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

Abstract

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.

1. Introduction

There seems to be rather compelling evidence that global warming is an issue that we seriously need to be concerned about today. One of the important facts on global warming is the result of greenhouse gases (GHGs) trapping heat and making the planet warmer. A GHG is one of several gases in an atmosphere that absorbs and emits infrared radiation in a planetary atmosphere. Many important GHGs are naturally occurring, such as water vapor, carbon dioxide, methane, nitrous oxide, and ozone, but others can also be added to the atmosphere by human activities, such as hydrofluorocarbons, perfluorocarbons, and sulfur hexafluoride. The United Nation’s Intergovernmental Panel on Climate Change (IPCC) claims that the only way they can get their computerized climate models to produce the observed warming is with anthropogenic pollution. Undoubtedly, the problem of global warming has been arising for more than 150 years and will get worse as time goes on.

According to CDIAC, top 20 emitting countries by total fossil-fuel CO₂ emissions for 2009 were China, the United States, India, the Russian Federation, Japan, Germany, Iran, Canada, South Korea, South Africa, United Kingdom, Indonesia, Mexico, Saudi Arabia, Italy, Australia, Brazil, France, Poland, and Spain. Under the Kyoto Protocol of 2010, 37 developed countries and European Community commit themselves to a reduction of GHGs in the period of 2008–2012. The United States signed but did not ratify the protocol, and Canada withdrew from it in 2011. To respond to the extension of the Kyoto Protocol beyond 2012, several developed countries have communicated their intentions to set quantified economy-wide emission reduction targets up to 2020. From China’s newest “12th Five-year Plan on Greenhouse Emission Control (Guofa [2011] No. 41),” China aims to reduce the carbon intensity by 17 percent by 2015 compared with 2010 levels, though it is still a developing country.

A lot of researchers have paid much attention to the problem of GHG emission reduction without game theory. Williams et al. [1] proposed a technology path to deep GHG emissions cuts by 2050. Bastianoni et al. [2] analyzed the problem of assigning responsibility for GHG emissions. Parton et al. [3] studied agriculture’s role in cutting GHG emissions. Tol [4] presented the marginal costs of GHG emissions. Ansuategi and Escapa [5] analyzed the relationship between economic growth and GHG emissions.

Other researchers have employed game theory to study the pollution management, including GHG emission reduction. Van der Ploeg and de Zeeuw studied noncooperative and cooperative pollution strategies and outcomes in a transboundary pollution in [6, 7]. Long [8] analyzed a two-player transboundary differential game. Rubio and Casino [9] show that it is necessary that the initial value of the stock of pollution to be higher than the cooperative solution stock. Yeung [10] presented a cooperative game of transboundary industrial pollution. Bahn and Haurie [11] designed equilibrium solutions with coupled constraints in dynamic games of greenhouse gas emissions abatement. Jørgensen et al. [12–14] designed a cooperative agreement which was time consistent, at no instant of time, no player or group of players wishes to defect from the agreement. Smala Fanokoa et al. [15] considered a two-player asymmetric differential game of pollution control. Breton et al. [16–18] proposed the differential game models to analyze joint implementation in environmental projects, one of the flexible mechanisms considered in the Kyoto Protocol. Petrosjan and Zaccour [19] studied time-consistent Shapley value allocation of pollution cost reduction. Dockner et al. [20] used differential games into economics and management science, including pollution management.

In fact, stock of GHGs not only depends on the amounts of GHG emission or natural absorption, but also is affected by the other stochastic factors, such as random climate, natural disaster, and man-made factor. So the stochastic differential game model should be more appropriate than the deterministic differential game model to describe the above problems.

