Linear quadratic social optima and mean field games

Abstract

The thesis explores linear quadratic (LQ) mean field (MF) large population (LP) systems. Three topics are considered: 1. The MF social optima (which is also called MF team (MFT)) problem for a LP system. 2. The MFT problem for a major-minor LP system. 3. The relation among the MF type control (MFC) problem, the MF game (MFG) problem and the MFT problem. In the first topic, the MF approximation method is applied to the social optimal problem. In this problem, the agents' states and strategies access the diffusion terms. In addition, the control weight for the cost functional might be indefinite. Firstly, we consider the convexity of the social cost functional. We derive some low-dimensional criteria to determine this convexity via algebra analysis. Secondly, under the person-by-person optimality principle, we apply some stochastic variational techniques and MF approximation to obtain the decentralized auxiliary control. Thirdly, to resolve the solvability of the consistency condition, which is represented as a MF forward-backward stochastic differential equations (MF-FBSDEs) system, we apply the decentralizing method to convert it to a general FBSDEs system. Furthermore, we apply the decoupling method and obtain a Riccati equation. Lastly, because the agent states access the diffusion term, we should consider the convergence of the average of a series of weakly coupled BSDEs. Through the decoupling method, we obtain two Lyapunov equations, and their uniform boundness ensure the convergence. In the second topic, the social optima of the major-minor LP system are considered. In our model, a considerable number of minor agents are cooperative to minimize the social cost as the sum of individual costs, while the major agent and minor agents competitively aim for Nash equilibrium. Moreover, as in topic one, the agents' states and strategies access the diffusion terms, and this brings essential difficulty to the proof of asymptotic optimality. In our research, we firstly study the decentralized control of the major agent. By freezing the minor state-average, we obtain the auxiliary control problem for the major agent. Furthermore, under the person-by-person optimality principle and applying some MF approximation, we obtain the auxiliary control problems for the minor agents. The consistency condition is also a MF-FBSDEs system. The well-posedness of the consistency condition system is obtained by the discounting method. The related asymptotic optimality is also verified. In the third topic, we study the relation among the MFC, MFG and MFT problems. Notably, the individual admissible controls are constrained in a linear subset. By introducing a new type of Riccati equation, we obtain a uniform convexity condition which is weaker than the "standard condition" widely used in previous literature. Also, using this new type of Riccati equation, we obtain the constrained feedback form optimal control and MF strategies for MFC, MFG and MFT problems respectively. Moreover, through analysing the corresponding Hamiltonian systems of these three problems, it can be concluded that under some mild conditions, the MF strategies of the MFG and MFT problems are equivalent to the optimal control of the MFC problem. Lastly, we also find that the MFT strategy obtained via the direct approaching method is identical to that obtained by the fixed-point approaching method.