Linear Quadratic Mean Field Games: Decentralized O(1/N)-Nash Equilibria

Huang, Minyi; Yang, Xuwei

doi:10.1007/s11424-021-1266-y

Cited by 8 publications

(6 citation statements)

References 36 publications

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“…It should be noted that most previous studies (see, e.g., earlier studies [10,[21][22][23][24] and references therein) are…”

Section: Det [mentioning

confidence: 99%

“…It should be noted that most previous studies (see, e.g., earlier studies [10, 21–24] and references therein) are

O\left(\sqrt{\varepsilon}\right)&#x0003D;O\left(1/\sqrt{N}\right)&#x0003D;O\left(\sqrt{\varepsilon}\right)

, not

O\left(\varepsilon \right)&#x0003D;O\left(1/N\right)&#x0003D;O\left(\varepsilon \right)

. While these proof methods are limited to showing

O\left(\sqrt{\varepsilon}\right)&#x0003D;O\left(1/\sqrt{N}\right)

, it is worth pointing out that this paper shows a stricter evaluation using the weakly couple system theory.…”

Section: Decentralized Strategy Setmentioning

confidence: 99%

“…Finally, compared with previous studies [10,[21][22][23][24], the real cost-functional degradation has never been numerically evaluated in Table 3, where 𝜙 ∶= ||P ic − X ic ||∕𝜀 is challenging to compute under a sufficiently large N. Therefore, the decomposition calculation is investigated. The following matrix can be defined for the same reason as (36).…”

Section: Numerical Examplementioning

confidence: 99%

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Sign‐indefinite static output feedback Nash strategy for mean‐field stochastic systems

Mukaidani

Sagara

2023

Asian Journal of Control

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We study static output feedback (SOF) Nash strategies for mean‐field stochastic systems. The study assumes a linear‐quadratic form of the cost functional, but the sign of the weighting matrix is indefinite. Moreover, the worst‐case disturbances are assumed to be of unconventional SOF form leading to a saddle‐point equilibrium. First, after defining the problem for the single‐player case, the necessary conditions for the existence of saddle‐point equilibrium are established. Then, we consider Nash games for mean‐field stochastic systems. Note that the cost functional associated with the mean‐field term is assumed to be of SOF form. We establish the necessary conditions for the Nash equilibrium using the cross‐coupled stochastic algebraic matrix equations (CCSAME). A new decomposition algorithm is proposed to avoid the challenges of solving high‐dimensional CCSAMEs. We also apply weakly coupled systems theory to propose a decentralized approximated reduced‐order Nash strategy. The degree of cost degradation of the decentralized strategy compared with the centralized strategy is derived more accurately than existing results. Finally, we illustrate the usefulness and effectiveness of the proposed strategy set through numerical examples.

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“…It should be noted that most previous studies (see, e.g., earlier studies [10,[21][22][23][24] and references therein) are…”

Section: Det [mentioning

confidence: 99%

“…It should be noted that most previous studies (see, e.g., earlier studies [10, 21–24] and references therein) are

O\left(\sqrt{\varepsilon}\right)&#x0003D;O\left(1/\sqrt{N}\right)&#x0003D;O\left(\sqrt{\varepsilon}\right)

, not

O\left(\varepsilon \right)&#x0003D;O\left(1/N\right)&#x0003D;O\left(\varepsilon \right)

. While these proof methods are limited to showing

O\left(\sqrt{\varepsilon}\right)&#x0003D;O\left(1/\sqrt{N}\right)

, it is worth pointing out that this paper shows a stricter evaluation using the weakly couple system theory.…”

Section: Decentralized Strategy Setmentioning

confidence: 99%

Section: Numerical Examplementioning

confidence: 99%

See 1 more Smart Citation

Sign‐indefinite static output feedback Nash strategy for mean‐field stochastic systems

Mukaidani

Sagara

2023

Asian Journal of Control

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“…The agents in the social optimization problem are cooperative with a common objective. A different solution notion is to solve a mean field game where each agent optimizes for its own cost J i ; this has been developed in a companion paper [29]. This subsection compares the two solutions by demonstrating the efficiency gain of social optimization with respect to the mean field game.…”

Section: Decentralized Controlmentioning

confidence: 99%

Linear quadratic mean field social optimization: Asymptotic solvability

Huang

Yang

2019

2019 IEEE 58th Conference on Decision and Control (CDC)

Self Cite

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This paper considers a linear-quadratic (LQ) mean field control problem involving a major player and a large number of minor players, where the dynamics and costs depend on random parameters. The objective is to optimize a social cost as a weighted sum of the individual costs under decentralized information. We apply the person-by-person optimality principle in team decision theory to the finite population model to construct two limiting variational problems whose solutions, subject to the requirement of consistent mean field approximations, yield a system of forwardbackward stochastic differential equations (FBSDEs). We show the existence and uniqueness of a solution to the FBSDEs and obtain decentralized strategies nearly achieving social optimality in the original large but finite population model. ).

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“…This is not surprising because, in terms of mathematics, the theory involves a variety of fields: probabilistic analysis, calculus of variations, the analysis of partial differential equations, and so on. Especially, the linear quadratic (LQ) MFG theory has been treated by many researchers with fruitful results; Huang et al [1] developed a state aggregation technique to obtain decentralized control laws for a large but finite population with the infinite horizon discounted costs; Bardi [23] provided a quadratic Gaussian solution to a system of HJB and FPK equations in the case of one‐dimensional state space; Moon and Başar [24] investigated the LQ risk‐sensitive MFG and obtained

\epsilon

‐Nash equilibria; Huang and Zhou [25] analyzed the connection and difference of the direct approach and the fixed point approach; Huang and Yang [26] studied an asymptotic solvability problem for LQ‐MFGs and established a decentralized

O\left(1/N\right)

‐Nash strategy.…”

Section: Introductionmentioning

confidence: 99%

Decentralized ϵ‐Nash strategy for linear quadratic mean field games using a successive approximation approach

Shen

2023

Asian Journal of Control

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This paper presents a successive approximation method for decentralized strategy design in the large‐scale linear quadratic (LQ) Gaussian game. The strategy consists of transforming the original mean field game (MFG) problem into solving decoupled ordinary differential equations (ODEs) whose numerical solutions are obtained by a new two‐loop iteration algorithm. It should be noted that we employ the augmented model technique and the LQ framework to derive these low‐dimensional solvable ODEs, which is the cornerstone of constructing the decentralized ‐Nash strategy. In addition, the quadratic ODEs contained therein are approximately solved for by a sequence of iterative linear ordinary differential equations (LODEs) with guaranteed convergence. A numerical example is given to show the effectiveness of the proposed algorithm.

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Linear Quadratic Mean Field Games: Decentralized O(1/N)-Nash Equilibria

Cited by 8 publications

References 36 publications

Sign‐indefinite static output feedback Nash strategy for mean‐field stochastic systems

Sign‐indefinite static output feedback Nash strategy for mean‐field stochastic systems

Linear quadratic mean field social optimization: Asymptotic solvability

Decentralized ϵ‐Nash strategy for linear quadratic mean field games using a successive approximation approach

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