Differential Game Approach Research Articles

Towards Cyber–Physical Systems Robust to Communication Delays: A Differential Game Approach

Collaboration between interconnected cyber-physical systems is becoming increasingly pervasive. Time-delays in communication channels between such systems are known to induce catastrophic failure modes, like high frequency oscillations in robotic manipulators in bilateral teleoperation or string instability in platoons of autonomous vehicles. This paper considers nonlinear time-delay systems representing coupled robotic agents, and proposes controllers that are robust to time-varying communication delays. We introduce approximations that allow the delays to be considered as implicit control inputs themselves, and formulate the problem as a zero-sum differential game between the stabilizing controllers and the delays acting adversarially. The ensuing optimal control law is finally compared to known results from Lyapunov-Krasovskii based approaches via numerical experiments.

A differential game approach to intrinsic formation control

This paper addresses the formation control problem of a multi-agent system in a non-cooperative differential game framework. Both finite horizon and infinite horizon games are considered and their Nash equilibria are studied. The desired formation patterns are achieved by Nash equilibrium strategies in an intrinsic way in the sense that they are only attributed to the inter-agent interaction and geometric properties of the network, where the desired formations are not designated directly in the controller. The whole formation manifold of the desired relative pattern is studied by allowing all orientations of the formation and all permutations of the agents. For finite horizon games the terminal formation of Nash equilibrium trajectories is shown to converge to desired pattern as the length of the time interval tends to infinity. Furthermore the asymptotic stability of the desired formation manifold is also guaranteed in infinite horizon games. Relative patterns of regular polyhedra and antipodal formations are achieved by designing the interaction graph while inter-agent collisions are avoided. Finally, numerical simulations are provided to demonstrate the effectiveness and feasibility of the proposed methods.

Advertising in an oligopoly with differentiated goods under general demand and cost functions: A differential game approach

AbstractWe present an analysis of advertising activities in a dynamic oligopoly with differentiated goods using a differential game approach under general demand and cost functions. The main conclusion is that the memoryless closed‐loop solution and the feedback solution are equivalent when there is no spillover effect of advertising activities. We also show that the comparison of the open‐loop solution and the memoryless closed‐loop solution depends on whether the firms’ outputs are strategic substitutes or strategic complements.

Differential game approach to pricing and advertising decisions

This study proposes a model to make concurrent decisions on dynamic pricing and advertising to maximise firms' profitability over an infinite time horizon in a duopoly market. To this end, the Nerlove-Arrow pricing and advertising model is designed in the presence of shifting costs in a dynamic duopolistic competition as a differential game. The Nash equilibrium solution is defined based upon a set of Hamilton–Jacobi–Bellman. Four scenarios are applied for economic interpretations and the efficacy of the model.

The Role of Cultural Capital on the Voluntary Contributions to Cultural Goods: A Differential Game Approach

This study aims to offer a formal analysis which relates provision of cultural goods to the society’s level of cultural capital. Such a level is assumed to already exist in society and is increasable thanks to individual support for the offer of cultural goods. The achievement of the highest levels of cultural capital increases satisfaction coming from cultural goods consumption, and then voluntary contributions. Social approval, deriving from donations, is positively related to society’s existing cultural capital and triggers a positive externality for donators, thus increasing contributions and generating a positive externality for the whole society. The dynamic analysis provided in this study requires the adoption of a differential game where individuals interact, making their choices on their voluntary contribution level. We find that, under certain conditions, the solution obtained for the Nash equilibrium with closed-loop strategies provides optimal level of cultural capital that exceed the Pareto efficient solution obtained through open-loop strategies.

Robust multi-agent differential games with application to cooperative guidance

Robust multi-agent differential games and its application to cooperative guidance are investigated in this paper. The multi-agent system is modeled by general linear dynamics with norm-bound parameter uncertainties. The players in the game are divided into two teams. One team consists of a leader, and the other team consists of a fixed number of followers agents that aim to achieve leader-following consensus. The two teams constitute the adversaries of a two-player zero-sum differential game. Based on quadratic stabilization techniques, a set of saddle point strategies of the game are designed to stabilize the closed-loop multi-agent system. By modifying the optimal solution of the multi-agent differential game of the nominal model, the sufficient conditions are presented such that the stabilization of the uncertain system is guaranteed. It is proved that the modified solution achieves quasi-optimality. It is also shown that L2-bounded consensus error can be achieved, if the control of the leader deviates from its quasi-optimal strategy. The model of cooperative guidance problem is established, based on which the robust multi-agent differential games approach is applied to design the cooperative guidance laws. Numerical examples are given to verify the effectiveness of the theoretical results and the performance of the proposed cooperative guidance law.

