Feedback Nash Equilibrium Research Articles

A Stochastic Differential Game Theoretic Study of Multipath Routing in Heterogeneous Wireless Networks

In heterogeneous wireless networks (HWNs), paths constituting multipath routing are characterized by selfish rationality. Each path's intentions of pursuing individual profits may cause unreasonable competition for limited wireless resources, which leads to the unreliable data transport problem. Therefore, multipath routing optimization is a challenge issue in HWNs. This paper provides a novel approach to study this issue by employing game theory. By taking the utility maximization as the design goal, limited bandwidth resources as the constraint and path reliability as the key metric, a noncooperative stochastic differential game model is constructed for HWNs. With the feedback Nash equilibrium solution, an optimal multipath routing strategy is obtained. Theoretical derivations and simulation results verify the validity of the method present in this paper.

Intrinsic Comparative Dynamics of Locally Differentiable Feedback Stackelberg Equilibria

The intrinsic comparative dynamics of locally differentiable feedback Stackelberg equilibria are derived for the ubiquitous class of autonomous and exponentially discounted infinite horizon differential games. It is shown that the follower’s intrinsic comparative dynamics agree in their form and qualitative properties with those of every player in a feedback Nash equilibrium, while those of the leader differ in form. The difference allows, in principle, an empirical test of the leader-follower role in a differential game. Separability conditions are identified on the instantaneous payoff and transition functions under which the intrinsic comparative dynamics of feedback Nash equilibria, feedback Stackelberg equilibria, and those in the corresponding optimal control problem are qualitatively identical.

INTERNATIONAL ENVIRONMENTAL AGREEMENTS WITH DEVELOPED AND DEVELOPING COUNTRIES IN A DYNAMIC APPROACH

AbstractThis paper is devoted to study the stability of international environmental agreements (IEAs) in a pollution abatement model. We focus our attention on two asymmetric agents: developed countries characterized by a strong measure of environmental awareness and developing ones characterized by less attention to environmental preservation. We model a differential game in order to determine both the optimal path of the abatement levels and stock pollutant as results of open‐loop and feedback Nash equilibria. By means of the concepts of internal and external stability, we explain how the environmental awareness of two types of countries and the number of cooperators and defectors influence the size of stable coalitions. The results establish that the asymmetry assumption does not have consequences for the stability of the grand coalition, which can be obtained only with transfers.

Oligopolistic spectrum allocation game via market competition under spectrum broker

For future dynamic spectrum access (DSA) systems, new spectrum management methodologies will be adopted such that the licensed primary spectrum holders can reallocate part of their available bandwidth to the unlicensed secondary service providers for profits, based on market-driven mechanisms. This is known as spectrum market or spectrum trading. This paper aims to propose a dynamic spectrum market model where each spectrum holder has a limited amount of spectrum and is allowed to enter a portion of its available bandwidth into the market managed by a spectrum broker, besides its primary services. To model the price dynamics, a continuous-time price adjustment process governed by a differential equation is considered. Subsequently, we show that the problem is a dynamic N-player oligopoly differential game, subject to the bandwidth constraint. We analyze the feedback Nash equilibrium (NE) solutions for the general game and provide a complete, closed-form solution for the special symmetric case. The solution can be characterized into three distinct regions and transitions between these regions may occur as time evolves, which will be thoroughly investigated. In addition, we propose a discrete-time price adjustment implemented at the spectrum broker. Extensive numerical studies are provided to investigate various aspects of the proposed competition.

Approximating Solutions for Nonlinear Dynamic Tracking Games

This paper presents the OPTGAME algorithm developed to iteratively approximate equilibrium solutions of ‘tracking games’, i.e. discrete-time nonzero-sum dynamic games with a finite number of players who face quadratic objective functions. Such a tracking game describes the behavior of decision makers who act upon a nonlinear discrete-time dynamical system, and who aim at minimizing the deviations from individually desirable paths of multiple states over a joint finite planning horizon. Among the noncooperative solution concepts, the OPTGAME algorithm approximates feedback Nash and Stackelberg equilibrium solutions, and the open-loop Nash solution, and the cooperative Pareto-optimal solution.

Battle for Climate and Scarcity Rents: Beyond the Linear-Quadratic Case

Industria imports oil, produces final goods and wishes to mitigate global warming. Oilrabia exports oil and buys final goods from the other country. Industria uses the carbon tax to impose an import tariff on oil and steal some of Oilrabia’s scarcity rent. Conversely, Oilrabia has monopoly power and sets the oil price to steal some of Industria’s climate rent. We analyze the relative speeds of oil extraction and carbon accumulation under these strategic interactions for various production function specifications and compare these with the efficient and competitive outcomes. We prove that for the class of HARA production functions the oil price is initially higher and subsequently lower in the open-loop Nash equilibrium than in the efficient outcome. The oil extraction rate is thus initially too low and in later stages too high. The HARA class includes linear, loglinear and semi-loglinear demand functions as special cases. For non-HARA production functions Oilrabia may in the open-loop Nash equilibrium initially price oil lower than the efficient level, thus resulting in more oil extraction and climate damages. We also contrast the open-loop Nash and efficient outcomes numerically with the feedback Nash outcomes. We find that the optimal carbon tax path in the feedback Nash equilibrium is flatter than in the open-loop Nash equilibrium. It turns out that for certain demand functions using the carbon tax as an import tariff may hurt consumers’ welfare as the resulting user cost of oil is so high that the fall in welfare wipes out the gain from higher tariff revenues.

