Mean-field linear-quadratic stochastic differential games

Sun, Jingrui; Wang, Hanxiao

doi:10.1016/j.jde.2021.06.004

Cited by 17 publications

(18 citation statements)

References 32 publications

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“…where Φ i (•) is determined by the following Riccati-type ordinary differential equation (ODE, for short): When the weighting matrices G 3 = 0 and g = 0, then Φ i ≡ 0 for i = 3, 4, 5, 7, and the Riccati equation (4.4) can be simplified. The following result shows that the LQ problem for FBSDEs is closely related to the so-called mean-field LQ optimal control problems (see [76,78,60], for example). Let A, B, C, D, M, N ≡ 0, G 3 = 0, g = 0 and H = I n .…”

Section: Linear-quadratic Problemsmentioning

confidence: 96%

Optimal Controls for Forward-Backward Stochastic Differential Equations: Time-Inconsistency and Time-Consistent Solutions

Wang¹,

Yong²,

Zhou³

2022

Preprint

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This paper is concerned with an optimal control problem for a forward-backward stochastic differential equation (FBSDE, for short) with a recursive cost functional determined by a backward stochastic Volterra integral equation (BSVIE, for short). It is found that such an optimal control problem is timeinconsistent in general, even if the cost functional is reduced to a classical Bolza type one as in Peng [50], Lim-Zhou [41], and Yong [74]. Therefore, instead of finding a global optimal control (which is time-inconsistent), we will look for a time-consistent and locally optimal equilibrium strategy, which can be constructed via the solution of an associated equilibrium Hamilton-Jacobi-Bellman (HJB, for short) equation. A verification theorem for the local optimality of the equilibrium strategy is proved by means of the generalized Feynman-Kac formula for BSVIEs and some stability estimates of the representation for parabolic partial differential equations (PDEs, for short). Under certain conditions, it is proved that the equilibrium HJB equation, which is a nonlocal PDE, admits a unique classical solution. As special cases and applications, the linear-quadratic problems, a mean-variance model, a social planner problem with heterogeneous Epstein-Zin utilities, and a Stackelberg game are briefly investigated. It turns out that our framework can cover not only the optimal control problems for FBSDEs studied in [50,41,74], and so on, but also the problems of the general discounting and some nonlinear appearance of conditional expectations for the terminal state, studied in Yong [75,77] and .

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Section: Linear-quadratic Problemsmentioning

confidence: 96%

Optimal Controls for Forward-Backward Stochastic Differential Equations: Time-Inconsistency and Time-Consistent Solutions

Wang¹,

Yong²,

Zhou³

2022

Preprint

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“…Last, it is worth pointing out that other possible forms of zero-sum DGs exist in the literature, such as the case with continuous and/or impulse controls [217], mean-field DGs [218], [219], risk-sensitive zero-sum DGs [204] and so forth.…”

Section: Zero-sum Differential Gamesmentioning

confidence: 99%

A Survey of Decision Making in Adversarial Games

Li¹,

Mao²,

Ye³

et al. 2022

Preprint

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Game theory has by now found numerous applications in various fields, including economics, industry, jurisprudence, and artificial intelligence, where each player only cares about its own interest in a noncooperative or cooperative manner, but without obvious malice to other players. However, in many practical applications, such as poker, chess, evader pursuing, drug interdiction, coast guard, cyber-security, and national defense, players often have apparently adversarial stances, that is, selfish actions of each player inevitably or intentionally inflict loss or wreak havoc on other players. Along this line, this paper provides a systematic survey on three main game models widely employed in adversarial games, i.e., zero-sum normal-form and extensive-form games, Stackelberg (security) games, zero-sum differential games, from an array of perspectives, including basic knowledge of game models, (approximate) equilibrium concepts, problem classifications, research frontiers, (approximate) optimal strategy seeking techniques, prevailing algorithms, and practical applications. Finally, promising future research directions are also discussed for relevant adversarial games.

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“…A particularly attractive subclass of mean-field/McKean-Vlasov type differential games is stochastic linear quadratic (LQ) mean field games. This is due to its tractable analytical structure [1,3,9,11,16,18,23,25]. In this paper we will focus particularly on LQ zero-sum differential games driven by SDEs with McKean-Vlasov type.…”

Section: Introductionmentioning

confidence: 99%

“…Knowing that these sign conditions play a key role in the solution process, this represents in our opinion a strong limitation for such results. Very recently, [23] addressed such a challenging problem under the general framework of uniform convexity/concavity. The solvability conditions proposed by [23] take explicitly into account the sign constraints on the quadratic terms of the Riccati equations.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

An Addendum to the Problem of Zero-Sum LQ Stochastic Mean-Field Dynamic Games\\ (Extended version)

Aberkane¹,

Drăgan²

2023

Preprint

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In this paper, we first address a linear quadratic mean-field game problem with a leader-follower structure. By adopting a Riccati-type approach, we show how one can obtain a state-feedback representation of the pairs of strategies which achieve an open-loop Stackelberg equilibrium in terms of the global solutions of a system of coupled matrix differential Riccati-type equations. In the second part of this paper, we obtain necessary and sufficient conditions for the solvability of the involved coupled generalized Riccati equations.

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Mean-field linear-quadratic stochastic differential games

Cited by 17 publications

References 32 publications

Optimal Controls for Forward-Backward Stochastic Differential Equations: Time-Inconsistency and Time-Consistent Solutions

Optimal Controls for Forward-Backward Stochastic Differential Equations: Time-Inconsistency and Time-Consistent Solutions

A Survey of Decision Making in Adversarial Games

An Addendum to the Problem of Zero-Sum LQ Stochastic Mean-Field Dynamic Games\\ (Extended version)

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