Noncooperative Differential Games Research Articles

Coalitional Pareto optimal solution of one differential game

This paper is devoted to the differential positional coalitional games with non-transferable payoffs (games without side payments). We believe that the researches of the objection and counter-objection equilibrium for non-cooperative differential games that have been carried out over the last years allow to cover some aspects of non-transferable payoff coalitional games. In this paper we consider the issues of the internal and external stability of coalitions in the class of positional differential games. For a differential positional linear-quadratic six-player game with a two-coalitional structure, the coefficient constraints are obtained which provide an internal and external stability of the coalitional structure.

Ordering stability of Nash equilibria for a class of differential games

Abstract This study is concerned with the stability of Nash equilibria for a class of n n -person noncooperative differential games. More precisely, due to a preorder induced by a convex cone on a real linear normed space, we define a new concept called ordering stability of equilibria against the perturbation of the right-hand side functions of state equations for the differential game. Moreover, using the set-valued analysis theory, we present the sufficient conditions of the ordering stability for such differential games.

Differential Game-Based Optimal Control of Autonomous Vehicle Convoy

Group control of connected and autonomous vehicles on automated highways is challenging for the advanced driver assistance systems (ADAS) and the automated driving systems (ADS). This paper investigates the differential game-based approach to autonomous convoy control with the aim of deployment on automated highways. Under the noncooperative differential games, the coupled vehicles make their decisions independently while their states are interdependent. The receding horizon Nash equilibrium of the linear-quadratic differential game provides the convoy a distributed state-feedback control strategy. This approach suffers a fundamental issue that neither a Nash equilibrium’s existence nor the uniqueness is guaranteed. We convert the individual dynamics-based differential game to a relative dynamics-based optimal control problem that carries all the features of the differential game. The existence of a unique Nash control under the differential game corresponds to a unique solution to the optimal control problem. The latter is shown, as well as the asymptotic stability of the closed-loop system. Simulations illustrate the effectiveness of the presented convey control scheme and how it well suits automated highway driving scenarios.

Seeking Nash Equilibrium in Non-Cooperative Differential Games

This paper aims at investigating the problem of fast convergence to the Nash equilibrium (NE) for N-Player noncooperative differential games. The proposed method is such that the players attain their NE point without steady-state oscillation (SSO) by measuring only their payoff values with no information about payoff functions, the model and also the actions of other players are not required for the players. The proposed method is based on an extremum seeking (ES) method, and moreover, compared to the traditional ES approaches, in the presented algorithm, the players can accomplish their NE faster. In fact, in our method, the amplitude of the sinusoidal excitation signal in classical ES is adaptively updated and exponentially converges to zero. In addition, the analysis of convergence to NE is provided in this paper. Finally, a simulation example confirms the effectiveness of the proposed method.

Sufficient conditions for the existence of equilibrium in the form of Hamilton-Jacobi equations

The article discusses the application of the theory of optimal control for solving Hamilton-Jacob equations with phase constraints. A method for constructing generalized solutions using optimal control problems is proposed. The results and analysis of numerical experiments, conditions for the existence of equilibrium situations in noncooperative differential games of several persons, namely the conditions for the existence of equilibrium situations in noncooperative differential games of several persons, defining the action according to Hamilton, are stated. Necessary conditions in the form of Hamilton-Jacobi equations are obtained. Game theory as an applied mathematical theory is used to understand and explain the mechanisms that are used when people make decisions. The theory contributes to the functioning of the logic of strategic planning and the relationship between individuals. Game theory as a method of applied mathematics is used for behavioral studies in various conditions, and helps understand the behavior of economic agents. The theory has many applications and can be used in different areas such as: strategy games, administration, economics and artificial intelligence research. The article describes a mathematical method for studying optimal situations in game theory.

Distributed Differential Games for Control of Multi-Agent Systems

Motivated by the challenges arising in the field of multi-agent system (MAS) control, we consider linear heterogeneous MAS subject to local communication and investigate the problem of designing distributed controllers for such systems. We provide a game theoretic framework for systematically designing distributed controllers, taking into account individual objectives of the agents and their possibly incomplete knowledge of the MAS. Linear state-feedback control laws are obtained via the introduction of a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">distributed differential game</i> , namely, the combination of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">local non-cooperative differential games</i> , which are solved in a decentralized fashion. Conditions for stability of the MAS are provided for the special cases of acyclic and strongly connected communication graph topologies. These results are then exploited to provide stability conditions for general graph topologies. The proposed framework is demonstrated on a tracking synchronization problem associated with the design of a distributed secondary voltage controller for microgrids and on a numerical example.

