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The environmental deterioration caused by carbon emissions has enhanced people’s awareness of environmental protection. Governments, businesses and consumers are expected to reduce carbon emissions while ensuring their daily activities. Therefore, based on the background of carbon emission reduction, this paper explores the impact of different low-carbon subsidy methods of government on supply chain decision-making. Through analysis and solutions, the main research conclusions are as follows: the government’s subsidy model will not affect the retailer’s low-carbon publicity level and the manufacturer’s cost-sharing rate. However, the government subsidy coefficient will affect the carbon emission reduction level, carbon emission reduction yield, low-carbon goodwill, the manufacturer’s profit and the retailer’s profit. If the manufacturer and the retailer aim only to maximize profits, the government subsidies based on carbon emission reduction yield will not be the optimal choice, which is mainly because the government only considers carbon emission reduction yield and not carbon emission reduction cost. On the contrary, if both take into account carbon emission reduction, the government subsidy to consumers will not be the optimal choice. Finally, according to the analysis and discussion, we can get some management enlightenment and suggestions.

In this paper, we develop a differential game model of firms’ clean technology (cleantech) research and development (R&D) with spillover effects and learning-by-doing in a duopoly market. A significant feature of our study is that each firm’s instantaneous investment cost function in cleantech R&D depends not only on the R&D investment itself but also on the accumulation of knowledge in cleantech R&D. Furthermore, the rate of change in each firm’s accumulation of cleantech R&D knowledge is treated as a state variable. The main objective of this paper is to investigate firms’ cleantech investment behavior and the effects of cleantech spillover and learning-by-doing on cleantech progress, both in cases of R&D competition and R&D cooperation. Additionally, we derive the general solutions of the model and discuss the results using numerical examples. We demonstrate that private and social incentives towards R&D cooperation align across all admissible levels of the clean technological spillover, which characterizes innovative activity, as well as the learning rate and growth rate of knowledge accumulation in cleantech investment, which characterizes learning-by-doing activity.

As a large mineral resource consuming country, it is of great strategic significance to effectively improve the efficiency of China’s mineral resource exploitation and reduce the waste of exploitation to achieve sustainable development. The purpose of this paper is to examine how governments can incentivize enterprises to invest in mining technology to improve mineral resource recovery rates under both process subsidy policy and outcome incentive policy, and to comparatively analyze which policy is more effective in increasing mineral resource enterprises’ investment in mining technology. We construct a differential game model of the government’s incentive policy for mining technology investment in mineral resource enterprise. The results show the following. (1) For mineral resources with low marginal returns, both process subsidy and outcome incentive policies can encourage the enterprise to increase investment in mining technology. (2) For scarce strategic resources, the process subsidy policy has a better incentive effect when the marginal returns of mineral resources are high, and the outcome incentive policy has a better incentive effect when the marginal returns are low. (3) Mineral resource enterprise has higher recovery rates under the process subsidy policy.

This paper documents the first attempt to apply the differential game theory for investigating some properties of a low-carbon supply chain coordination by employing the cost-sharing mechanism. The manufacturer is the brand owner and the corresponding low-carbon goodwill is increased with respect to the reference emission reduction effect and consumption promotion. Three dynamic games are well-studied by considering the multiple effects of price and non-price factors on the market. It can be observed that the manufacturer always prefers to employ the coordination mechanism. When the manufacturer opts for a cost-sharing program, the manufacturer and retailer under Model-D are always economically better off, and therefore a cost-sharing program is always profit-Pareto-improving.

In this paper, we analyse a differential game describing the interactions between a potential offender and the law enforcement agency. We assume that both players want to maximise their welfare expressed in monetary units, and compare the results obtained by applying the Nash equilibrium concept under symmetric with that under asymmetric information. The comparison reveals that under asymmetric information the offence rate is lower, due to the deterrence caused by the activities of the law enforcement agency. Both players' controls start at a steady state value and stick to it until close to the end of the planning horizon, when they leave the steady state to take into account the scrap value; this can be interpreted as a turnpike property of Nash equilibria. Furthermore, a sensitivity analysis is carried out. Among others, it turns out that a myopic offender tends to a higher offence level.

We give complete proofs to the verification theorems announced recently by the author for the "pairs of relatively optimal feedback strategies" of an autonomous differential game. These concepts are considered to describe the possibly optimal solutions of a differential game while the corresponding value functions are used as "instruments" for proving the relative optimality and also as "auxiliary characteristics" of the differential game. The 6 verification theorems in the paper are proved under different regularity assumptions accompanied by suitable differential inequalities verified by the generalized derivatives, mainly of contingent type, of the value function.

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.

A planar interception problem of a maneuverable target is considered using the linearized kinematic model with variable velocities and first-order dynamics of the interceptor and target. The maneuverabilities of the interceptor and target are assumed to be variable. By using an auxiliary zero-sum linear-quadratic differential game with cheap controls of both players, an interception guidance law, linear with respect to the state vector, is derived. An analytical description of the set of initial positions (capture set) is obtained, from which this guidance law provides zero miss distance, subject to given maneuverabilities of the interceptor and target. A numerical example illustrating the analytical results is presented.

