Non-cooperative Dynamic Game Research Articles

Synchronization of Coupled Oscillators is a Game

The purpose of this paper is to understand phase transition in noncooperative dynamic games with a large number of agents. Applications are found in neuroscience, biology, and economics, as well as traditional engineering applications. The focus of analysis is a variation of the large population linear quadratic Gaussian (LQG) model of Huang et al. 2007, comprised here of a controlled N-dimensional stochastic differential equation model, coupled only through a cost function. The states are interpreted as phase angles for a collection of heterogeneous oscillators, and in this way the model may be regarded as an extension of the classical coupled oscillator model of Kuramoto. A deterministic PDE model is proposed, which is shown to approximate the stochastic system as the population size approaches infinity. Key to the analysis of the PDE model is the existence of a particular Nash equilibrium in which the agents opt out' of the game, setting their controls to zero, resulting in the incoherence' equilibrium. Methods from dynamical systems theory are used in a bifurcation analysis, based on a linearization of the partial differential equation (PDE) model about the incoherence equilibrium. A critical value of the control cost 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.

Strategic upfront marketing channel integration as an entry barrier

This paper investigates an organizational design problem concerning whether duopolistic firms competing in a product market should vertically integrate or separate their marketing channels in a dynamic noncooperative game setting. Previous operational research models have shown that the separation of the marketing channel with the adoption of a two-part tariff contract is the dominant strategy compared with integration for each firm if the two firms face retail price competition, and thereby constitutes the subgame perfect Nash equilibrium (SPNE). Contrary to this previous insight, this paper demonstrates that if exogenous parameters that characterize fixed costs, product substitutability, and a demand function fall into a specific region, marketing channel integration dominates the separation strategy when one of the two firms is the incumbent firm while the other is a potential entrant. In other words, the well-known result in the price-setting game can be reversed when we take entry threats into consideration. Specifically, we show that upfront vertical integration of the marketing channel enables the incumbent to deter the entry of the potential competitor and to monopolize the market in the SPNE. This result has operational implications for a firm confronting the threat of potential rivals entering the market, in that the firm can use this apparently inferior strategy as a commitment device, which creates a virtual entry barrier.

On the Core of Dynamic Cooperative Games

We consider dynamic cooperative games, where the worth of coalitions varies over time according to the history of allocations. When defining the core of a dynamic game, we allow the possibility for coalitions to deviate at any time and thereby to give rise to a new environment. A coalition that considers a deviation needs to take the consequences into account because from the deviation point on, the game is no longer played with the original set of players. The deviating coalition becomes the new grand coalition which, in turn, induces a new dynamic game. The stage games of the new dynamical game depend on all previous allocation including those that have materialized from the deviating time on.We define three types of core solutions: fair core, stable core and credible core. We characterize the first two in case where the instantaneous game depends on the last allocation (rather than on the whole history of allocations) and the third in the general case. The analysis and the results resemble to a great extent the theory of non-cooperative dynamic games.

A game-theoretic framework for the security system of visible watermarking

Perfect digital watermarking systems are contained two characteristics. One is the robustness that it need be resistant to geometric attacks in addition to common image processing tasks, such as JPEG compression. The other one is to preserve the image quality after embedding. However, the requirement of robustness and quality for watermark are conflicted with each other. How to determine the equilibrium of optimal security strategies between encoder and attacker and the optimal tradeoff between the intensity of embedded watermark and the perceptual translucence for visible watermark is still remained as one of the most challenging research topics in image watermarking. Therefore, in order to achieve the best tradeoff between the embedding energy of watermark, the quality of perceptual watermark translucence and the image fidelity after attacks, we propose a system architecture which is based on the game-theoretic approach that provides an optimum solution for the decision maker by studying the intensity and perceptual efficiency. The game-theoretic approach determines the transmission strategy using utility optimization according to the fluctuation of watermark states. The watermark embedding problem is formulated as a dynamic non-cooperative game with complete information while the optimal strategy is defined by the Nash equilibrium of the game. The experimental results demonstrate the feasibility of the proposed approach which allows the watermark encoder to obtain the best adaptive watermarking strategy in the different texture under attacks. Additionally, we demonstrate that the proposed system could help each user to choose the optimal transmission power to maximize its utility based on other constant parameters and resolve security issue of visual communication.