Cooperative differential game for model energy-bandwidth efficiency tradeoff in the Internet of Things

Internet of Things (IoT) refers to an infrastructure which enables the forms of communication and collaboration between people and things, and between things themselves. In order to improve its performance, we present a tradeoff between bandwidth and energy consumption in the IoT in this paper. A service providing model is built to find the relationship between bandwidth and energy consumption using a cooperative differential game model. The game solution is gotten in the condition of grand coalition, feedback Nash equilibrium and intermediate coalitions and an allocation policy is obtain by Shapley theory. The results are shown as follows. Firstly, the performance of IoT decreases with the increasing of bandwidth cost or with the decreasing of energy cost; secondly, all the nodes in the IoT composing a grand coalition can save bandwidth and energy consumption; thirdly, when the factors of bandwidth cost and energy cost are equal, the obtained number of provided services is an optimised value which is the trade-off between energy and bandwidth consumption.

Optimal advertising decisions for promoting retail and online channels in a dynamic framework

AbstractAdvertising plays an important role in sales promotion in supply chains. In this paper, advertising decisions for retail and online channels respectively, by retailer and manufacturer will be analyzed. The competition between retailer and manufacturer will be modeled by differential game theory, and feedback Nash equilibrium of the game will be obtained. Some insights are provided by using comparative statics. The obtained results show that compatibility factor of a product with online marketing, difference between manufacturer's online price and wholesale price, effectiveness of advertising, marginal profits, and cost of advertising all have significant effects on the advertising decisions in the equilibrium.

Uncertain differential games with application to capitalism

Uncertainty theory provides us an alternative for modeling human uncertainty without the scope of probability theory. In a dynamic game, the system dynamics often has noise by human disturbance and can be described by an uncertain differential equation. In order to investigate such a problem, an uncertain differential game model and its solution concept of feedback Nash equilibrium is proposed in this paper. Moreover, a sufficient condition is established to guarantee a feedback Nash equilibrium of the game. As an application, the capitalism problem is analyzed by uncertain differential game. The results show that uncertain differential game is a powerful tool for dealing with dynamic game.

Feedback Nash Equilibrium for Dynamic Brand and Channel Advertising in Dual Channel Supply Chain

This paper investigates dynamic brand and channel advertising and their effects on market expansion and market share in a dual channel supply chain by using differential game theory. Nash equilibrium of the differential game will be obtained by the feedback approach. A sensitivity analysis was performed to assess the impact of key parameters on optimal strategies and to characterize behaviors of both the manufacturer and retailer over time. Comparative statistics show that a higher compatibility of a product with online marketing leads to a higher advertising effort for the online channel by the manufacturer, an enhanced steady state for demand of the brand as well as greater sales in the steady state through the online channel.

Feedback saddle point solution of counterterror measures and economic growth game

In this paper, the feedback Nash equilibrium solutions of the differential game between counterterror measures and economic growth are investigated. The Hamilton–Jacobi–Isaacs (HJI) equation is used to obtain the feedback saddle point of this zero-sum game. Moreover, the characteristics of the feedback strategies for the government and terrorist organization as well as their relationships with both resource states are analyzed.

Concurrent learning-based approximate feedback-Nash equilibrium solution of N-player nonzero-sum differential games

This paper presents a concurrent learning-based actor-critic-identifier architecture to obtain an approximate feedback-Nash equilibrium solution to an infinite horizon N-player nonzero-sum differential game. The solution is obtained online for a nonlinear control-affine system with uncertain linearly parameterized drift dynamics. It is shown that under a condition milder than persistence of excitation (PE), uniformly ultimately bounded convergence of the developed control policies to the feedback-Nash equilibrium policies can be established. Simulation results are presented to demonstrate the performance of the developed technique without an added excitation signal.

The intrinsic comparative dynamics of locally differentiable feedback Nash equilibria of autonomous and exponentially discounted infinite horizon differential games

The comparative dynamics of locally differentiable feedback Nash equilibria are derived for the ubiquitous class of autonomous and exponentially discounted infinite horizon differential games. The resulting refutable implications are intrinsic to the said class of differential games, and thus form their basic, empirically testable, properties. Their relationship with extant results in the optimal control theory and the static game theory is discussed. Separability conditions are identified on the instantaneous payoff and transition functions under which the intrinsic comparative dynamics collapse, in form, to those in optimal control problems. Applications of the results to capital accumulation and sticky-price games are provided.

Fighting corruption: To precommit or not?