Coordinating Carbon Emissions via Production Quantities: A Differential Game Approach

Production emissions in the industrial sector are a major source of environmental pollution. In this paper, we explore how emission considerations are integrated with production decisions. We develop a dynamic model consisting of two firms located in the same industrial park, which satisfies exogenously given demands in separate markets. The two firms can build up or rundown stocks (full backlogging), both of which are costly. The emission cost depends on the total output of the two firms. We develop Nash equilibrium feedback strategies, where each firm decides on its output based on its inventory or the inventories of both. We also develop a social planning solution where decisions are centralized. We present the analytic results for the total profits in these settings. The results show the benefits of a decentralized approach over a centralized one, provided there is a mechanism for coordination. Finally, emission costs are compared for the various solution concepts.

Water taxes and fines imposed on legal and illegal firms exploiting groudwater

<p style='text-indent:20px;'>This paper uses a differential game approach to investigate a model that represents the exploitation of groundwater, taking into account the strategic and dynamic interactions among users of the resource and public authority. Agents' behaviour may influence their gains but also the overexploitation of the aquifer. The effects of legal and illegal firms' actions and the contribution of taxes and penalties imposed by public authorities, are analysed by studying Feedback equilibria in order to capture the problem of non-compliance with resource management regimes and to discuss policy options in a non-cooperative and cooperative context. We show that illegal extractions can be a significant stumbling block on the path towards implementing of better management and environmental policies and we explain how, in order to fight this phenomenon, the public authority must increase controlled activity rather than taxation, but also encourage cooperation between legal firms under appropriate conditions.</p>

Dynamic Equilibrium of Market Making with Price Competition

In this paper, we discuss the dynamic equilibrium of market making with price competition and incomplete information. The arrival of market sell/buy orders follows a pure jump process with intensity depending on bid/ask spreads among market makers and having a looping countermonotonic structure. We solve the problem with the nonzero-sum stochastic differential game approach and characterize the equilibrium value function with a coupled system of Hamilton–Jacobi nonlinear ordinary differential equations. We prove, do not assume a priori, that the generalized Issac’s condition is satisfied, which ensures the existence and uniqueness of Nash equilibrium. We also perform some numerical tests that show our model produces tighter bid/ask spreads than those derived using a benchmark model without price competition, which indicates the market liquidity would be enhanced in the presence of price competition of market makers.

Optimal capital structure and the debtholder-manager conflicts of interests: a management decision model

PurposeDetermining the optimal capital structure becomes more complicated by the presence of an agency problem. The issuance of debt as a corporate governance mechanism introduces the asset substitution problem – the agency cost of debt. Thus, there is a recognized need for models that can resolve the agency problem between the debtholder and the manager who acts on behalf of the shareholder, leading to optimal capital structure choice, and enhanced firm value. The purpose of this paper is to model the debtholder-manager agency problem as a dynamic game, resolve the conflicts of interests and determine the optimal capital structure.Design/methodology/approachAs there is no satisfactory model for dealing with the above issues, this paper uses a differential game framework to analyze the incongruity of interests between the debtholder and the manager as a non-cooperative dynamic game and further resolves the conflicts of interests as a cooperative game via a Pareto-efficient outcome.FindingsThe optimal capital structure required to minimize the marginal cost of the agency problem is a higher use of debt, lower cost of equity and withheld capital distributions. The debtholder is also able to enforce cooperation from the manager by providing a lower and stable cost of debt and a greater debt facility in the overtime framework.Originality/valueThe study develops a new dynamic contract theory model based on the integrated issues of capital structure, corporate governance and agency problems and applies the differential game approach to minimize the agency problem between the debtholder and the manager.

Leveraging online customer reviews in new product development: a differential game approach

Large volumes of online product reviews generated by customers have important strategic values for new product development. We consider a duopoly setting where two manufacturers aim to develop their own new products and services. Applying a differential game framework, we examine how online customer reviews can be leveraged as external knowledge for manufacturers to develop new products. In our base models, we assume that the products supplied by the manufacturers are homogenous. First, we consider a closed innovation setting as a benchmark case in which both manufacturers develop new products by their internal R&D without leveraging online customer reviews. Second, we propose a model in which one manufacturer leverages online customer reviews as external knowledge, while the other manufacturer only relies on internal R&D effort. We derive analytical equilibrium solutions to both models. We find that when one manufacturer uses online customer reviews, if the manufacturer’s R&D process becomes more effective in improving its new product performance or reducing its cost, it certainly hurts the other manufacturer, but it may sometimes hurt this particular manufacturer as well. Furthermore, we demonstrate that when the manufacturer utilizes online customer reviews more in R&D, both manufacturers’ profits can either increase or decrease. In an extended model, we relax the product homogeneity assumption and obtain the equilibrium solution analytically. We show that main managerial insights still hold in the extend model.

