Search

Pairwise-interaction graphical games have been widely used in the study and design of strategic interaction in multiagent systems. With regard to this issue, one entitative problem is actually to understand how the interaction structure of agents affects the strategy configuration of Nash equilibria. This paper intends to study the effect of interaction networks on Nash equilibria in pairwise-interaction graphical games. We first show that interaction networks may induce new strategy equilibria in pairwise-interaction graphical games and then provide graphical conditions for the existence of these network-induced equilibria. Furthermore, to determine Nash equilibria of pairwise-interaction graphical games, a graphical replicator dynamics model is formulated, and its connection with graphical games is established. In detail, it is shown that every Nash equilibrium of the graphical games corresponds to a fixed point of the graphical replicator dynamics and that every asymptotically stable fixed point of the graphical replicator dynamics corresponds to a strict pure Nash equilibrium of the graphical games. The obtained results are applied in understanding coordination in complex networks and determination of structural conflicts in signed graphs. This work may provide new insights into understanding and designing strategy equilibria and dynamics in games on networks. [ABSTRACT FROM AUTHOR]

Mean field game theory has been developed largely following two routes. One of them, called the direct approach, starts by solving a large-scale game and next derives a set of limiting equations as the population size tends to infinity. The second route is to apply mean field approximations and formalize a fixed point problem by analyzing the best response of a representative player. This paper addresses the connection and difference of the two approaches in a linear quadratic (LQ) setting. We first introduce an asymptotic solvability notion for the direct approach, which means for all sufficiently large population sizes, the corresponding game has a set of feedback Nash strategies in addition to a mild regularity requirement. We provide a necessary and sufficient condition for asymptotic solvability 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 role. We next compare with the fixed point approach which determines a two-point boundary value (TPBV) problem, and show that asymptotic solvability implies feasibility of the fixed point approach, but the converse is not true. We further address non-uniqueness in the fixed point approach and examine the long time behavior of the non-symmetric Riccati ODE in the asymptotic solvability problem. [ABSTRACT FROM AUTHOR]

This paper presents a discrete-time distributed and noncooperative routing algorithm, which is proved, via Lyapunov arguments, to asymptotically converge to a specific equilibrium condition among the traffic flows over the network paths, known as Wardrop equilibrium. This convergence result improves the discrete-time algorithms in the literature, which achieve approximate convergence to the Wardrop equilibrium. Numerical simulations show the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

This paper studies the linear–quadratic mean-field game (MFG) for a class of stochastic delayed systems. We consider a large-population system, where the dynamics of each agent is modeled by a stochastic differential delayed equation. The consistency condition is derived through an auxiliary system, which is an anticipated forward–backward stochastic differential equation with delay (AFBSDDE). The wellposedness of such an AFBSDDE system can be obtained using a continuation method. Thus, the MFG strategies can be defined on an arbitrary time horizon, not necessary on a small time horizon by a commonly used contraction mapping method. Moreover, the decentralized strategies are verified to satisfy the $\epsilon$ -Nash equilibrium property. For illustration, three special cases of delayed systems are further explored, for which the closed-loop and open-loop MFG strategies are derived, respectively. [ABSTRACT FROM AUTHOR]

Within the realm of smart buildings and smart cities, dynamic response management is playing an ever-increasing role, thus attracting the attention of scientists from different disciplines. Dynamic demand response management involves a set of operations aiming at decentralizing the control of loads in large and complex power networks. Each single appliance is fully responsive and readjusts its energy demand to the overall network load. A main issue is related to mains frequency oscillations resulting from an unbalance between supply and demand. In a nutshell, this paper contributes to the topic by equipping each consumer with strategic insight. In particular, we highlight three main contributions and a few other minor contributions. First, we design a mean-field game for a population of thermostatically controlled loads, study the mean-field equilibrium for the deterministic mean-field game, and investigate on asymptotic stability for the microscopic dynamics. Second, we extend the analysis and design to uncertain models, which involve both stochastic or deterministic disturbances. This leads to robust mean-field equilibrium strategies guaranteeing stochastic and worst-case stability, respectively. Minor contributions involve the use of stochastic control strategies rather than deterministic and some numerical studies illustrating the efficacy of the proposed strategies. [ABSTRACT FROM PUBLISHER]

