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In this paper, we develop a dynamic control to investigate the hospitals’ quality improvement and congestion reduction, along with corresponding knowledge accumulations. The main features of our work are: (i) developing a dynamic analysis model of the hospitals’ quality improvement and congestion reduction, considering knowledge accumulations in health care markets; (ii) the demand function for medical treatment depends on treatment quality, congestion degree, price, and traveling cost; (iii) each hospital’s instantaneous costs of quality improvement and congestion reduction depend on treatment quality, quality-improving investment, corresponding knowledge accumulation as well as congestion degree, congestion-reducing investment and the corresponding knowledge accumulation; (iv) the hospitals’ dynamic competitions are not only in treatment quality and congestion degree but also in treatment prices. Our results show that (i) there exists a unique saddle-stable steady-state equilibrium under hospital optimum and social optimum; (ii) knowledge accumulation and the complementarity or substitutability affect the hospitals’ decision behavior; (iii) whether the price and the hospitals’ investments in quality improvement and congestion reduction are higher or lower under hospital optimum than that under social optimum depends on the parameter regions of complementarity or substitutability.

The issues of global climate warming and waste product recycling are increasingly severe, highlighting the need for solutions in closed-loop supply chain green innovation and the choice of recycling and sales modes on online platforms. This paper considers the dynamic impact of green innovation and advertising investment on goodwill, using a dynamic game model to explore equilibrium solutions under different modes. Numerical simulations analyze the optimal goodwill trajectory and the impact of commission rates on profits. The motivation of this paper is to provide decision-making support for the selection of closed-loop supply chain business operation modes, to reveal the relationship between green innovation and changes in goodwill, and thus to promote sustainable green closed-loop development. The study finds that the goodwill evolution paths in four decision modes initially decline and then stabilize. As sales commission rates increase, system profits in agency selling and platform agency recycling model (Model AP) and the agency selling and commissioned recycling model (Model AC) gradually decrease. As recycling commission rates rise, profits in model AP and reselling and platform agency recycling model (Model RP) gradually decrease. Under mode AC, manufacturers and network platforms achieve a “Pareto improvement”.

This paper analyzes noncooperative dynamic game behavior in a two-echelon supply chain composed of one supplier and one retailer. The aim is to give supplier reasonable suggestions on decision making of marketing strategy. The challenge of dealing with this problem is the discreteness of demand. The main contributions of this paper are twofold: one is to address the necessary conditions in which supplier can choose quantity discount policy as marketing strategy; the second is to provide the solving methods of the equilibrium strategy and discount rate under quantity discount policy. Two types of quantity discount marketing strategies are considered. It is concluded that the strategy sets of production and procurement are independent of discount rate via analyzing the optimality properties. Numerical tests illustrate that incremental quantity discount policy is more widely considered than all-unit quantity discount policy, and whether the quantity discount policy can be chosen as the marketing strategy not only relates to supplier's own cost structure but also retailer's cost structure.

In this paper, we establish a cold chain dynamic game model including a milk manufacturer and two downstream oligopoly supermarkets under the wholesale price contract in the real world. The manufacturer is responsible for the production and cold transportation, and the two retailers sell the product. The Nash equilibrium points and the complexity of the system are discussed. The influence of the decision parameters and the stability of the system are studied by using complexity theory. We reveal the stable regions for the dynamic system. In addition, revenue sharing contract and profit sharing contract are two valuable contracts. In order to see how the two contracts would impact on the system’s equilibrium solution and the profits, we establish and analyze two new dynamic models for the cold chain. By the comparison of the analyses under three contracts, we find that the manufacturer’s effort of cold transportation will change under different contracts, and the profit distribution of the whole cold chain will be affected. Chaos control is also studied by the method of delay feedback control, in order to provide some management advice.

The multichannel sales mode is employed by more and more manufacturers to maximize their profits or gain much more than market share. This paper develops a dynamic pricing game model of multichannel supply chain, and focuses on the impacts of channel competition on the complex characteristics of the evolutionary dynamic game, using analyses based on stability domain, bifurcation diagram, the largest Lyapunov exponent, attractor domain and time series. The results indicate that the system will gradually enter chaotic state through flip bifurcation from stable state with the increase of price adjustment speed, and the two-dimensional stability domain will shrink as the competition coefficient increases. Interestingly, although an increase in commission rate will lead to a decrease in profits, it will widen the stability range of the whole system. In the end, for further analysis, the chaotic behavior of the multichannel supply chain is controlled by using the feedback control method. The manufacturer is suggested to adopt a cautious pricing strategy in the stability region, and minimize the direct competition between channels in the sales process and diversify the productions in different channels.

This study investigates the dynamic game behaviors of dual-channel supply chains involving an oligopoly manufacturer selling low-carbon products to online and offline retailers. The price game models under government subsidy are discussed under three scenarios: (1) simultaneous decision, (2) manufacturer dominates the market, and (3) retailer dominates the market. The equilibrium strategies are compared under the government subsidy policy. Using numerical simulation, complex characteristics of the dual-channel supply chain under the carbon subsidy policy are investigated. The complexity of wholesale price and sales commission of each channel are analyzed by bifurcation, largest Lyapunov exponent and basin of attraction diagrams. Furthermore, parameter adjustment and delayed feedback control methods are proven to be effective approaches to chaos control.

