linear-quadratic-Gaussian Games Research Articles

Linear-quadratic Gaussian Games with Asymmetric Information: Belief Corrections Using the Opponents Actions

Linear-quadratic Gaussian Games with Asymmetric Information: Belief Corrections Using the Opponents Actions

ϵ-Nash mean-field games for linear-quadratic systems with random jumps and applications

This paper studies linear-quadratic-Gaussian (LQG) games of stochastic large population system with jump-diffusion processes. The most distinguishing feature, compared with the well-studied mean-field LQG games, is that the game system follows linear SDEs driven by random jumps. The individual agents of large population system are coupled both in their state dynamics and in their individual cost functionals. Each agent in large population system has negligible impact on others, but their collective behaviors will impose some significant impact on all agents. The so-called NCE methodology is introduced to deal with the dimensionality difficulty. This methodology derives a set of decentralized strategies, which is an ϵ-Nash equilibrium for a finite N population system where . A numerical example is provided to illustrate the consistency of the mean field estimation and the impact of the population's collective behaviors. As applications, a pricing problem is studied and the decentralized suboptimal price strategy is obtained.

Backward Mean-Field Linear-Quadratic-Gaussian (LQG) Games: Full and Partial Information

This paper introduces the backward mean-field (MF) linear-quadratic-Gaussian (LQG) games (for short, BMFLQG) of weakly coupled stochastic large-population system. In contrast to the well-studied forward mean-field LQG games, the individual state in our large-population system follows the backward stochastic differential equation (BSDE) whose terminal instead initial condition should be prescribed. Two classes of BMFLQG games are discussed here and their decentralized strategies are derived through the consistency condition. In the first class, the individual agents of large-population system are weakly coupled in their state dynamics and the full information can be accessible to all agents. In the second class, the coupling structure lies in the cost functional with only partial information structure. In both classes, the asymptotic near-optimality property (namely, $\epsilon$ - Nash equilibrium) of decentralized strategies are verified. To this end, some estimates to BSDE, are presented in the large-population setting.

Mean-Field Linear-Quadratic-Gaussian (LQG) Games for Stochastic Integral Systems

In this paper we discuss a class of mean field linear-quadratic-Gaussian (LQG) games for large population system which has never been addressed by existing literature. The features of our works are sketched as follows. First of all, our state is modeled by stochastic Volterra-type equation which leads to some new study on stochastic "integral" system. This feature makes our setup significantly different from the previous mean field games where the states always follow some stochastic "differential" equations. Actually, our stochastic integral system is rather general and can be viewed as natural generalization of stochastic differential equations. In addition, it also includes some types of stochastic delayed systems as its special cases. Second, some new techniques are explored to tackle our mean-field LQG games due to the special structure of integral system. For example, unlike the Riccati equation in linear controlled differential system, some Fredholm-type equations are introduced to characterize the consistency condition of our integral system via the resolvent kernels. Third, based on the state aggregation technique, the Nash certainty equivalence (NCE) equation is derived and the set of associated decentralized controls are verified to satisfy the $\epsilon$-Nash equilibrium property. To this end, some new estimates of stochastic Volterra equations are developed which also have their own interests.

Social Optima in Mean Field LQG Control: Centralized and Decentralized Strategies

We study a class of linear-quadratic-Gaussian (LQG) control problems with N decision makers, where the basic objective is to minimize a social cost as the sum of N 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 N → ∞. 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.

Large-Population LQG Games Involving a Major Player: The Nash Certainty Equivalence Principle

We consider linear-quadratic-Gaussian (LQG) games with a major player and a large number of minor players. The major player has a significant influence on others. The minor players individually have negligible impact, but they collectively contribute mean field coupling terms in the individual dynamics and costs. To overcome the dimensionality difficulty and obtain decentralized strategies, the so-called Nash certainty equivalence methodology is applied. The control synthesis is preceded by a state space augmentation via a set of aggregate quantities giving the mean field approximation. Subsequently, within the population limit the LQG game is decomposed into a family of limiting two-player games as each is locally seen by a representative minor player. Next, when solving these limiting two-player games, we impose certain interaction consistency conditions such that the aggregate quantities initially assumed coincide with the ones replicated by the closed loop of a large number of minor players. This procedure leads to a set of decentralized strategies for the original LQG game, which is an $\varepsilon$-Nash equilibrium.