We investigate a class of zero-sum linear-quadratic stochastic differential games on a finite time horizon governed by multiscale state equations. The multiscale nature of the problem can be leveraged to reformulate the associated generalised Riccati equation as a deterministic singular perturbation problem. In doing so, we show that, for small enough $ε$, the existence of solution to the associated generalised Riccati equation is guaranteed by the existence of a solution to a decoupled pair of differential and algebraic Riccati equations with a reduced order of dimensionality. Furthermore, we are able to formulate a pair of asymptotic estimates to the value function of the game problem by constructing an approximate feedback strategy and observing the limiting value function.
We study a general linear quadratic mean field type control problem and connect it to mean field games of a similar type. The solution is given both in terms of a forward/backward system of stochastic differential equations and by a pair of Riccati equations. In certain cases, the solution to the mean field type control is also the equilibrium strategy for a class of mean field games. We use this fact to study an economic model of production of exhaustible resources. Keywords: mean field type control, mean field games, linear-quadratic, optimal control, riccati equations, exhaustible resource …
This paper studies an asymptotic solvability problem for linear quadratic (LQ) mean field games with controlled diffusions and indefinite weights for the state and control in the costs. We employ a rescaling approach to derive a low dimensional Riccati ordinary differential equation (ODE) system, which characterizes a necessary and sufficient condition for asymptotic solvability. The rescaling technique is further used for performance estimates, establishing an $O(1/N)$-Nash equilibrium for the obtained decentralized strategies.
This paper focuses on linear-quadratic (LQ for short) mean-field games described by forward-backward stochastic differential equations (FBSDEs for short), in which the individual control region is postulated to be convex. The decentralized strategies and consistency condition are represented by a kind of coupled mean-field FBSDEs with projection operators. The well-posedness of consistency condition system is obtained using the monotonicity condition method. The $ε$-Nash equilibrium property is discussed as well.
We consider a controlled linear-quadratic (LQ) large-population system with mixture of three types agents: major leader, minor leaders and minor followers. The Stackelberg-Nash-Cournot (SNC) approximate equilibrium is studied by a major-minor mean-field game (MFG) coupled with a leader-follower Stackelberg game. By variational method, the SNC approximate equilibrium strategy can be represented by some forward-backward-stochastic-differential-equations (FBSDEs) in the open-loop sense. And we pay great effort to give the feedback form of the open-loop strategy by some Riccati equations.
This paper establishes that $N$-player stochastic games with singular controls, either of bounded velocity or of finite variation, can both be approximated by mean field games (MFGs) with singular controls of bounded velocity. More specifically, it shows i) the optimal control to an MFG with singular controls of a bounded velocity $θ$ is shown to be an $ε_N$-NE to an $N$-player game with singular controls of the bounded velocity, with $ε_N = O(\frac{1}{\sqrt{N}})$, and (ii) the optimal control to this MFG is an $(ε_N + ε_θ)$-NE to an $N$-player game with singular controls of finite variation, …
This paper is concerned with a linear quadratic stochastic Stackelberg differential game with time delay. The model is general, in which the state delay and the control delay both appear in the state equation, moreover, they both enter into the diffusion term. By introducing two Pseudo-Riccati equations and a special matrix equation, the state feedback representation of the open-loop Stackelberg strategy is derived, under some assumptions. Finally, two examples are given to illustrate the applications of the theoretical results.
This paper is concerned with a linear quadratic stochastic two-person zero-sum differential game with constant coefficients in an infinite time horizon. Open-loop and closed-loop saddle points are introduced. The existence of closed-loop saddle points is characterized by the solvability of an algebraic Riccati equation with a certain stabilizing condition. A crucial result makes our approach work is the unique solvability of a class of linear backward stochastic differential equations in an infinite horizon.
Mean field games with a major player were introduced in (Huang, 2010) within a linear-quadratic (LQ) modeling framework. Due to the rich structure of major-minor player models, the past ten years have seen significant research efforts for different solution notions and analytical techniques. For LQ models, we address the relation between three solution frameworks: the Nash certainty equivalence (NCE) approach in (Huang, 2010), master equations, and asymptotic solvability, which have been developed starting with different ideas. We establish their equivalence relationships.