In recent years, stochastic theory has been playing an important role in the study of in science, engineering, and sociology, such as multiobjective stochastic production-distribution planning problem [21], stochastic stabilization and control [22–28], market portfolio problem in a jump diffusion market [29], partial information problem [30], dynamic stochastic route choice model [31], stochastic heat dynamics systems [32], and stochastic Fokker-Planck-type problems [33]. In particular, the stochastic differential game has been successfully applied into the economics and management. For example, Leong and Huang [34] developed a stochastic differential game of capitalism to analyze the role of uncertainty. Yeung and Petrosyan [35] presented a cooperative stochastic differential game of transboundary industrial pollution with two novel features and analyzed the pollution management in the framework. Wang and Ewald [36] developed a stochastic differential fishery game for a two-species fish population with ecological interaction. Taksar and Zeng [37] studied stochastic differential games between two insurance companies who employ reinsurance to reduce risk exposure.

The remainder of this paper is organized as follows. In Section 2, we propose cooperative and noncooperative stochastic differential game models to describe greenhouse gas emissions decision makings of developed and developing countries. In Section 3, we calculate feedback Nash equilibrium and the Pareto optimal solution, characterize parameter spaces that developed and developing countries can cooperate, and design cooperative conditions under which participants buy the cooperative payoff. In Section 4, numerical simulations are employed to illustrate the above results. In Section 5, we study the assign the cooperative payoff with Nash bargaining solution. Finally, conclusions in Section 6 close the paper.

2. Stochastic Differential Game Models

For the sake of simplicity, global countries emitting GHGs can be divided into two interest groups: developing countries and developed countries, which are, respectively, denoted by player 1 and player 2. Inspired by [8, 15], we assume the two participants have asymmetric environmental behaviors, which mean player 1 is not affected by the pollution, while player 2 is affected by the pollution, or player 1 has no interest in sacrificing economic growth to reduce GHGs emission, whereas player 2 would like to take necessary measures to protect and improve the environment through GHG emission reduction. In order to clarify the above problem, we further assume the following.(1)𝑞_𝑖(𝑡) is the output of player 𝑖  (𝑖=1,2) with 𝑡∈[0,∞). 𝑒_𝑖(𝑡)=ℎ_𝑖(𝑞_𝑖(𝑡)) denotes the quantity of GHG emission of player 𝑖. 𝑟_𝑖(𝑞_𝑖(𝑡)) represents net revenue of player 𝑖 with 𝑞_𝑖(𝑡).(2)ℎ_𝑖(𝑞_𝑖(𝑡)) is a strictly increasing function, that is, (𝑑/𝑑𝑞)(ℎ_𝑖(𝑞_𝑖(𝑡)))>0. So we can denote 𝑟_𝑖(𝑞_𝑖(𝑡))=𝑟_𝑖(ℎ_𝑖⁻¹(𝑒_𝑖(𝑡))) and 𝑅_𝑖(𝑒_𝑖(𝑡))=𝑟_𝑖(ℎ_𝑖⁻¹(𝑒_𝑖(𝑡))).(3)𝑅_𝑖(𝑒_𝑖(𝑡)) is a monotonically increasing concave function of 𝑒_𝑖(𝑡), that is, 𝑅^′_𝑖(𝑒_𝑖)≥0, 𝑅^′′_𝑖(𝑒_𝑖)≥0, 𝑅_𝑖(0)=0. 𝑠(𝑡) is the stock of GHGs. 𝐷_𝑖(𝑠(𝑡)) is a monotone increasing convex function of 𝑠(𝑡), which is the cost of player 𝑖 caused by the stock of GHGs 𝑠(𝑡).(4)𝜇≥0 is a constant that means the marginal impact on GHG emission. 𝜀>0 is a natural absorption rate. Without loss of generality, we assume that the marginal impact of GHG emission on both players is the same. 𝜎≥0 is a constant affected by the weather, natural disasters, human factors, unpredictable factors, and so on. 𝐵(𝑡) is a Wiener process.(5)Player 𝑖 will, even if alone, maximize its expected payoff in case no agreement is reached between both players.

According to the above assumptions, we get cooperative and noncooperative stochastic differential game models for asymmetric GHG emission problems over an infinite horizon.

