We consider linear quadratic Gaussian (LQG) games in large population systems where the agents evolve according to nonuniform dynamics and are coupled via their individual costs. A state aggregation technique is developed to obtain a set of decentralized control laws for the individuals which possesses an -Nash equilibrium property. A stability property of the mass behavior is established, and the effect of inaccurate population statistics on an isolated agent is also analyzed by variational techniques.

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Abstract-We study a class of linear-quadratic-Gaussian (LQG) control problems with decision makers, where the basic objective is to minimize a social cost as the sum of individual costs containing mean field coupling. The exact socially optimal solution (determining a particular Pareto optimum) requires centralized information for each agent and has high implementational complexity. As an alternative we subsequently exploit a mean field structure in the centralized optimal control problem to develop decentralized cooperative optimization so that each agent only uses its own state and a function which may be computed offline; the resulting set of strategies asymptotically achieves the social optimum as . A key feature in this scheme is to let each agent optimize a new cost as the sum of its own cost and another component capturing its social impact on all other agents. We also discuss the relationship between the decentralized cooperative solution and the so-called Nash Certainty Equivalence based solution presented in previous work on mean field LQG games.

We consider stochastic dynamic games in large population conditions where multiclass agents are weakly coupled via their individual dynamics and costs. We approach this large population game problem by the so-called Nash Certainty Equivalence (NCE) Principle which leads to a decentralized control synthesis. The McKean-Vlasov NCE method presented in this paper has a close connection with the statistical physics of large particle systems: both identify a consistency relationship between the individual agent (or particle) at the microscopic level and the mass of individuals (or particles) at the macroscopic level. The overall game is decomposed into (i) an optimal control problem whose Hamilton-Jacobi-Bellman (HJB) equation determines the optimal control for each individual and which involves a measure corresponding to the mass effect, and (ii) a family of McKean-Vlasov (M-V) equations which also depend upon this measure. We designate the NCE Principle as the property that the resulting scheme is consistent (or soluble), i.e. the prescribed control laws produce sample paths which produce the mass effect measure. By construction, the overall closed-loop behaviour is such that each agent's behaviour is optimal with respect to all other agents in the game theoretic Nash sense.

We study large population stochastic dynamic games where the so-called Nash certainty equivalence based control laws are implemented by the individual players. We first show a martingale property for the limiting control problem of a single agent and then perform averaging across the population; this procedure leads to a constant value for the martingale which shows an invariance property of the population behavior induced by the Nash strategies.

We consider dynamic games in large population conditions where the agents evolve according t o non-uniform dynamics and are weakly coupled via their dynamics and the individual costs. A state aggregation technique is developed to obtain a set of decentralized control laws for the individuals which possesses an E-Nash equilibrium property. An attraction property of the mass behaviour is established. The methodology and the results contained in this paper reveal novel behavioural properties of the relationship of any given individual with respect to the mass of individuals in large-scale noncooperative systems of weakly coupled agents.

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Abstract-We study large population stochastic dynamic games where each agent assigns individually determined coupling strengths (with possible spatial interpretation) to the states of other agents in its performance function. The mean field methodology [14] yields a set of decentralized controls which generates an -Nash equilibrium for the population of size . A key feature of the mean field approximation (here with localized interactions) is that the resulting th individual agent's control law depends on that agent's state and the precomputable weighted average trajectory of the collection of all agents each applying a decentralized control law.