This paper focuses on a kind of linear quadratic non-zero sum differential game driven by backward stochastic differential equation with asymmetric information, which is a natural continuation of Wang and Yu [IEEE TAC (2010) 55: 1742-1747, Automatica (2012) 48: 342-352]. Different from Wang and Yu [IEEE TAC (2010) 55: 1742-1747, Automatica (2012) 48: 342-352], novel motivations for studying this kind of game are provided. Some feedback Nash equilibrium points are uniquely obtained by forward-backward stochastic differential equations, their filters and the corresponding Riccati equations with …
This paper investigates the so-called asymptotic solvability problem in linear quadratic (LQ) mean field games. The model has asymptotic solvability if for all sufficiently large population sizes, the corresponding game has a set of feedback Nash strategies subject to a mild regularity requirement. We provide a necessary and sufficient condition 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 rol…
The paper is concerned with a zero-sum Stackelberg stochastic linear-quadratic (LQ, for short) differential game over finite horizons. Under a fairly weak condition, the Stackelberg equilibrium is explicitly obtained by first solving a forward stochastic LQ optimal control problem (SLQ problem, for short) and then a backward SLQ problem. Two Riccati equations are derived in constructing the Stackelberg equilibrium. An interesting finding is that the difference of these two Riccati equations coincides with the Riccati equation associated with the zero-sum Nash stochastic LQ differential game, w…
This paper revisits well-studied dynamic decisions of weakly coupled large-population (LP) systems. Specifically, three types of LP decision problems: mean-field game (MG), mean-field team (MT), and mean-field-type control (MC), are completely analyzed in a general stochastic linear-quadratic setting with controlled-diffusion in state dynamics and indefinite weight in cost functional. More importantly, interrelations among MG, MT and MC are systematically discussed; some relevant interesting findings are reported that may be applied to a structural analysis of general LP decisions.
The paper is concerned with two-person zero-sum mean-field linear-quadratic stochastic differential games over finite horizons. By a Hilbert space method, a necessary condition and a sufficient condition are derived for the existence of an open-loop saddle point. It is shown that under the sufficient condition, the associated two Riccati equations admit unique strongly regular solutions, in terms of which the open-loop saddle point can be represented as a linear feedback of the current state. When the game only satisfies the necessary condition, an approximate sequence is constructed by solvin…
In this paper we establish a new connection between a class of 2-player nonzero-sum games of optimal stopping and certain $2$-player nonzero-sum games of singular control. We show that whenever a Nash equilibrium in the game of stopping is attained by hitting times at two separate boundaries, then such boundaries also trigger a Nash equilibrium in the game of singular control. Moreover a differential link between the players' value functions holds across the two games.
This paper establishes the existence of equilibria result of a class of mean field games with singular controls. The interaction takes place through both states and controls. A relaxed solution approach is used. To circumvent the tightness issue, we prove the existence of equilibria by first considering the corresponding mean field games with continuous controls instead of singular controls and then taking approximation.
This article is dedicated to the study of mixed zero-sum two-player stochastic differential games in the situation when the player's cost functionals are modeled by doubly controlled reflected backward stochastic equations with two barriers whose coefficients have quadratic growth in Z. This is a generalization of the risk-sensitive payoffs. We show that the lower and the upper value function associated with this stochastic differential game with reflection are deterministic and they are also the unique viscosity solutions for two Isaacs equations with obstacles.
This paper studies singular mean field control problems and singular mean field stochastic differential games. Both sufficient and necessary conditions for the optimal controls and for the Nash equilibrium are obtained. Under some assumptions the optimality conditions for singular mean-field control are reduced to a reflected Skorohod problem, whose solution is proved to exist uniquely. Applications are given to optimal harvesting of stochastic mean-field systems, optimal irreversible investments under uncertainty and to mean-field singular investment games. In particular, a simple singular me…
This paper considers a class of mean field linear-quadratic-Gaussian (LQG) games with model uncertainty. The drift term in the dynamics of the agents contains a common unknown function. We take a robust optimization approach where a representative agent in the limiting model views the drift uncertainty as an adversarial player. By including the mean field dynamics in an augmented state space, we solve two optimal control problems sequentially, which combined with consistent mean field approximations provides a solution to the robust game. A set of decentralized control strategies is derived by…
This paper is concerned with the closed-loop solvability of one kind of linear-quadratic Stackelberg stochastic differential game, where the coefficients are deterministic. The notion of the closed-loop solvability is introduced, which require to be independent of the initial state. The follower's problem is solved first, and the closed-loop optimal strategy is characterized by a Riccati equation, together with an adapted solution to a linear backward stochastic differential equation. Then the necessary conditions of the existence of the leader's nonanticipating closed-loop optimal strategy is…
We consider a class of linear-quadratic-Gaussian mean-field games with a major agent and considerable heterogeneous minor agents in the presence of mean-field interactions. The individual admissible controls are constrained in closed convex subsets $Γ_{k}$ of $\mathbb{R}^{m}.$ The decentralized strategies for individual agents and consistency condition system are represented in an unified manner through a class of mean-field forward-backward stochastic differential equations involving projection operators on $Γ_{k}$. The well-posedness of consistency system is established in both the local and…
Game theory is playing more and more important roles in understanding complex systems and in investigating intelligent machines with various uncertainties. As a starting point, we consider the classical two-player zero-sum linear-quadratic stochastic differential games, but in contrast to most of the existing studies, the coefficient matrices of the systems are assumed to be unknown to both players, and consequently it is necessary to study adaptive strategies of the players, which may be termed as adaptive games and which has rarely been explored in the literature. In this paper, it will be s…
This paper establishes the existence of relaxed solutions to mean field games (MFGs for short) with singular controls. We also prove approximations of solutions results for a particular class of MFGs with singular controls by solutions, respectively control rules, for MFGs with purely regular controls. Our existence and approximation results strongly hinge on the use of the Skorokhod $M_1$ topology on the space of càdlàg functions.