2.1. Noncooperative Stochastic Differential Game Model

  

Consider 𝑑𝑠(𝑡)=𝜇𝑒₁(𝑡)+𝑒₂(𝑡)(𝑡)(𝑡)(𝑡),𝑠(0)−𝜀𝑠𝑑𝑡+𝜎𝑠𝑑𝐵=𝑠₀𝐽_𝑖=max_𝑒𝑖𝐸^∞₀𝑒^−𝑟𝑡𝑅_𝑖𝑒_𝑖(𝑡)−𝐷_𝑖(𝑠(𝑡))𝑑𝑡,𝑖=1,2, (1)(2) where 𝑠₀ is the initial GHGs stock with 𝑡=0, 𝑟 is the discount rate, and 𝑒^−𝑟𝑡 is the discount factor.

Without consensus agreement between both players, they will, alone, maximize their payoffs as follows: 𝐽₁=max_𝑒1𝐸^∞₀𝑒^−𝑟𝑡𝑅₁𝑒₁(𝑡)𝐽𝑑𝑡,₂=max_𝑒2𝐸^∞₀𝑒^−𝑟𝑡𝑅₂𝑒₂(𝑡)−𝐷₂(𝑠(𝑡))𝑑𝑡, (3)(4) which mean that player 1 maximizes its payoffs through its industrial activities without internalizing the damage of GHGs emission on the basis of (3), and player 2 maximizes its payoff based on (4).

We use Γ(𝑠₀,∞−0) to denote the noncooperative stochastic differential game Equations (1) and (3) and (1) and (4). In addition, we use Γ(𝑠_𝜏,∞−𝜏) to represent the above game with the initial state 𝑠_𝜏∈𝑆 and the time 𝜏∈[0,∞).

The rational economic explanation for the noncooperative game is that (3) is the payoff of player 1 and implies its prior strategy of economic development, which means that developing countries will always maximize their economic benefit rather than give environmental protection a high priority.

2.2. Cooperative Stochastic Differential Game Model

In order to reach a cooperative agreement of GHG emission, we adopt a cooperative stochastic differential game to get its Nash equilibrium, that is, a cooperative agreement, which is a solution concept of a cooperative game involving two players, such that no player has an incentive to unilaterally change its GHG emission decision. In other words, players are in an equilibrium if a GHG emission decision change in strategies by any one of them would lead that player to earn less than if it remained with its current GHG emission strategy [38]. Two players’ joint payoff is 𝐽₁+𝐽₂=max_𝑒1,𝑒2𝐸²_𝑖=1^∞₀𝑒^−𝑟𝑡𝑅_𝑖𝑒_𝑖(𝑡)−𝐷_𝑖(𝑠(𝑡))𝑑𝑡. (5)

We use Ψ(𝑠₀,∞−0) to denote the cooperative stochastic differential game Equations (1) and (5). In addition, we use Ψ(𝑠_𝜏,∞−𝜏) to represent the above game with the initial state 𝑠_𝜏∈𝑆 and the time 𝜏∈[0,∞).

3. Resolving Models

In order to reduce the difficulty of mathematical handling, we assume that 𝑅_𝑖(𝑒_𝑖(𝑡)) is a quadratic function of 𝑒_𝑖(𝑡), 𝑖=1,2, and 𝐷_𝑖(𝑠(𝑡)) is a quadratic function of 𝑠(𝑡), and cost parameters 𝛽₂≥𝛽₁≥0. Consider 𝑅_𝑖𝑒_𝑖(𝑡)=𝛼_𝑖𝑒_𝑖(𝑡)−12𝑒_𝑖(𝑡)²𝐷,𝑖=1,2,_𝑖(𝑠(𝑡=1))2𝛽_𝑖𝑠²(𝑡),𝑖=1,2, (6)𝛼_𝑖>0, 𝑖=1,2 and 𝛽_𝑖≥0, 𝑖=1,2 are constants.