We consider a differential game with a corrupt government and civil society as its players. We characterize open-loop and feedback Nash equilibria and find that, whereas it is in the best interest of the government not to commit to a repression policy, civil society is better off precommitting to fight corruption.

Characterization of feedback Nash equilibria for multi-channel systems via a set of non-fragile stabilizing state-feedback solutions and dissipativity inequalities

We consider the problem of state-feedback stabilization for a multi-channel system in a game-theoretic framework, where the class of admissible strategies for the players is induced from a solution set of the individual objective functions that are associated with certain dissipativity properties of the system. In such a framework, we characterize the feedback Nash equilibria via a set of non-fragile stabilizing state-feedback solutions corresponding to a family of perturbed multi-channel systems. Moreover, we show that the existence of a weak-optimal solution to a set of constrained dissipativity problems is a sufficient condition for the existence of a feedback Nash equilibrium, whereas the set of non-fragile stabilizing state-feedbacks solutions is described in terms of a set of dilated linear matrix inequalities.

Governance Mechanism for Global Greenhouse Gas Emissions: A Stochastic Differential Game Approach

Today developed and developing countries have to admit the fact that global warming is affecting the earth, but the fundamental problem of how to divide up necessary greenhouse gas reductions between developed and developing countries remains. In this paper, we propose cooperative and noncooperative stochastic differential game models to describe greenhouse gas emissions decision makings of developed and developing countries, calculate their feedback Nash equilibrium and the Pareto optimal solution, characterize parameter spaces that developed and developing countries can cooperate, design cooperative conditions under which participants buy the cooperative payoff, and distribute the cooperative payoff with Nash bargaining solution. Lastly, numerical simulations are employed to illustrate the above results.

A two-period dynamic game for a substitutable product inventory control problem

We consider the equilibrium strategies for substitutable product inventory control systems with a random demand in a two-period stationary environment between two retailers. This stationary scenario can be viewed as a dynamic game in a duopoly setting. We formulate the single period game and extend it to the two-period dynamic game. We investigate the existence and uniqueness of the feedback Nash equilibrium with two periods to go. We also suggest a threshold inventory level with two periods to go below which the usual substitution effect on the equilibrium may not be observed. We prove the uniqueness of the equilibrium by imposing more structure on the density function of the demand.

Governments, Large Corporations and Labour: A Differential Game Model of Class Conflict

This paper attempts to analyse the problem of how a government allocates resources to different social classes. We combine and extend two existing lines of research in game theoretical studies of political economy, building on Buenrostro et al. [3] who analysed how protests affect governments’ decisions, but not including class conflict, and Acemouglu [1] who does include class conflict but does not take into account protests. We develop a continuous time framework and obtain a stochastic differential game, where the government as well as the upper and lower social classes represent the players involved in the game. The stationary feedback Nash-equilibrium strategies are derived. Furthermore, we compare the results with another two cases where the government and the upper social class maximize their joint utility and the government transfers some resources to the lower social class. We find that under some specific conditions, the government which transfers some resources to the lower social class is more successful in creating a more harmonious society.

On feedback nash equilibrium and cooperation in the neoclassical growth model

Lejos de cualquier punto de vista ideológico, nuestro objetivo en este trabajo es estudiar, dentro de un enfoque diferencial de la teoría de juegos, el modelo neoclásico de crecimiento económico. Nuestro modelo base viene de Kaitala y Pohjola (2009) sobre la base de las ideas originales de Lancaster (1973). Nos centramos en el cálculo de las estrategias de información y el uso de las estrategias histórico-dependientes, como las estrategias de gatillo en el que empezar por cooperar y colaborar se da siempre y cuando los rivales lo hacen, pero al observar una deserción de tal situación implicará entrar de inmediato a un período de castigo a longitud especificada en donde todo el mundo juega de forma no-cooperativa. Se demuestra que las estrategias de gatillo son empleadas por ambos grupos, firmas y trabajadores, para mantener la cooperación como un equilibrio. Entonces llegamos a la conclusión de que las estrategias con memoria de los jugadores son la clave para lograr el máximo crecimiento económico.

Dynamic DTN Routing Strategies Based on Knowledge

This paper presents a novel based-knowledge dynamic routing model with rate allocation for delay-tolerant networks (DTNs). The objectives are how to embody knowledge about destination offered by joint nodes and how to apply knowledge about destination to transfer data when the instantaneous end-to-end paths do not exist in DTNs. The first problem is addressed through a rate allocation model. For rate allocation, taking into account the dynamic feature of rate control as well as the selfish nature of joint nodes, we design a non-cooperative differential game model to offer efficient rate allocation scheme with the constraint of limited bandwidth, and obtain a feedback Nash equilibrium solution of the game. In this study, knowledge is valued information and provides advisable solutions to problems. For the second problem, assuming that the obtained value function in rate allocation model is the available knowledge about destination of joint node; with this assumption, a polynomial time routing algorithm is proposed to describe the procedure of the routing model.