Groundwater extraction among overlapping generations: a differential game approach

Groundwater is a common resource that has been wasted for years. Today, we pay the consequences of such inappropriate exploitation and we are aware that it is necessary to realize policies in order to guarantee the use of this resource for future generations. In fact, the irrational exploitation of water by agents, nevertheless it is a renewable resource, may cause its exhaustion. In our paper, we develop a differential game to determine the efficient extraction of groundwater resource among overlapping generations. We consider intragenerational as well as intergenerational competition between extractors that exploit the resource in different time intervals, and so the horizons of the players in the game are asynchronous. Feedback equilibria have been computed in order to determine the optimal extraction rate of “young” and “old” agents that coexist in the economy. The effects of the withdrawal by several generations are numerically and graphically analyzed in order to obtain results on the efficiency of the groundwater resource.

Optimal extraction and taxation of strategic natural resources: a differential game approach

This paper studies the optimal extraction and taxation of nonrenewable natural resources. It is well known that the market values of the main strategic resources such as oil, natural gas, uranium, copper,..., etc, fluctuate randomly following global and seasonal macroeconomic parameters, these values are modeled using Markov switching L\'evy processes. We formulate this problem as a differential game. The two players of this differential game are the mining company whose aim is to maximize the revenues generated from its extracting activities and the government agency in charge of regulating and taxing natural resources. We prove the existence of a Nash equilibrium. The corresponding Hamilton Jacobi Isaacs equations are completely solved and the value functions as well as the optimal extraction and taxation rates are derived in closed-form. A Numerical example is presented to illustrate our findings.

Linear quadratic differential game approach for attitude takeover control of failed spacecraft

Spacecraft is mostly abandoned after its fuel exhausts or actuator fails. Using microsatellites to implement attitude takeover control for failed spacecraft is a cost efficient way to recycle on-board valuable payloads at a low cost. In order to achieve collaboration between independent microsatellites, a differential game based method is proposed to stabilize the combined spacecraft composed of multiple microsatellites and the failed spacecraft, in which each microsatellite can independently calculate its own control strategy. Firstly, based on the attitude model of combined spacecraft and the performance index function of each microsatellite, the linear quadratic differential game model is formulated. By optimizing individual performance index functions, the control strategies of multiple microsatellites can be obtained. Secondly, in order to find the Nash equilibrium strategies of microsatellites, the coupled Hamilton-Jacobi(HJ) equations are transferred to a set of coupled Riccati differential equations over finite time horizon. In order to avoid computational overhead, a set of coupled algebraic Riccati equations over infinite horizon is also derived, which can be solved easily with high precision by the Lyapunov iterations method. Finally, numerical simulations verify the effectiveness of the differential game method for the attitude takeover control of failed spacecraft.

Partially-observed decentralized optimal control for large population two-wheeled vehicles: A differential game approach

In this paper, we consider the decentralized noncooperative optimal control problem for a large population of two-wheeled unmanned vehicles with partial observations. Specifically, an individual vehicle is controlled only by its local noisy information measured from a local sensor in order to follow the average behavior (mean field) of the entire population while achieving the overall optimal control performance. We solve the partially-observed linear-quadratic mean field game to obtain decentralized optimal controls for two-wheeled vehicles. These controls are decentralized since they are functions of the local estimated state from the local Kalman filter. We show that the set of the decentralized optimal controls constitutes an ϵ-Nash equilibrium, where ϵ converges to zero as the number of vehicles becomes large. Finally, the theoretical results are validated through simulations and experiments with various operation scenarios for a large population of two-wheeled vehicles.

