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Mean field game theory has been developed largely following two routes. One of them, called the direct approach, starts by solving a large-scale game and next derives a set of limiting equations as the population size tends to infinity. The second route is to apply mean field approximations and formalize a fixed point problem by analyzing the best response of a representative player. This paper addresses the connection and difference of the two approaches in a linear quadratic (LQ) setting. We first introduce an asymptotic solvability notion for the direct approach, which means for all sufficiently large population sizes, the corresponding game has a set of feedback Nash strategies in addition to a mild regularity requirement. We provide a necessary and sufficient condition for asymptotic solvability and show that in this case the solution converges to a mean field limit. This is accomplished by developing a re-scaling method to derive a low-dimensional ordinary differential equation (ODE) system, where a non-symmetric Riccati ODE has a central role. We next compare with the fixed point approach which determines a two-point boundary value (TPBV) problem, and show that asymptotic solvability implies feasibility of the fixed point approach, but the converse is not true. We further address non-uniqueness in the fixed point approach and examine the long time behavior of the non-symmetric Riccati ODE in the asymptotic solvability problem. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to show how insight obtained from a mean-field model can be used to create an architecture for approximate dynamic programming (ADP) for a certain class of games comprising of a large number of agents. The general technique is illustrated with the aid of a mean-field oscillator game model introduced in our prior work. The states of the model are interpreted as the phase angles for a collection of nonhomogeneous oscillators, and in this way the model may be regarded as an extension of the classical coupled oscillator model of Kuramoto. The paper introduces ADP techniques for design and adaptation (learning) of approximately optimal control laws for this model. For this purpose, a parameterization is proposed, based on an analysis of the mean-field PDE model for the game. In an offline setting, a Galerkin procedure is introduced to choose the optimal parameters while in an online setting, a steepest descent algorithm is proposed. The paper provides a detailed analysis of the optimal parameter values as well as the Bellman error with both the Galerkin approximation and the online algorithm. Finally, a phase transition result is described for the large population limit when each oscillator uses the approximately optimal control law. A critical value of the control penalty parameter is identified: above this value, the oscillators are incoherent; and below this value (when control is sufficiently cheap) the oscillators synchronize. These conclusions are illustrated with results from numerical experiments. [ABSTRACT FROM AUTHOR]

This paper considers decentralized control and optimization methodologies for large populations of systems, consisting of several agents with different individual behaviors, constraints and interests, and influenced by the aggregate behavior of the overall population. For such large-scale systems, the theory of aggregative and mean field games has been established and successfully applied in various scientific disciplines. While the existing literature addresses the case of unconstrained agents, we formulate deterministic mean field control problems in the presence of heterogeneous convex constraints for the individual agents, for instance arising from agents with linear dynamics subject to convex state and control constraints. We propose several model-free feedback iterations to compute in a decentralized fashion a mean field Nash equilibrium in the limit of infinite population size. We apply our methods to the constrained linear quadratic deterministic mean field control problem. [ABSTRACT FROM PUBLISHER]

In this paper, we study a class of risk-sensitive mean-field stochastic differential games. We show that under appropriate regularity conditions, the mean-field value of the stochastic differential game with exponentiated integral cost functional coincides with the value function satisfying a Hamilton –Jacobi– Bellman (HJB) equation with an additional quadratic term. We provide an explicit solution of the mean-field best response when the instantaneous cost functions are log-quadratic and the state dynamics are affine in the control. An equivalent mean-field risk-neutral problem is formulated and the corresponding mean-field equilibria are characterized in terms of backward-forward macroscopic McKean–Vlasov equations, Fokker–Planck–Kolmogorov equations, and HJB equations. We provide numerical examples on the mean field behavior to illustrate both linear and McKean–Vlasov dynamics. [ABSTRACT FROM PUBLISHER]

The purpose of this paper is to synthesize initial mean consensus behavior of a set of agents from the fundamental optimization principles of i) stochastic dynamic games, and ii) optimal control. In the stochastic dynamic game model each agent seeks to minimize its individual quadratic discounted cost function involving the mean of the states of all other agents. In this formulation we derive the limiting infinite population mean field equation system and explicitly compute its unique solution. The resulting mean field (MF) control strategies drive each agent to track the overall population's initial state distribution mean, and by applying these decentralized strategies, any finite population system reaches mean consensus asymptotically as time goes to infinity. Furthermore, these control laws possess an \varepsilonN-Nash equilibrium property where \varepsilonN goes to zero as the population size N goes to infinity. Finally, the analysis is extended to the case of random mean field couplings. In the social cooperative formulation the basic objective is to minimize a social cost as the sum of the individual cost functions containing mean field coupling. In this formulation we show that for any individual agent the decentralized mean field social (MF Social) control strategy is the same as the mean field Nash (MF Nash) equilibrium strategy. Hence $MF-Nash\ Controls\ UNash^N=\rm MF-Social\ Controls\ USoc^N.$On the other hand, the solution to the centralized LQR optimal control formulation yields the Standard Consensus (SC) algorithm whenever the graph representing the corresponding topology of the network is Completely Connected (CC). Hence $Cent.\ LQR\ Controls\ UCent^N=\rm SC-CC\ Controls\ USC^N.$Moreover, a system with centralized control laws reaches consensus on the initial state distribution mean as time and population size N go to infinity. Hence, asymptotically in time $\eqalignno&MF-Nash\ Controls\ UNash^N\!=\!\,\rm MF-Social\ Controls\ USoc^N\cr&\!=\!\,\rm Cent.\ \rm LQR\ Controls\ UCent^\infty \!=\! \,\rm SC-CC\ Controls\ USC^\infty.$Finally, the analysis is extended to the long time average (LTA) cost functions case. [ABSTRACT FROM AUTHOR]