The purpose of this paper is to show how insight obtained from a mean-field model can be used to create an architecture for approximate dynamic programming (ADP) for a certain class of games comprising of a large number of agents. The general technique is illustrated with the aid of a mean-field oscillator game model introduced in our prior work. The states of the model are interpreted as the phase angles for a collection of nonhomogeneous oscillators, and in this way the model may be regarded as an extension of the classical coupled oscillator model of Kuramoto. The paper introduces ADP techniques for design and adaptation (learning) of approximately optimal control laws for this model. For this purpose, a parameterization is proposed, based on an analysis of the mean-field PDE model for the game. In an offline setting, a Galerkin procedure is introduced to choose the optimal parameters while in an online setting, a steepest descent algorithm is proposed. The paper provides a detailed analysis of the optimal parameter values as well as the Bellman error with both the Galerkin approximation and the online algorithm. Finally, a phase transition result is described for the large population limit when each oscillator uses the approximately optimal control law. A critical value of the control penalty parameter is identified: above this value, the oscillators are incoherent; and below this value (when control is sufficiently cheap) the oscillators synchronize. These conclusions are illustrated with results from numerical experiments. [ABSTRACT FROM AUTHOR]

Huang (2010) and Nguyen and Huang (2012) solved the linear quadratic mean field systems and control problem in the case where there is a major agent (i.e. non-asymptotically vanishing as the population size goes to infinity) together with a population of minor agents (i.e. individually asymptotically negligible). The new feature in this case is that the mean field becomes stochastic and then, by minor agent state extension, the existence of $\epsilon$-Nash equilibria together with the individual agents' control laws that yield the equilibria may be established. This paper presents results initially announced by Caines and Kizilkale (2013, 2014) where it is shown that if the major agent's state is partially observed by the minor agents, and if the major agent completely observes its own state, all agents can recursively generate estimates (in general individually distinct) of the major agent's state and the mean field, and thence generate feedback controls yielding $\epsilon$ -Nash equilibria. [ABSTRACT FROM PUBLISHER]

This paper considers two classes of large population stochastic differential games connected to optimal and robust decentralized control of large-scale multiagent systems. The first problem (P1) is one where each agent minimizes an exponentiated cost function, capturing risk-sensitive behavior, whereas in the second problem (P2 ) each agent minimizes a worst-case risk-neutral cost function, where the “worst case” stems from the presence of an adversary entering each agent’s dynamics characterized by a stochastic differential equation. In both problems, the individual agents are coupled through the mean field term included in each agent’s cost function, which captures the average or mass behavior of the agents. We solve both P1 and P2 via mean field game theory. Specifically, we first solve a generic risk-sensitive optimal control problem and a generic stochastic zero-sum differential game, where the corresponding optimal controllers are applied by each agent to construct the mean field systems of P1 and P2. We then characterize an approximated mass behavior effect on an individual agent via a fixed-point analysis of the mean field system. For each problem, P1 and P2, we show that the approximated mass behavior is in fact the best estimate of the actual mass behavior in various senses as the population size, $N$, goes to infinity. Moreover, we show that for finite $N$, there exist $\epsilon$- Nash equilibria for both P1 and P2, where the corresponding individual Nash strategies are decentralized in terms of local state information and the approximated mass behavior. We also show that $\epsilon$ can be taken to be arbitrarily small when $N$ is sufficiently large. We show that the $\epsilon$- Nash equilibria of P1 and P2 are partially equivalent in the sense that the individual Nash strategies share identical control laws, but the approximated mass behaviors for P1 and P2 are different, since in P2, the mass behavior is also affected by the associated worst-case disturbance. Finally, we prove that the Nash equilibria for P1 and P2 both feature robustness, and as the parameter characterizing this robustness becomes infinite, the two Nash equilibria become identical and equivalent to that of the risk-neutral case, as in the one-agent risk-sensitive and robust control theory. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper considers decentralized control and optimization methodologies for large populations of systems, consisting of several agents with different individual behaviors, constraints and interests, and influenced by the aggregate behavior of the overall population. For such large-scale systems, the theory of aggregative and mean field games has been established and successfully applied in various scientific disciplines. While the existing literature addresses the case of unconstrained agents, we formulate deterministic mean field control problems in the presence of heterogeneous convex constraints for the individual agents, for instance arising from agents with linear dynamics subject to convex state and control constraints. We propose several model-free feedback iterations to compute in a decentralized fashion a mean field Nash equilibrium in the limit of infinite population size. We apply our methods to the constrained linear quadratic deterministic mean field control problem. [ABSTRACT FROM PUBLISHER]