Stability of international pollution control games: A potential game approach

&lt;p style='text-indent:20px;'&gt;In this paper, stabilization problems for &lt;i&gt;n-&lt;/i&gt;player noncooperative differential games of international pollution control (IPC) are analysed via the concept of the potential differential game (PDG) introduced by Fonseca-Morales and Hernández-Lerma (2018). By first identifying a game of IPC as a PDG, an associated optimal control problem (OCP) is obtained, whose optimal solution is a Nash equilibrium (NE) for the game of IPC. Thus, the problem of finding conditions for which the NE stabilizes the game of IPC reduces to finding conditions for which the optimal solution stabilizes the associated OCP. The concept not only yields mild conditions for saddle point stability analysed in the literature but also for the overtaking optimality of the NE of the game of IPC.&lt;/p&gt;

Non-autonomous Linear Quadratic Non-cooperative Differential Games with Continuous Updating

The subject of this paper is a non-autonomous linear quadratic case of a differential game model with continuous updating. This class of differential games is essentially new where it is assumed that, at each time instant, players have or use information about the game structure defined on a closed time interval with a fixed duration. During the interval information about motion equations and payoff functions of players updates. It is non-autonomy that simulates this effect of updating information. A linear quadratic case for this class of games is particularly important for practical problems arising in the engineering of human-machine interaction. Here we define the Nash equilibrium as an optimality principle and present an explicit form of Nash equilibrium for the linear quadratic case. Also, the case of dynamic updating for the linear quadratic differential game is studied and uniform convergence of Nash equilibrium strategies and corresponding trajectory for a case of continuous updating and dynamic updating is demonstrated.

A dynamic theory of spatial externalities

We characterize the shape of spatial externalities in a continuous time and space differential game with transboundary pollution. We posit a realistic spatiotemporal law of motion for pollution (diffusion and advection), and tackle spatiotemporal non-cooperative (and cooperative) differential games. Precisely, we consider a circle partitioned into several states where a local authority decides autonomously about its investment, production and depollution strategies over time knowing that investment/production generates pollution, and pollution is transboundary. The time horizon is infinite. We allow for a rich set of geographic heterogeneities across states. We solve analytically the induced non-cooperative differential game and characterize its long-term spatial distributions. In particular, we prove that there exist a Perfect Markov Equilibrium, unique among the class of the affine feedbacks. We further provide with a full exploration of the free riding problem and the associated border effect.

Learning feedback Nash strategies for nonlinear port-Hamiltonian systems

This paper presents an adaptive control strategy for solving multi-player noncooperative differential games with dynamics modelled as general nonlinear input-state-output port-Hamiltonian systems. The proposed controller is obtained by extending an existing single-player feedback Nash strategy to N players and by using the Hamiltonian of the port-Hamiltonian system as an admissible control-Lyapunov function for each player. Necessary and sufficient conditions for the stability of the resulting controlled system are provided by employing Lyapunov stability theory. Furthermore, the N player feedback strategy is extended by adaptively weighting the individual value functions to ensure convergence to the Nash solution. Finally, numerical simulations demonstrate the effectiveness of the proposed explicit control laws.

A differential game of N persons in which there is Pareto equilibrium of objections and counterobjections and no Nash equilibrium

A linear-quadratic positional differential game of N persons is considered. The solution of a game in the form of Nash equilibrium has become widespread in the theory of noncooperative differential games. However, Nash equilibrium can be internally and externally unstable, which is a negative in its practical use. The consequences of such instability could be avoided by using Pareto maximality in a Nash equilibrium situation. But such a coincidence is rather an exotic phenomenon (at least we are aware of only three cases of such coincidence). For this reason, it is proposed to consider the equilibrium of objections and counterobjections. This article establishes the coefficient criteria under which in a differential positional linear-quadratic game of N persons there is Pareto equilibrium of objections and counterobjections and at the same time no Nash equilibrium situation; an explicit form of the solution of the game is obtained.