3.1. Noncooperative Game

3.1.1. Noncooperative Game Solutions

Definition 1. When the state variable 𝑠(𝑡)=𝑠_𝑡, the value function 𝑉^(𝜏)𝑖(𝑡,𝑠_𝑡), 𝑖=1,2 of a Nash equilibrium for the non-cooperation stochastic differential game Γ(𝑠_𝜏,∞−𝜏), that is, the current value of the expected payoff for player 𝑖, 𝑖=1,2 with the interval [𝑡,∞) is given by 𝑉^(𝜏)𝑖𝑡,𝑠_𝑡=𝐸^∞_𝑡𝑒^{−𝑟(𝑦−𝜏)}𝑅_𝑖𝑒_𝑖𝑦−𝐷_𝑖𝑠𝑦𝑑𝑦, (7) for 𝜏∈[0,∞), 𝑡∈[𝜏,∞), 𝑖=1,2.

Theorem 2 (see [39]). The strategy {𝜙^∗₁(𝑡),𝜙^∗₂(𝑡)} is a feedback Nash equilibrium solution to the game Γ(𝑠₀,∞−0). If there exist continuous functions 𝑉_𝑖(𝑠)∶𝑅→𝑅, 𝑖=1,2 and continuous derivatives 𝑉^′_𝑖(𝑠), 𝑉^′′_𝑖(𝑠), 𝑖=1,2 satisfy the following Hamilton-Jacobi-Bellman equation: 𝑟𝑉_𝑖(𝑠)−𝜎²𝑠²2𝑉^′′_𝑖(𝑠)=max_𝑒𝑖𝛼_𝑖𝑒_𝑖(𝑡)−12𝑒²_𝑖(𝑡)−12𝛽_𝑖𝑠²(𝑡)+𝑉^′_𝑖(𝑠)𝜇𝑒_𝑖(𝑡)+𝜙^∗_𝑗(𝑡)−𝜀𝑠(𝑡), (8) where 𝜙^∗_𝑗(𝑡),𝑗=1,2 is the optimal control of the player 𝑗,𝑗=1,2.

Theorem 3. If above conditions are satisfied, its feedback Nash equilibrium is 𝑒^𝑁₁(𝑡)=𝛼₁,𝑒^𝑁₂=𝛼₂+𝜇𝑉^′₂(𝑠)=𝛼₂+𝜇𝐴₂𝑠+𝐵₂, (9) where 𝐴₂=𝑟+2𝜀−𝜎²−𝑟+2𝜀−𝜎²²+4𝛽₂𝜇²2𝜇²𝐵<0,₂=𝛼₁+𝛼₂𝜇𝐴₂𝑟+𝜀−𝜇²𝐴₂<0. (10)