Coordinating pricing and inventory decisions in a multi-level supply chain: A game-theoretic approach

This paper concerns coordination of enterprise decisions such as suppliers and components selection, pricing and inventory in a multi-level supply chain composed of multiple suppliers, a single manufacturer and multiple retailers. The problem is modeled as a three-level dynamic non-cooperative game. Analytical and computational methods are developed to determine the Nash equilibrium of the game. Finally, a numerical study in computer industry is conducted to understand the influence of the market scale parameter and the components selection strategy on the optimal decisions and profits of the supply chain as well as its constituent members. Several research findings have been obtained.

Supply quality management with wholesale price and revenue-sharing contracts under horizontal competition

In a number of industries (e.g., the airplane industry, aerospace industry, auto industry, or computer industry), certain suppliers essentially have a monopoly on the production technology for key components, and inevitably manufacturers in these industries have common suppliers. A key part of manufacturers’ work with suppliers concerns improving the quality of their respective products, which gives rise to a collaborative activity usually termed as “supply quality management”. When the manufacturers are competitors, they do not wish to see a common supplier dividing his involvement in quality improvement unequally between themselves and their rivals. However, as the suppliers collaborate with several manufacturers, it is highly questionable whether their efforts will be strictly equivalent for each manufacturer. In this paper, a non-cooperative dynamic game is formulated in which a single supplier collaborates with two manufacturers on design quality improvements for their respective products. The manufacturers compete for market demand both on price and design quality. The paper analyzes how each party should allocate resources for quality improvement over time. In order to take into account the potential coordinating power of the compensation scheme adopted in this type of decentralized setting, we compare the possible outcomes under a wholesale price contract and a revenue-sharing contract.

Search for maximum points of a vector criterion based on decomposition properties

A multiplayer dynamic noncooperative game is considered, in which the players make decisions by maximizing individual utility functions. In each round of the game, information exchange is performed by means of a mechanism similar to the Walras auction. The notion of market equilibrium is introduced, which combines the properties of a Nash equilibrium and Pareto maximum. A theorem on the existence of such an equilibrium is proved. An algorithm for searching for a market equilibrium is proposed, which shifts a Nash competitive equilibrium to a Pareto cooperative maximum. The algorithm is interpreted in the form of a repeated auction, in which the auctioneer has no information about the utility functions of the players. The players, in turn, have no information about the utility functions of other participants. In each round of the stepwise auction, individual interest rates are proposed to the players, based on which they maximize their utility functions. Then, the players give their best replies to the auctioneer. The auctioneer’s strategies of forming interest rates that provide conditions for reaching a market equilibrium are considered. From the game-theoretical point of view, the repeated auction describes the learning process in a noncooperative repeated game under uncertainty.

Pursuit Evasion: The Herding Non-Cooperative Dynamic Game

A class of pursuit evasion problems is studied. This problem involves a dog agent herding a agent in order to take the sheep to a pen. The problem is stated in terms of the allowable sequential actions of the two agents and the game being played because of the choices each agent has. The solution is obtained using the dynamic programming principle applied in the game setting. The algorithm is analyzed and simulation results are presented.

A Model of Non-Cooperative Dynamic Game to Conflict Resolution Among Common Natural Resource Operators

A Model of Non-Cooperative Dynamic Game to Conflict Resolution Among Common Natural Resource Operators

The Strategic Value of Unilateral Abatement in Games of Climate Change Policy

Unilateral abatement of greenhouse gas emissions is not often explicitly considered in game-theoretic models of climate policy adoption and in reality is often justified on grounds of non-climatic benefits. We develop one non-cooperative static game and two non-cooperative dynamic games that show how allowing players to respond to commitments to unilaterally abate emissions expands the range of parameter values that support joint abatement. Importantly, joint abatement occurs only via initial unilateral abatement whenever the parameter values support joint abatement in the two dynamic games but not in the static game. The dynamic games also show that if unilateral abatement lowers costs for later abaters, then still more states of the world support joint abatement. When assessing the climatic benefits of unilateral abatement, we should consider not only the direct reductions in the abater's emissions but also the effect on other emitters' future adoption of climate policies. Unilateral abaters may want to design their climate policy instruments so as to encourage future abatement by others, which could mean focusing on the promotion of technological innovation and diffusion and on providing policy models that others could adapt to their own contexts.