In this paper, we consider a linear quadratic (LQ) leader-follower stochastic differential game for regime switching diffusions with mean-field interactions. One of the salient features of this paper is that conditional mean-field terms are included in the state equation and cost functionals. Based on stochastic maximum principles (SMPs), the follower's problem and the leader's problem are solved sequentially and an open-loop Stackelberg equilibrium is obtained. Further, with the help of the so-called four-step scheme, the corresponding Hamiltonian systems for the two players are decoupled and…
This paper is concerned with the stochastic linear quadratic Stackelberg differential game with overlapping information, where the diffusion terms contain the control and state variables. Here the term "overlapping" means that there are common part between the follower's and the leader's information, while they have no inclusion relation. Optimal controls of the follower and the leader are obtained by the stochastic maximum principle, the direct calculation of the derivative of the cost functional and stochastic filtering. A new system of Riccati equations is introduced to represent the state …
We study a zero-sum game where the evolution of a spectrally one-sided Levy process is modified by a singular controller and is terminated by the stopper. The singular controller minimizes the expected values of running, controlling and terminal costs while the stopper maximizes them. Using fluctuation theory and scale functions, we derive a saddle point and the value function of the game. Numerical examples under phase-type Levy processes are also given.
All-Against-One (AAO) games are a special class of multi-player games where all players except one are in direct conflict with the remaining player. In the case of Linear Quadratic Differential (LQD) games, the AAO structure can be used to describe a situation where all players except one are trying to regulate the state of the system (drive it towards the origin) while the opposing player is trying to de-regulate the state (drive it away from the origin). Similar to the standard LQD games, the closed-loop Nash strategies in the AAO LQD games are expressed in terms of the solutions of a set of…
We formulate and analyze game-theoretic problems for systems governed by integral equations. For Volterra integral equations, we obtain and prove necessary and sufficient conditions for linear-quadratic problems, and for problems that are linear-quadratic in the control. Also, we obtain necessary conditions for one type of pursuit-evasion Volterra games.
This paper is concerned with a three-level stochastic linear-quadratic Stackelberg differential game with asymmetric information, in which three players participate credited as Player 1, Player 2 and Player 3. Player 3 acts as the leader of Player 2 and Player 1, Player 2 acts as the leader of Player 1 and Player 1 acts as the follower. The asymmetric information considered is: the information available to Player 1 is based on the sub-$σ$-algebra of the information available to Player 2, and the information available to Player 2 is based on the sub-$σ$-algebra of the information available to P…
In this paper, we consider a linear quadratic stochastic two-person zero-sum differential game. The controls for both players are allowed to appear in both drift and diffusion of the state equation. The weighting matrices in the performance functional are not assumed to be definite/non-singular. A necessary and sufficient condition for the existence of a closed-loop saddle point is established in terms of the solvability of a Riccati differential equation with certain regularity. It is possible that the closed-loop saddle point fails to exist, and at the same time, the corresponding Riccati eq…
This paper considers linear quadratic (LQ) mean field games with a major player and analyzes an asymptotic solvability problem. It starts with a large-scale system of coupled dynamic programming equations and applies a re-scaling technique introduced in Huang and Zhou (2018a, 2018b) to derive a set of Riccati equations in lower dimensions, the solvability of which determines the necessary and sufficient condition for asymptotic solvability. We next derive the mean field limit of the strategies and the value functions. Finally, we show that the two decentralized strategies can be interpreted as…
This paper investigates the linear-quadratic-Gaussian (LQG) mean-field game (MFG) for a class of stochastic delay systems. We consider a large population system in which the dynamics of each player satisfies some forward stochastic differential delay equation (SDDE). The consistency condition or Nash certainty equivalence (NCE) principle is established through an auxiliary mean-field system of anticipated forward-backward stochastic differential equation with delay (AFBSDDE). The wellposedness of such consistency condition system can be further established by some continuation method instead t…
This paper studies a class of dynamic Stackelberg games under open-loop information structure with constrained linear agent dynamics and quadratic utility functions. We show two important properties for this class of dynamic Stackelberg games. First, we prove that under mild conditions, the optimal control of individual agents at the solution of the Stackelberg game coincides with the solution to the team problem, where all agents cooperatively achieve the coordinator's objective. Second, we show that the the agent's control at each time step is non-increasing with respect to the coordinator's…