Generic stability of Nash equilibria for noncooperative differential games

The stability of Nash equilibria against the perturbation of the right-hand side functions of state equations for noncooperative differential games is investigated. By employing the set-valued analysis theory, we show that the differential games whose equilibria are all stable form a dense residual set, and every differential game can be approximated arbitrarily by a sequence of stable differential games, that is, in the sense of Baire’s category most of the differential games are stable.

Inverse Optimal Control for Identification in Non-Cooperative Differential Games

This paper presents research on inverse optimal control in non-cooperative differential games. An inverse optimization framework for the identification of an unknown objective function of a game’s participant is introduced. The identified objective function yields the same trajectories as the ground truth objective function. We apply the presented methods in a human-machine cooperation scenario, where identification of human behavior is indispensable for automation design. The framework’s methods are evaluated through simulation of a driver cooperating with a driving assistance system based on a realistic vehicle dynamics model. The identification results are highly accurate and calculation performances motivate real-time applications.

Stability of Feedback Solutions for Infinite Horizon Noncooperative Differential Games

We consider a noncooperative game in infinite time horizon, with linear dynamics and exponentially discounted quadratic costs. Assuming that the state space is one-dimensional, we prove that the Nash equilibrium solution in feedback form is stable under nonlinear perturbations. The analysis shows that, in a generic setting, the linear-quadratic game can have either one or infinitely many feedback equilibrium solutions. For each of these, a nearby solution of the perturbed nonlinear game can be constructed.

Routing Algorithm Based on Non-cooperative Differential Games in Deep Space Networks

Most of the existing routing protocols from those in the terrestrial networks can not be directly applied in Deep Space Information Networks (DSINs), because of its inherent outstanding characteristics, such as bandwidth constraints, high latency, asymmetric data rates, and heterogeneous network architectures. Therefore, designing an optimal routing algorithm that is suitable for DSINs becomes a challenging and interesting problem. In this paper, a bandwidth allocation scheme based on non-cooperative differential game is proposed. Our scheme takes the selfness of nodes into consideration. It is shown that our proposed scheme can effectively allocate bandwidth with bandwidth constraints, and is more feasible and adaptive to DSINs with bandwidth constraints.

Noncooperative Differential Games

This paper, having a tutorial character, is intended to provide an introduction to the theory of noncooperative differential games.

A deterministic linear quadratic time-inconsistent optimal control problem

A time-inconsistent optimal control problem is formulated and studied for a controlled linear ordinary differential equation with quadratic cost functional. A notion of equilibrium control is introduced, which can be regarded as a time-consistent solution to the original time-inconsistent problem. Under certain conditions, we constructively prove the existence of such an equilibrium control which is represented via a forward ordinary differential equation coupled with a backward Riccati--Volterra integral equation. Our constructive approach is based on the introduction of a family of $N$-person non-cooperative differential games.

Infinite horizon noncooperative differential games with nonsmooth costs

For a noncooperative differential game, the value functions of the various players satisfy a system of Hamilton–Jacobi equations. In the present paper, we study a class of infinite-horizon scalar games with either piecewise linear or piecewise smooth costs, exponentially discounted in time. By the analysis of the value functions, we find that results about existence and uniqueness of admissible solutions to the HJ system, and therefore of Nash equilibrium solutions in feedback form, can be recovered as in the smooth costs case, provided the costs are globally monotone. On the other hand, we present examples of costs such that the corresponding HJ system has infinitely many admissible solutions or no admissible solutions at all, suggesting that new concepts of equilibria may be needed to study games with general nonlinear costs.

THE PEDESTRIAN PRINCIPLE FOR DIFFERENTIAL GAMES

A dynamic normal formulation for differential games is introduced and the "pedestrian principle" is discussed as a means of dynamically implementing the equilibrium strategy in a single game. Our formulation emphasizes the distinction between a player's rational prediction and the actual evolution of the game dynamics. To model the free will of players, a randomized strategy is introduced which serves as the justification of mixed strategies and the bridge from a static analysis to a dynamic one. Existence of Nash equilibrium in the class of mixed strategies is proved for non-cooperative deterministic differential games.

Infinite horizon noncooperative differential games

For a noncooperative differential game, the value functions of the various players satisfy a system of Hamilton–Jacobi equations. In the present paper, we consider a class of infinite horizon games with nonlinear costs exponentially discounted in time. By the analysis of the value functions, we establish the existence of Nash equilibrium solutions in feedback form and provide results and counterexamples on their uniqueness and stability.