For achieving the goals of carbon peaking and carbon neutrality, the government will further strengthen subsidies to enterprises for carbon emission reduction. This paper investigates the long-term repeated game behavior of manufacturer and retailer in multichannel supply chains under three different forms of government subsidies. The long-term repeated game behavior of manufacturer and retailer is studied under the manufacturer green innovation subsidy model (IS model), manufacturer production subsidy model (MS model) and retailer sales subsidy model (RS model). Under three situations, excessive price adjustment can easily lead to system instability and sharp fluctuations in profits. Two methods are used for chaos control and management, and the parameter adjustment control method is found having more advantages for system re-stabilization.

The purpose of this paper is to present some simple properties and applications of dynamic games with discrete time and a continuum of players. For such games relations between dynamic equilibria and families of static equilibria in the corresponding static games, as well as between dynamic and static best response sets are examined and an equivalence theorem is proven. The existence of a dynamic equilibrium is also proven. These results are counterintuitive since they differ from results that can be obtained in similar games with a finite number of players.

The theoretical results are illustrated with examples describing voting and exploitation of ecological systems.

In this paper we consider dynamic games with continuum of players which can constitute a framework to model large financial markets. They are called semi-decomposable games.

In semi-decomposable games the system changes in response to a (possibly distorted) aggregate of players' decisions and the payoff is a sum of discounted semi-instantaneous payoffs. The purpose of this paper is to present some simple properties and applications of these games. The main result is an equivalence between dynamic equilibria and families of static equilibria in the corresponding static perfect-foresight games, as well as between dynamic and static best response sets. The existence of a dynamic equilibrium is also proven. These results are counterintuitive since they differ from results that can be obtained in games with a finite number of players.

The theoretical results are illustrated with examples describing large financial markets: markets for futures and stock exchanges.

In the case of natural duopoly, we suggest a finitely repeated game between two incumbent firms and a potential entrant that limits the incumbent firms' power and compels them to approach economic efficiency. We prove that such a game admits a perfect subgame equilibrium along which the incumbent firms maintain with equal quantities while preventing entry. Moreover, the incumbents' strategies along this path converge to average cost pricing quantities as the number of periods goes to infinity.

Constructing a dynamic game model of trade of an exhaustible resource, this paper compares feedback Nash and Stackelberg equilibria. We consider two different leadership scenarios: leadership by the importing country, and leadership by the exporting country. We numerically show that as compared to the Nash equilibrium, both countries are better off if the importing country is a leader, but that the follower is worse off if the exporting country is a leader. Consequently, the world welfare is highest under the importing country's leadership and lowest under the exporting country's leadership.

This paper presents a general class of dynamic network games to analyze trade with technology spillover. Due to the fact that the benefits of technology spillover are not fully accrued to the technology developer, the positive externalities are under-exploited. The cooperative solution yields an optimal outcome. To reflect the contributions of individual agents to the network, the Shapley value is used as a solution optimality principle in sharing the cooperative gains. A time-consistent payoff imputation procedure is derived to maintain the Shapley value at each stage of the cooperation. A representative model based on the general class of network games with explicit functional form is given. This is the first time that trade with technology spillover is studied in the framework of dynamic network games, further studies along this line are expected.

We consider a discrete-time, infinite-horizon dynamic game of groundwater extraction. A Water Agency charges an extraction cost to water users and controls the marginal extraction cost so that it depends not only on the level of groundwater but also on total water extraction (through a parameter $n$ that represents the degree of strategic interactions between water users) and on rainfall (through parameter $m$). The water users are selfish and myopic, and the goal of the agency is to give them incentives so as to improve their total discounted welfare. We look at this problem in several situations. In the first situation, the parameters $n$ and $m$ are considered to be fixed over time. The first result shows that when the Water Agency is patient (the discount factor tends to 1), the optimal marginal extraction cost asks for strategic interactions between agents. The contrary holds for a discount factor near 0. In a second situation, we look at the dynamic Stackelberg game where the Agency decides at each time what cost parameter they must announce. We study theoretically and numerically the solution to this problem. Simulations illustrate the possibility that threshold policies are good candidates for optimal policies.

This paper formulates a dynamic game between the governments of two countries that share a common stock of natural capital (such as environmental quality). The objective of each government is to find a sustainable utility path that satisfies the sequential maximin property. A utility path is a sequential maximin if it survives all successive rounds of eliminating Pareto inferior outcomes based on the maximin criterion. It is shown that, under the sequential maximin objective, there exists a Markov perfect Nash equilibrium in which utility is constant over time. This equilibrium turns out to be Pareto efficient, in sharp contrast to the typical inefficiency of Markov perfect Nash equilibria under the usual discounted utilitarian objective. We also find a Stackelberg equilibrium that results in sustainable development, in which the welfare of the leader is lower (and that of the passive follower is higher) than the symmetric welfare level in the Nash equilibrium.

In this paper we consider a game modelling a market consisting of two firms with market power and a continuum of consumers. A specific feature of a market for toys is considered with each firm producing two kinds of distinguishable goods. The problem of finding a Nash equilibrium implies firms' optimal advertising and production plans over time, where the aggregate of demands of consumers may depend on firms' past decisions. Equilibria at this market may have strange properties, like oscillatory production and advertising strategies.

The decisions made by petroleum producers in the world oil market are both dynamic and strategic, and are thus best modeled as a dynamic game. In this chapter, we review the literature on the world oil market and discuss our research on econometric modeling of the world oil market as a dynamic game. Our research builds on the previous literature by combining three erstwhile separate dimensions of modeling the world oil market: dynamic optimization, game theory, and econometrics. Our results show that dynamic behavior and strategic interactions are important aspects of the world oil market that must be accounted for in empirical analyses of the world oil market.