Proof. For player 1, 𝑟𝑉₁(𝑠)−𝜎²𝑠²2𝑉^′′₁(𝑠)=max_𝑒1𝛼₁𝑒₁(𝑡)−12𝑒²₁(𝑡)+𝑉^′₁(𝑠)𝜇𝑒₁(𝑡)+𝜙^∗₂(𝑡)−𝜀𝑠(𝑡). (11) Differentiating (11) with respect to 𝑒₁ and using extreme conditions, we can get 𝑒^∗₁(𝑡)=𝛼₁+𝜇𝑉^′₁(𝑠). (12) For player 2, 𝑟𝑉₂(𝑠)−𝜎²𝑠²2𝑉^′′₂(𝑠)=max_𝑒2𝛼₂𝑒₂(𝑡)−12𝑒²₂(𝑡)−12𝛽₂𝑠²(𝑡)+𝑉^′₂(𝑠)𝜇𝑒₂(𝑡)+𝜙^∗₁(𝑡)−𝜀𝑠(𝑡). (13) Differentiating (13) with respect to 𝑒₂ and using the extreme value theorem, we can get 𝑒^∗₂(𝑡)=𝛼₂+𝜇𝑉^′₂(𝑠). (14) Substituting 𝑒^∗₁(𝑡),𝑒^∗₂(𝑡) to (11), 𝑟𝑉₁(𝑠)−𝜎²𝑠²2𝑉^′′₁(𝑠)=𝛼₁𝛼₁+𝜇𝑉^′₁(𝑠)−12𝛼₁+𝜇𝑉^′₁(𝑠)²+𝑉^′₁(𝑠)𝜇𝛼₁+𝜇𝑉^′₁(𝑠)+𝛼₂+𝜇𝑉^′₂(𝑠)−𝜀𝑠(𝑡), (15) hence, 𝑉₁(𝑠)=𝐴₁, 𝑒^∗₁(𝑡)=𝛼₁, 𝐴₁=𝛼²₁/2𝑟.
Substituting 𝑒^∗₁(𝑡) and 𝑒^∗₂(𝑡) to (13), 𝑟𝑉₂(𝑠)−𝜎²𝑠²2𝑉^′′₂(𝑠)=𝛼₂𝛼₂+𝜇𝑉^′₂(𝑠)−12𝛼₁+𝜇𝑉^′₁(𝑠)²−12𝛽₂𝑠²+𝑉^′₂(𝑠)𝜇𝛼₁+𝜇𝑉^′₁(𝑠)+𝛼₂+𝜇𝑉^′₂(𝑠)−𝜀𝑠(𝑡), (16) hence, 𝑉₂(𝑠)=(1/2)𝐴₂𝑠²+𝐵₂𝑠+𝐶₂, 𝑉^′₂(𝑠)=𝐴₂𝑠+𝐵₂, 𝑉^′′₂(𝑠)=𝐴₂.
As [40], we have 𝐴₂=𝑟+2𝜀−𝜎²−𝑟+2𝜀−𝜎²²+4𝛽₂𝜇²2𝜇²𝐵<0,₂=𝛼₁+𝛼₂𝜇𝐴₂𝑟+𝜀−𝜇²𝐴₂𝐶<0,₂=𝛼²₂+2𝜇𝛼₁𝐵₂+2𝜇𝛼₂𝐵₂+𝜇²𝐵²₂,2𝑟 (17) hence, 𝑒^𝑁₂=𝛼₂+𝜇𝑉^′₂(𝑠)=𝛼₂+𝜇(𝐴₂𝑠+𝐵₂).

Remark 4. (1) The first-order extreme value conditions show that player 2 determines its GHG emissions whose marginal profit (𝛼₂−𝑒₂) is equal to marginal cost −𝜇𝑉^′₂(𝑠)=−𝜇(𝐴₂𝑠+𝐵₂). Since 𝑉^′₂(𝑠)=𝐴₂𝑠+𝐵₂<0, if 𝑒^𝑁₂=𝛼₂+𝜇𝑉^′₂(𝑠)=𝛼₂+𝜇(𝐴₂𝑠+𝐵₂)⇔𝛼₂>−𝜇(𝐴₂𝑠+𝐵₂), the feedback Nash equilibrium is interior.
(2) From Theorem 3, one can know that the feedback Nash equilibrium strategy {𝑒^𝑁₁(𝑡),𝑒^𝑁₂(𝑡)} possesses the Markov property, because it depends only upon the present state, not on the sequence of events that preceded it.
(3) Theorem 3 shows that optimal strategies of the subgame and the original game are the same at the same time and conditions.
The individual gross payoff in the noncooperative game can be written as 𝑊^𝑁₁=𝑉₁𝑠₀=𝛼²₁,𝑊2𝑟_𝑁2=𝑉₂𝑠₀=12𝐴₂𝑠₀²+𝐵₂𝑠₀+𝐶₂. (18)
From Definition 1, one can obtain 𝑉^(𝜏)𝑖𝑡,𝑠_𝑡=𝐸^∞_𝑡𝑒^{−𝑟(𝑦−𝜏)}𝑅_𝑖𝑒_𝑖𝑦−𝐷_𝑖𝑠𝑦𝑑𝑦=𝑒^{−𝑟(𝑡−𝜏)}𝐸^∞_𝑡𝑒^{−𝑟(𝑦−𝑡)}𝑅_𝑖𝑒_𝑖𝑦−𝐷_𝑖𝑠𝑦𝑑𝑦=𝑒^{−𝑟(𝑡−𝜏)}𝑉^(𝑡)𝑖𝑡,𝑠_𝑡. (19)

Remark 5. For player 𝑖, 𝑖=1,2 in the noncooperative stochastic differential game Γ(𝑠_𝜏,∞−𝜏) with the state variable 𝑠(𝑡)=𝑠_𝑡; the discounted value of the subgame value function is equal to the current value of the expected payoff at the time 𝑡∈[𝑡,∞).

