Last but not least we give conditions under which participants do not need to instantaneously know the state of the whole system, but can "learn" it day after day, observing others' behaviors.

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Section: The Hamiltonianmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Mean field game of controls and an application to trade crowding

2017

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“…is always non-positive, we conclude that ∂ b2 (b 1 , b 2 ) must be negative whenever 0 20 We know that…”

Section: Numerical Resultsmentioning

confidence: 69%

High-Roller Impact: A Large Generalized Game Model of Parimutuel Wagering

Bayraktar

Munk

2016

SSRN Journal

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Abstract. How do large-scale participants in parimutuel wagering events affect the house and ordinary bettors? A standard narrative suggests that they may temporarily benefit the former at the expense of the latter. To approach this problem, we begin by developing a model based on the theory of large generalized games. Constrained only by their budgets, a continuum of diffuse (ordinary) players and a single atomic (large-scale) player simultaneously wager to maximize their expected profits according to their individual beliefs. Our main theoretical result gives necessary and sufficient conditions for the existence and uniqueness of a pure-strategy Nash equilibrium. Using this framework, we analyze our question in concrete scenarios. First, we study a situation in which both predicted effects are observed. Neither is always observed in our remaining examples, suggesting the need for a more nuanced view of large-scale participants.

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“…There is also a model for price impact in the book of Carmona and Delarue which we take as one of our example problems in Section 4.4.2. Mean field games where players interact through the law of their controls is sometimes referred to as extended mean field games [20], see also Chapter 4 in [8]. We are interested in testing our numerical methods on a certain class of mean field games of control: those in which the interaction is through the marginal distributions, L(X t ) and L(α t ).…”

Section: Mean Field Games Of Controlmentioning

confidence: 99%

Cemracs 2017: numerical probabilistic approach to MFG

Angiuli

Graves

et al. 2019

ESAIM: ProcS

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This project investigates numerical methods for solving fully coupled forward-backward stochastic differential equations (FBSDEs) of McKean-Vlasov type. Having numerical solvers for such mean field FBSDEs is of interest because of the potential application of these equations to optimization problems over a large population, say for instance mean field games (MFG) and optimal mean field control problems. Theory for this kind of problems has met with great success since the early works on mean field games by Lasry and Lions, see [27], and by Huang, Caines, and Malhamé, see [24]. Generally speaking, the purpose is to understand the continuum limit of optimizers or of equilibria (say in Nash sense) as the number of underlying players tends to infinity. When approached from the probabilistic viewpoint, solutions to these control problems (or games) can be described by coupled mean field FBSDEs, meaning that the coefficients depend upon the own marginal laws of the solution. In this note, we detail two methods for solving such FBSDEs which we implement and apply to five benchmark problems. The first method uses a tree structure to represent the pathwise laws of the solution, whereas the second method uses a grid discretization to represent the time marginal laws of the solutions. Both are based on a Picard scheme; importantly, we combine each of them with a generic continuation method that permits to extend the time horizon (or equivalently the coupling strength between the two equations) for which the Picard iteration converges.

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Extended Deterministic Mean-Field Games

Cited by 70 publications

References 18 publications

Mean field game of controls and an application to trade crowding

Mean field game of controls and an application to trade crowding

High-Roller Impact: A Large Generalized Game Model of Parimutuel Wagering

Cemracs 2017: numerical probabilistic approach to MFG

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