We investigate a simple motion pursuit-evasion differential game of one Pursuer and one Evader. Maximal speeds of the players are equal. The Evader moves along a given curve without self-intersection. There is no phase constraint for the Pursuer. Necessary and sufficient conditions to complete pursuit from both fixed initial position and all initial positions are obtained.

We consider pursuit and evasion differential game problems described by an infinite system of differential equations with countably many Pursuers in Hilbert space. Integral constraints are imposed on the controls of players. In this paper an attempt has been made to solve an evasion problem under the condition that the total resource of the Pursuers is less then that of the Evader and a pursuit problem when the total resource of the Pursuers greater than that of the Evader. The strategy of the Evader is constructed.

A game-theoretic model of territorial environmental production under Cournot competition is studied. The process is modeled as cooperative differential game. The stable distribution mechanism of the common cooperative benefit among players is proposed. We proved that the cooperative total stock of accumulated pollution is strictly less then the pollution under Nash equilibrium for the whole duration of the game. We design a stable Shapley value as a cooperative solution, which is time-consistent. The Shapley value is also strategic stable and satisfies the irrational-behavior-proofness condition. The numerical example is given.

A pursuit–evasion differential game of prescribed duration with bounded controls is considered. The evader has two possible dynamics, while the pursuer dynamics is fixed. The evader can change its dynamics once during the game and its initial lateral acceleration is nonzero. The pursuer knows the possible dynamics of the evader, but not the actual one. Due to the different information sets of the players the game is not zero-sum. The robust capture zone of the pursuer and the robust escape zone of the evader are constructed and analyzed. Since the game is not zero-sum the robust capture/escape zones do not complement each other. Illustrative examples are presented.

We formulate an overlapping generations model on optimal emissions with continuous age structure. We compare the noncooperative solution to the cooperative one and obtain fundamental differences in the optimal strategies. Also including an altruistic motive does not avoid the problem of the myopic noncooperative solution. Finally we define a time-consistent tax scheme to obtain the cooperative solution in the noncooperative case.

A linear nonstationary pursuit problem in which a group of pursuers and a group of evaders are involved is considered under the condition that the group of pursuers includes participants whose admissible controls set coincides with that of the evaders and participants whose admissible controls sets belong to interior of admissible controls set of the evaders. The aim of the group of pursuers is to capture all the evaders. The aim of the group of evaders is to prevent the capture, that is, to allow at least one of the evaders to avoid the rendezvous. It is shown that, if in the game in which all the participants have equal capabilities at least one of the evaders avoids the rendezvous on an infinite time interval, then as a result of the addition of any number of pursuers with less capabilities, at least one of the evaders will avoid the rendezvous on any finite time interval.

New approach to the definition of solution in cooperative differential games is considered. The approach is based on artificially truncated information about the game. It assumed that at each time, instant players have information about the structure of the game (payoff functions, motion equations) only for the next fixed time interval. Based on this information they make the decision. Looking Forward Approach is applied to the cases when the players are not sure about the dynamics of the game on the whole time interval $[0,T]$ and orient themselves on the game dynamics defined on the smaller time interval $\overline{T}$ ($0<\overline{T}<T$), on which they surely know that the game dynamics is not changing.

This paper handles the problem of intercepting a superior missile using a formation of lower-capabilities pursuers. The problem is formulated as multi-player differential game and solved using optimal control techniques. This approach makes use of the Apollonius circles as a final path constraint of the problem in order to achieve the capturing conditions. Additionally, a dimensionality reduction technique based on differential flatness property of the formation is investigated.

In finite-dimensional Euclidean space, an analysis is made of the problem of pursuit of a single evader by a group of pursuers, which is described by a system of the form

In this paper, a differential game of kind of several pursuers and one evader is studied. All the players move only along the edges of a simplex of dimension $d$. The maximal speed of each pursuer is less than that of the evader. If the state of a pursuer coincides with the state of the evader, then pursuit is completed. An exact mathematical formulation of the problem is given by introducing special classes of strategies adapted for games on graphs. Sufficient conditions for completion of pursuit and possibility of evasion are obtained. In the case where the simplex is regular we obtained a condition. If this condition is satisfied, then pursuit can be completed, else evasion is possible.

A differential game described by a nonlinear system of differential equations is considered in a finite-dimensional Euclidean space. The value set of the pursuer control is a finite set. The value set of the evader control is a compact set. The purpose of the pursuer is a translation of the system in a finite time to any given neighborhood of zero. The pursuer uses a piecewise open-loop strategy constructed only by using information on the state coordinates and the velocity in the partition points of a time interval. In the past work, sufficient conditions were obtained for existence of a neighborhood of zero from which the capture occurs. The statement of the capture theorem contains such a condition that some vectors set up a positive basis. In this research, we consider the case when these vectors set up a one-sided set. For this case, sufficient conditions are obtained for existence of a set of initial position, from which the capture occurs.

Differential games with additive payoffs are considered. A new approach to constructing the characteristic function for such games is defined. Properties of optimality principles such as the core and the Shapley value constructed by applying the new characteristic function are studied. The solutions obtained are demonstrated by means of an example.