Consensus in Noncooperative Dynamic Games: A Multiretailer Inventory Application

We focus on Nash equilibria and Pareto optimal Nash equilibria for a finite horizon noncooperative dynamic game with a special structure of the stage cost. We study the existence of these solutions by proving that the game is a potential game. For the single-stage version of the game, we characterize the aforementioned solutions and derive a consensus protocol that makes the players converge to the unique Pareto optimal Nash equilibrium. Such an equilibrium guarantees the interests of the players and is also social optimal in the set of Nash equilibria. For the multistage version of the game, we present an algorithm that converges to Nash equilibria, unfortunately, not necessarily Pareto optimal. The algorithm returns a sequence of joint decisions, each one obtained from the previous one by an unilateral improvement on the part of a single player. We also specialize the game to a multiretailer inventory system.

Game theory and power control in ultrawideband networks

This paper describes a theoretical framework for the design and analysis of power control algorithms for high-throughput wireless networks using ultrawideband (UWB) technologies. The tools of game theory are shown to be expedient for deriving scalable, energy-efficient, distributed power control schemes to be applied to a population of battery-operated user terminals in a rich multipath environment. In particular, the power control issue is modeled as a dynamic noncooperative game in which each user chooses its transmit power so as to maximize its own utility, which is defined as the ratio of throughput to transmit power. Although distributed (noncooperative) control is known to be suboptimal with respect to the optimal centralized (cooperative) solution, it is shown via large-system analysis that the game-theoretic distributed algorithm based on Nash equilibrium exhibits negligible performance degradation with respect to the centralized socially optimal configuration. The framework described here is general enough to also encompass the analysis of code division multiple access (CDMA) systems and to show that UWB slightly outperforms CDMA in terms of achieved utility at the Nash equilibrium.

HUMAN FERTILITY DECISIONS AND COMMON PROPERTY RESOURCES: A DYNAMIC ANALYSIS

ABSTRACT. In rural areas of developing countries, parental decisions on number of offspring may be made on the basis of the role of children in harvesting local common property renewable resources. It has been argued that this may lead to a cycle of human over‐population and resource over‐exploitation. To investigate the plausibility of this argument, we present a discrete dynamic model with two state variables representing human population level N and resource stock level S. The model is similar to one given by Nerlove and Meyer but differs in several important respects. It is assumed that, in each over‐lapping generation of parents and children, parents decide how many children to have based on their resulting share of the local resource harvest and the costs associated with child‐rearing. Using simulation and analytical methods, the long term steady state population and resource stock levels for this dynamic noncooperative game are contrasted with the steady state when parental fertility decisions are made in a cooperative manner.

Capacity utilization and investment in environmental quality

This paper characterizes the optimal use of productive capacity and optimal investment in environmental quality when the latter has a positive impact on the production process. For the case of a single country, we find conditions under which capital should or should not be fully utilized, and investment in environmental quality should be positive or zero. We then extend the model to the case of two countries playing a non-cooperative dynamic game. The Nash equilibrium turns out to be a dominant-strategy equilibrium. Since this equilibrium is not Pareto efficient, we show how one country may bribe the other country to achieve a better outcome, for example, by refraining from full utilization of capacity. Under certain conditions, the optimal solution requires that a constant fraction of one country's income be used to bribe the other country to scale down its production.

COARSE-GRAINED PARALLELIZATION OF THE ADVANCED NEUROSIMULATOR FAUN 1.0 WITH PVM AND THE ENHANCED CORNERED RAT GAME REVISITED

Today artificial neural networks are very useful to solve complex dynamic games of various types, i.e., to approximate optimal strategies with sufficient accuracy. Exemplarily four synthesis approaches for the solution of zero-sum, noncooperative dynamic games are outlined and discussed. Either value function, adjoint vector components or optimal strategies can be synthesized as functions of the state variables. In principle all approaches enable the solution of dynamic games. Nevertheless every approach has advantages and disadvantages which are discussed. The neural network training usually is very difficult and computationally very expensive. The coarse-grained parallelization FAUN 1.0-HPC-PVM of the advanced neurosimulator FAUN uses PVM subroutines and runs on heterogeneous and decentralized networks interconnecting general-purpose workstations, PCs and also high-performance computers. Computing times of days, weeks or months can be cut down to hours. An enhanced cornered rat game — formulated and analyzed in 1993 — serves as an example. Optimal strategies for cat and rat are synthesized. For this purpose open-loop representations of optimal strategies on an equidistant grid in the state space are used. An important end game modification is presented.

