Mean field LQG games with mass behavior responsive to a major player

Nguyen, Son Luu; Huang, Minyi

doi:10.1109/cdc.2012.6425984

Cited by 22 publications

(21 citation statements)

References 28 publications

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“…The scheme proposed in this paper differs from the one proposed in [24,25] as the control problem faced by the major player is here of the conditional McKean-Vlasov type, and the measure flow is endogenous to the controller. This makes the limiting problem a bona fide two-player game instead of a succession of two consecutive standard optimal control problems.…”

Section: A Concrete Examplementioning

confidence: 87%

“…However, the state of the major player does not enter the dynamics of the states of the minor players in [23]. The general finite-horizon case is solved in [24] by the use of the so-called nonanticipative variational calculus. It is important to point out that the notion of Nash equilibrium used in [24] corresponds to the Markovian feedback Nash equilibrium' while here, we work with open-loop Nash equilibriums.…”

Section: Mfg With Major-minor Agents: the Lqg Casementioning

confidence: 99%

“…The general finite-horizon case is solved in [24] by the use of the so-called nonanticipative variational calculus. It is important to point out that the notion of Nash equilibrium used in [24] corresponds to the Markovian feedback Nash equilibrium' while here, we work with open-loop Nash equilibriums. In what follows, we carry out the general systematic scheme introduced in the previous discussions and derive approximate Nash equilibria for the LQG major-minor mean field games.…”

Section: Mfg With Major-minor Agents: the Lqg Casementioning

confidence: 99%

“…In section 4, we prove that the solution of the limiting problem can actually be used to build approximate Nash equilibria for the finite-player games, justifying our scheme. In section 5, we apply the scheme to the case of Linear Quadratic Gaussian (LQG for whort) models, and find explicit approximate Nash equilibria for the finite-player games, and in section 6 a concrete example is given to show that the current scheme leads to different results from the scheme proposed in [24] and [25]. In the independent section 7, we prove a conditional version of propagation of chaos which plays a pivotal role in the construction of approximate Nash equilibria in section 4.…”

Section: Introductionmentioning

confidence: 99%

“…To be more specific, in [23,25], the major player influences the minor players solely via their cost functionals. [24] proposes a new scheme to solve the general case for linear-quadratic-Guassian (LQG for short) games in which the major player's state enters the dynamics of the minor players. The limiting control problem for the major player is solved by what the authors call "anticipative variational calculation".…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

A probabilistic approach to mean field games with major and minor players

Carmona¹,

Zhu²

2016

Ann. Appl. Probab.

108

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Abstract. We propose a new approach to mean field games with major and minor players. Our formulation involves a two player game where the optimization of the representative minor player is standard while the major player faces an optimization over conditional McKean-Vlasov stochastic differential equations. The definition of this limiting game is justified by proving that its solution provides approximate Nash equilibriums for large finite player games. This proof depends upon the generalization of standard results on the propagation of chaos to conditional dynamics. Because it is on independent interest, we prove this generalization in full detail. Using a conditional form of the Pontryagin stochastic maximum principle (proven in the appendix), we reduce the solution of the mean field game to a forward-backward system of stochastic differential equations of the conditional McKean-Vlasov type, which we solve in the Linear Quadratic setting. We use this class of models to show that Nash equilibriums in our formulation can be different from those of the formulations contemplated so far in the literature.

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Section: A Concrete Examplementioning

confidence: 87%