Linear–Quadratic Mean-Field-Type Games: A Direct Method

Duncan, Tyrone E.; Tembiné, Hamidou

doi:10.3390/g9010007

Cited by 51 publications

(46 citation statements)

References 38 publications

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“…The issue here is the fact that the controls of the other players appear in the best response functions of each player through the vectors (Y 1 , Z 1 ), ..., (Y n , Z n ). To solve this fixed point problem, we first rewrite (9) and the backward equations followed by (Y, Z) = ((Y 1 , Z 1 ), ..., (Y n , Z n )) in the following way (note that we omit the time dependence of the coefficients to make the notations less cluttered):…”

Section: Step 5: Search For a Fixed Pointmentioning

confidence: 99%

Linear-Quadratic McKean-Vlasov Stochastic Differential Games

Miller

Pham

2019

Modeling, Stochastic Control, Optimization, and Applications

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We consider a multi-player stochastic differential game with linear McKean-Vlasov dynamics and quadratic cost functional depending on the variance and mean of the state and control actions of the players in open-loop form. Finite and infinite horizon problems with possibly some random coefficients as well as common noise are addressed. We propose a simple direct approach based on weak martingale optimality principle together with a fixed point argument in the space of controls for solving this game problem. The Nash equilibria are characterized in terms of systems of Riccati ordinary differential equations and linear mean-field backward stochastic differential equations: existence and uniqueness conditions are provided for such systems. Finally, we illustrate our results on a toy example.MSC Classification: 49N10, 49L20, 91A13. the cost functional depend upon the law of the stochastic process. This corresponds to a Paretooptimum where a social planner/influencer decides of the strategies for each individual. The theory of McKV control problems, also called mean-field type control, has generated recent advances in the literature, either by the maximum principle [5], or the dynamic programming approach [14], see also the recent books [3] and [6], and the references therein, and linear quadratic (LQ) models provide an important class of solvable applications studied in many papers, see, e.g., [15], [11], [10], [2].In this paper, we consider multi-player stochastic differential games for McKean-Vlasov dynamics. This corresponds and is motivated by the competitive interaction of multi-population with a large number of indistinguishable agents. In this context, we are then looking for a Nash equilibrium among the multi-class of populations. Such problem, sometimes refereed to as mean-field-type game, allows to incorporate competition and heterogeneity in the population, and is a natural extension of McKean-Vlasov (or mean-field-type) control by including multiple decision makers. It finds natural applications in engineering, power systems, social sciences and cybersecurity, and has attracted recent attention in the literature, see, e.g., [1], [7], [8], [4]. We focus more specifically on the case of linear McKean-Vlasov dynamics and quadratic cost functional for each player (social planner). Linear Quadratic McKean-Vlasov stochastic differential game has been studied in [9] for a onedimensional state process, and by restricting to closed-loop control. Here, we consider both finite and infinite horizon problems in a multi-dimensional framework, with random coefficients for the affine terms of the McKean-Vlasov dynamics and random coefficients for the linear terms of the cost functional. Moreover, controls of each player are in open-loop form. Our main contribution is to provide a simple and direct approach based on weak martingale optimality principle developed in [2] for McKean-Vlasov control problem, and that we extend to the stochastic differential game, together with a fixed point argument in the space of open-loop c...

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Section: Step 5: Search For a Fixed Pointmentioning

confidence: 99%

Linear-Quadratic McKean-Vlasov Stochastic Differential Games

Miller

Pham

2019

Modeling, Stochastic Control, Optimization, and Applications

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“…Mean-field-type control and global optimization can be found in [134,36,133,172,173,175]. The models (1) and (3) are easily adapted to bargaining solution, cooperative and coalitional MFTGs and can be found in [135,176,177]. Psychological MFTG was recently introduced in [111,178] where spitefulness, altruism, selfishness, reciprocity of the agents are examined by means empathy, other-regarding behavior and psychological factors.…”

Section: A Basic Dynamic Mftg: Finite Regimementioning

confidence: 99%

Mean-Field-Type Games in Engineering

Djehiche

Tcheukam

Tembiné

2017

AIMS Electronics and Electrical Engineering

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“…Games with non-linearly distribution-dependent quantity-of-interest [1,2,3] are very attractive in terms of applications because the non-linear dependence of the payoff functions in terms of state distribution allow us to capture risk measures which are functionals of variance, inverse quantiles, and or higher moments. During the past, a significant amount of research on mean-fieldtype games has been performed [4,5,6,8,9,10]. In the time-dependent case, the analysis of mean-field-type games has several challenges.…”

Section: Introductionmentioning

confidence: 99%

“…In the time-dependent case, the analysis of mean-field-type games has several challenges. Previous works have devoted tremendous effort in terms of partial integro-differential system of linear-quadratic mean-field-type games (LQ-MFTG) [6], (ii) linear-exponentiated quadratic mean-field-type games (LEQ-MFTG) [7] , (ii) adversarial linear-quadratic mean-field-type games (minmax LQ, minmax LEQ-MFTG) [6]. In LQ-MFTG the base state dynamics has two components: drift and noise.…”

Section: Introductionmentioning

confidence: 99%

Semiexplicit Solutions to Some Nonlinear Nonquadratic Mean-Field-Type Games: A Direct Method

Barreiro-Gómez

Duncan

Pasik-Duncan

et al. 2020

IEEE Trans. Automat. Contr.

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This article examines mean-field-type game problems by means of a direct method. We provide various solvable examples beyond the classical linear-quadratic game problems. These include quadratic-quadratic games and games with power, logarithmic, sine square, hyperbolic sine square payoffs. Non-linear state dynamics such as log-state, controldependent regime switching, quadratic state, cotangent state and hyperbolic cotangent state are considered. We identify equilibrium strategies and equilibrium payoffs in state-and-conditional mean-field type feedback form. It is shown that a simple direct method can be used to solve broader classes of non-quadratic mean-field-type games under jumpdiffusion-regime switching Gauss-Volterra processes which include fractional Brownian motions and multi-fractional Brownian motions. We

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Linear–Quadratic Mean-Field-Type Games: A Direct Method

Cited by 51 publications

References 38 publications

Linear-Quadratic McKean-Vlasov Stochastic Differential Games

Linear-Quadratic McKean-Vlasov Stochastic Differential Games

Mean-Field-Type Games in Engineering

Semiexplicit Solutions to Some Nonlinear Nonquadratic Mean-Field-Type Games: A Direct Method

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