OPEN-LOOP AND CLOSED-LOOP MODELS OF DYNAMIC OLIGOPOLY IN THE CRUISE LINE INDUSTRY

In this paper, the competition of dynamic oligopoly in the cruise line industry is modeled as an N-person nonzero-sum noncooperative dynamic game where a finite number of cruise lines compete to maximize their profits over a fixed planning horizon. The noncooperative Nash equilibrium capacity investment strategies of cruise lines are theoretically analyzed under the open-loop and closed-loop information structures. The optimality conditions for open-loop and closed-loop Nash equilibrium solutions are derived using a Pontryagin-type maximum principle and given economic interpretations so as to demonstrate the differences between the open-loop and closed-loop Nash equilibrium solutions. The dynamic oligopolistic competition of three cruise lines in a hypothetical setting is numerically analyzed by using the iterative algorithms for open-loop and closed-loop models. Numerical results provide a number of important managerial guidelines for cruise capacity investment decisions. The paper concludes with a discussion on future research directions.

Generating unit maintenance scheduling under competitive market environments

A novel approach to a generating unit maintenance scheduling problem in competitive electricity markets is presented in this paper. The objective is to develop a game-theoretic framework for analyzing strategic behaviors of generating companies (Gencos) from the standpoint of the generating unit maintenance scheduling (GMS) game and for obtaining the equilibrium solution for the GMS game. The GMS problem is formulated as a dynamic non-cooperative game with complete information. The players correspond to profit maximizing individual Gencos, and the payoff of each player is defined as the profits from the energy market. The optimal schedule is defined by Nash equilibrium (equilibriums) of the game. Numerical results for two-Genco system are used to demonstrate that the proposed framework can be successfully applied to analyzing the strategic behaviors of each Genco and to obtaining the corresponding Nash equilibrium. The result indicates that generating unit maintenance schedule is one of the major strategic behaviors whereby Gencos maximize their profits in a competitive market environment.

Distributed Scheduling and Dynamic Pricing in a Communication Network

This paper studies dynamic resource allocation in a decentralized communication network. The temporal aspect in the decentralized resource allocation problem presents new challenges, e.g., in optimizing the delay-throughput trade-off under user-specific delay costs. A dynamic bandwidth allocation game modelling an agent-based network is presented. The dynamic noncooperative game achieves Pareto-efficient bandwidth allocation that can be implemented by a greedy algorithm with pricing. Optimal dynamic pricing is discussed for the efficient sharing of network resources. An ad hoc wireless network is an example of such self-organizing decentralized system: the mobile nodes need not be directly connected to a base station. Another application of the model is to consider distributed uplink scheduling, based on local information, in a WCDMA network. The discretized control variable of a mobile node is either the received power/QoS-level or the binary decision on packet transmission.

A dynamic game model of strategic capacity investment in the cruise line industry

In this paper, the cruise line industry is characterized as an oligopoly where a finite number of cruise lines compete to maximize their profits over a fixed planning horizon. The oligopolistic competition is modeled as a N-person nonzero-sum noncooperative dynamic game. The noncooperative Nash equilibrium capacity investment strategies of cruise lines are theoretically analyzed under the open-loop information structure. The necessary conditions for an open-loop Nash equilibrium solution are derived using a Pontryagin-type maximum principle and the sufficient conditions are also established. The economic interpretation of the optimality conditions is given so as to demonstrate the difference between the static and dynamic oligopoly problems. The dynamic oligopolistic competition of three cruise lines in a hypothetical setting is also numerically analyzed by using the iterative algorithm. Numerical results provide a number of important managerial guidelines for cruise capacity investment decisions. The paper concludes with a discussion on future research directions.

A new game-theoretic framework for maintenance strategy analysis

In this paper, a new approach to maintenance scheduling of generating units (MSU) in competitive electricity markets is presented. The main contribution of this study can be focused on modeling a game-theoretic framework for the MSU problem to analyze strategic behaviors of generating companies (Gencos). The MSU problem is formulated as a dynamic noncooperative game with complete information. The player corresponds to the profit maximizing individual Genco, and the payoff of each player is defined as the profit from the energy auction market. The optimal strategy profile is defined by the Nash equilibrium of the game. Numerical results for two-Genco systems are used to demonstrate the basic ideas of the proposed scheme, and large-scale IEEE reliability test systems are also considered to show the applicability of the proposed framework. The results obtained indicate that maintenance schedule can be one of the crucial strategic behaviors whereby Genco maximizes their profit in a competitive market environment.