In this paper, we study a class of risk-sensitive mean-field stochastic differential games. We show that under appropriate regularity conditions, the mean-field value of the stochastic differential game with exponentiated integral cost functional coincides with the value function satisfying a Hamilton –Jacobi– Bellman (HJB) equation with an additional quadratic term. We provide an explicit solution of the mean-field best response when the instantaneous cost functions are log-quadratic and the state dynamics are affine in the control. An equivalent mean-field risk-neutral problem is formulated and the corresponding mean-field equilibria are characterized in terms of backward-forward macroscopic McKean–Vlasov equations, Fokker–Planck–Kolmogorov equations, and HJB equations. We provide numerical examples on the mean field behavior to illustrate both linear and McKean–Vlasov dynamics. [ABSTRACT FROM PUBLISHER]

The purpose of this paper is to synthesize initial mean consensus behavior of a set of agents from the fundamental optimization principles of i) stochastic dynamic games, and ii) optimal control. In the stochastic dynamic game model each agent seeks to minimize its individual quadratic discounted cost function involving the mean of the states of all other agents. In this formulation we derive the limiting infinite population mean field equation system and explicitly compute its unique solution. The resulting mean field (MF) control strategies drive each agent to track the overall population's initial state distribution mean, and by applying these decentralized strategies, any finite population system reaches mean consensus asymptotically as time goes to infinity. Furthermore, these control laws possess an \varepsilonN-Nash equilibrium property where \varepsilonN goes to zero as the population size N goes to infinity. Finally, the analysis is extended to the case of random mean field couplings. In the social cooperative formulation the basic objective is to minimize a social cost as the sum of the individual cost functions containing mean field coupling. In this formulation we show that for any individual agent the decentralized mean field social (MF Social) control strategy is the same as the mean field Nash (MF Nash) equilibrium strategy. Hence $MF-Nash\ Controls\ UNash^N=\rm MF-Social\ Controls\ USoc^N.$On the other hand, the solution to the centralized LQR optimal control formulation yields the Standard Consensus (SC) algorithm whenever the graph representing the corresponding topology of the network is Completely Connected (CC). Hence $Cent.\ LQR\ Controls\ UCent^N=\rm SC-CC\ Controls\ USC^N.$Moreover, a system with centralized control laws reaches consensus on the initial state distribution mean as time and population size N go to infinity. Hence, asymptotically in time $\eqalignno&MF-Nash\ Controls\ UNash^N\!=\!\,\rm MF-Social\ Controls\ USoc^N\cr&\!=\!\,\rm Cent.\ \rm LQR\ Controls\ UCent^\infty \!=\! \,\rm SC-CC\ Controls\ USC^\infty.$Finally, the analysis is extended to the long time average (LTA) cost functions case. [ABSTRACT FROM AUTHOR]