The Intelligent Offense and Defense Mechanism of Internet of Vehicles Based on the Differential Game-IP Hopping

The Road Side Unit (RSU) is an important part of the Internet of Vehicles (IoV), which collects a large number of user's privacy information and connects vehicles to the Internet. The current RSU is running with static configurations, and the adversary can get enough time to analyze the vulnerability of the RSU and launch attacks, which poses a huge threat to the privacy security of the IoV. Under this asymmetrical condition, the attackers occupy a predominant position, and it is difficult to fully defend. To solve the above asymmetry offense-defense problem, we adopt the moving target defense (MTD) to dynamically change the IP address of the RSU, which increases the difficulty of the attacker's attack. However, due to the unreasonable selection of the RSU's IP hopping frequency, it is difficult for the RSU's IP hopping to exert its defense effect or it may introduce excessive defense overheads. Given the above issue, we propose an intelligent offense and defense mechanism based on a differential game approach, which can adaptively adjust the RSU's IP hopping frequency and maximize its defense benefits. First, based on the offense and defense relationship, we model the RSU's IP hopping frequency intelligent arrangement as a differential game (i.e., IP hopping game) in which the attacker attempts to maximize his potential benefits, and the RSU manages to maximize its defense benefits. On this basis, we calculate a potential Nash equilibrium of the IP hopping game. Finally, simulation results show our proposed mechanism can adaptively adjust the RSU's IP hopping frequency according to the attack frequency and effectively defend attacks. Compared with the mechanisms with a fixed IP hopping frequency, our proposed mechanism can make up for the ineffectiveness of the fixed low-frequency IP hopping defense and avoid the excessive defense overheads of the fixed high-frequency IP hopping defense.

A Game Theoretical Approach to Homothetic Robust Forward Investment Performance Processes in Stochastic Factor Models

This paper studies an optimal forward investment problem in an incomplete market with model uncertainty, in which the dynamics of the underlying stocks depends on the correlated stochastic factors. The uncertainty stems from the probability measure chosen by an investor to evaluate the performance. We obtain directly the representation of the power robust forward performance process in factor-form by combining the zero-sum stochastic differential game and ergodic BSDE approach. We also establish the connections with the risk-sensitive zero-sum stochastic differential games over an infinite horizon with ergodic payoff criteria, as well as with the classical power robust expected utility for long time horizons.

Ship differential game approach for multiple players: Stackelberg security games

SummaryCasting the problem as a pursuit‐evasion behavior in continuous time, this paper addresses a class of multiplayer ship differential Stackelberg security game. We are concerned with the conditions under which the defenders can capture the attackers. We represent the Stackelberg game as a Nash game for relaxing the interpretation of the noncooperative solution and the equilibrium selection problem. The weights of the players for the Nash solution are determined by their role in the Stackelberg game. The defenders try to minimize the capture condition. The attackers, knowing that they are being pursued by defenders, try to maximize the capture condition and minimize the distance to a certain target. For computing the equilibrium of the game, we employ a saddle‐point method approach. The method consists of two half‐steps iterated procedure where the functional of the game decrease and finally converges to an equilibrium point. We present the analysis of the convergence. Finally, we give a numerical example to illustrate the effectiveness and usefulness of our approach.

Optimal control for transboundary pollution under ecological compensation: A stochastic differential game approach

Abstract To account for previously ignored, yet widely observed uncertainty in nature's capability to replenish the natural environment in ways that should inform ideal design of ecological compensation (EC) regimes, this study constructs a stochastic differential game (SDG) model to analyze transboundary pollution control options between a compensating and compensated region. Equilibrium strategies in the stochastic, two player game inform optimal control theory and reveal a welfare distribution mechanism to form the basis of an improved cooperative game contract. A case-based numerical example serves to verify the theoretical results and supports three key insights. First, accounting for various random disturbance factors, the probabilistic pollutant stock in Stackelberg non-cooperative game exceeds that of a cooperative game situation. Second, the EC mechanism provides long-term, effective incentives only when the marginal losses of environmental damage in the compensating region are more than twice that of the compensated region. Achieving a Pareto optimal equilibrium relies upon the attainment of a dynamic allocation ratio derived from the analysis of a robust welfare allocation mechanism. Third, cross-region cooperation reliably outperforms Stackelberg non-cooperation due to either overwhelming incumbent economic interests or high abatement costs. This study illuminates the importance of balancing both parties' interests within an EC agreement while reducing uncertainty around unobserved environmental factors during ex-ante negotiations.

Political business cycles in a dynamic bipartisan voting model

We provide a microfoundation for the existence of political business cycles generalizing and extending the classic Nordhaus model. Using a differential games approach, we model the strategic interaction of legislators in an election period and show that cycles can exist in equilibrium. The existence and shape of cycles depend on the parameters of the model and the initial level of unemployment. In addition, we offer an extension of the model to incorporate presidential veto behavior.