Mean field games with imperfect information

In this thesis, three topics in Mean Field Games in the absence of complete information have been studied. The first part of the thesis focus on Mean Field Stackelberg Games between a large group of followers and a leader, in such a way that each follower is subject to a delay effect inherited from the leader. The case with delays being identical among the followers in the population is first considered. Under mild assumptions of regular enough coefficients, the whole Stackelberg game problem is solved via stochastic maximum principle. The solution could be represented by a system of six coupled forward backward stochastic differential equations. A comprehensive study on the particular Linear Quadratic case has been provided. By considering the corresponding linear functional, the time-independent sufficient condition which warrants the unique existence of the solution of the whole Stackelberg game is obtained. Several numerical examples are also demonstrated. The second work studies another class of Stackelberg games, under a Linear Quadratic setting, in the presence with an additional leader. Given the trajectories of the mean field term and two leaders, the follower's optimal control problem is first solved. Depending on whether or not the leaders cooperate, the solutions of the respective Pareto and Nash games between the leaders are obtained, which can be represented by systems of forward backward stochastic functional differential equations. To numerically implement the obtained results, explicit expression of solutions of the whole problem: Mean Field Game among the followers and Nash (and Pareto) Game between the leaders, are provided. Finally, several examples are given to study the impact of different games on the cost functionals of the followers. An interesting example shows that the population are worse off as the leaders cooperate. The last part of the thesis studies discrete time partially observable mean field systems in the presence of a common noise. Each player makes decision solely based on the observable processes but not the common noise. Both the mean field game and the associated mean field type stochastic control problem are formulated. The mean field type control problem is solved by adopting the classical discrete time Kalman filter with notable modifications; indeed, the unique existence of the resulting forward-backward stochastic difference system is then established by Separation Principle. The mean field game problem is also solved via an application of stochastic maximum principle, while the existence of the mean field equilibrium is shown by the Schauder's fixed point theorem.Open Acces

Optimization frameworks and sensitivity analysis of Stackelberg mean-field games

This paper proposes and studies a class of discrete-time finite-time-horizon Stackelberg mean-field games, with one leader and an infinite number of identical and indistinguishable followers. In this game, the objective of the leader is to maximize her reward considering the worst-case cost over all possible $\epsilon$-Nash equilibria among followers. A new analytical paradigm is established by showing the equivalence between this Stackelberg mean-field game and a minimax optimization problem. This optimization framework facilitates studying both analytically and numerically the set of Nash equilibria for the game; and leads to the sensitivity and the robustness analysis of the game value. In particular, when there is model uncertainty, the game value for the leader suffers non-vanishing sub-optimality as the perturbed model converges to the true model. In order to obtain a near-optimal solution, the leader needs to be more pessimistic with anticipation of model errors and adopts a relaxed version of the original Stackelberg game

Equilibrium in Functional Stochastic Games with Mean-Field Interaction

We consider a general class of finite-player stochastic games with mean-field interaction, in which the linear-quadratic cost functional includes linear operators acting on controls in $L^2$. We propose a novel approach for deriving the Nash equilibrium of the game explicitly in terms of operator resolvents, by reducing the associated first order conditions to a system of stochastic Fredholm equations of the second kind and deriving their closed form solution. Furthermore, by proving stability results for the system of stochastic Fredholm equations we derive the convergence of the equilibrium of the $N$-player game to the corresponding mean-field equilibrium. As a by-product we also derive an $\varepsilon$-Nash equilibrium for the mean-field game, which is valuable in this setting as we show that the conditions for existence of an equilibrium in the mean-field limit are less restrictive than in the finite-player game. Finally we apply our general framework to solve various examples, such as stochastic Volterra linear-quadratic games, models of systemic risk and advertising with delay, and optimal liquidation games with transient price impact.Comment: 48 page

Stackelberg+solution+of+first-order+mean+field+game+with+a+major+player

The paper is concerned with the study of the large system of identical players interacting with the environment. We model the environment as a major (exogenous) player. The main assumption of our model is that the minor players influence on each other and on the major (exogenous) player only via certain averaging characteristics. Such models are called mean field games with a major player. It is assumed that the game is considered in the continuous time and the dynamics of major and minor players is given by ordinary differential equations. We study the Stackelberg solution with the major player playing as a leader, i.e., it is assumed that the major player announces his/her control. The main result of the paper is the existence of the Stackelberg solution in the mean field game with the major player in the class of relaxed open-loop strategies

Wireless Resource Management in Industrial Internet of Things

Wireless communications are highly demanded in Industrial Internet of Things (IIoT) to realize the vision of future flexible, scalable and customized manufacturing. Despite the academia research and on-going standardization efforts, there are still many challenges for IIoT, including the ultra-high reliability and low latency requirements, spectral shortage, and limited energy supply. To tackle the above challenges, we will focus on wireless resource management in IIoT in this thesis by designing novel framework, analyzing performance and optimizing wireless resources. We first propose a bandwidth reservation scheme for Tactile Internet in the local area network of IIoT. Specifically, we minimize the reserved bandwidth taking into account the classification errors while ensuring the latency and reliability requirements. We then extend to the more challenging long distance communications for IIoT, which can support the global skill-set delivery network. We propose to predict the future system state and send to the receiver in advance, and thus the delay experienced by the user is reduced. The bandwidth usage is analysed and minimized to ensure delay and reliability requirements. Finally, we address the issue of energy supply in IIoT, where Radio frequency energy harvesting (RFEH) is used to charge unattended IIoT low-power devices remotely and continuously. To motivate the third-party chargers, a contract theory-based framework is proposed, where the optimal contract is derived to maximize the social welfare