3.1.2. The Expected Stock and Its Limitation

Theorem 6. The expectation stock and its limit in noncooperative game feedback Nash equilibrium satisfy 𝐸𝑠(𝑡)=𝑠₀𝑒^{(𝜇2𝐴2−𝜀)𝑡}+𝜇𝛼₁+𝜇𝛼₂+𝜇²𝐵₂𝜀−𝜇²𝐴₂1−𝑒^{(𝜇2𝐴2−𝜀)𝑡},lim_𝑡→∞𝐸𝑠(𝑡)=𝜇𝛼₁+𝜇𝛼₂+𝜇²𝐵₁𝜀−𝜇²𝐴₂. (20)

Proof. Substituting 𝑒₁(𝑡),𝑒₂(𝑡) to (1) leads to 𝑑𝑠(𝑡)=𝜇𝛼₁+𝜇𝛼₂+𝜇²𝐵₂+𝜇²𝐴₂−𝜀𝑠𝑑𝑡+𝜎𝑠𝑑𝐵(𝑡). (21) Solving the above stochastic differential equation leads to 𝑠(𝑡)=𝑠₀𝑒^{(𝜇2𝐴2−𝜀−(1/2)𝜎2)𝑡+𝜎𝐵(𝑡)}+^𝑡₀𝜇𝛼₁+𝜇𝛼₂+𝜇²𝐵₂×𝑒^{(𝜇2𝐴2−𝜀−(1/2)𝜎2)(𝑡−𝜏)+𝜎(𝐵(𝑡)−𝐵(𝜏))}𝑑𝜏, (22) whose expectation is 𝐸𝑠(𝑡)=𝐸𝑠₀𝑒^{(𝜇2𝐴2−𝜀−(1/2)𝜎2)𝑡+𝜎𝐵(𝑡)}+𝐸^𝑡₀𝜇𝛼₁+𝜇𝛼₂+𝜇²𝐵₂×𝑒^{(𝜇2𝐴2−𝜀−(1/2)𝜎2)(𝑡−𝜏)+𝜎(𝐵(𝑡)−𝐵(𝜏))}=𝑠𝑑𝜏₀𝑒^{(𝜇2𝐴2−𝜀)𝑡}+𝜇𝛼₁+𝜇𝛼₂+𝜇²𝐵₂𝜀−𝜇²𝐴₂1−𝑒^{(𝜇2𝐴2−𝜀)𝑡}. (23) Since 𝐴₂<0 and 𝜇²𝐴₂−𝜀<0 hold, one can obtain lim_𝑡→∞𝐸𝑠(𝑡)=(𝜇𝛼₁+𝜇𝛼₂+𝜇²𝐵₁)/(𝜀−𝜇²𝐴₂).

3.2. Cooperative Game

3.2.1. Cooperative Game Solutions

Theorem 7 (see [39]). The strategy {𝜑^∗₁(𝑡),𝜑^∗₂(𝑡)} is a Pareto optimal solution to the game Ψ(𝑠₀,∞−0), if there exist continuous function 𝑊_𝑖(𝑠)∶𝑅→𝑅 and continuous derivatives 𝑊^′_𝑖(𝑠), 𝑊^′′_𝑖(𝑠); 𝑖=1,2 satisfy the following Hamilton-Jacobi-Bellman equation: (𝑠)−𝜎𝑟𝑊²𝑠²2𝑊^′′(𝑠)=max_𝑒1,𝑒2²_𝑖=1𝛼_𝑖𝑒_𝑖(𝑡)−12𝑒²_𝑖(𝑡)−12𝛽_𝑖𝑠²(𝑡)+𝑊^′(𝑠)𝜇𝑒_𝑖(𝑡)+𝜑^∗_𝑗(𝑡)−𝜀𝑠(𝑡). (24)
Hence, the players will adopt the cooperative control {𝜑^∗₁(𝑡),𝜑^∗₂(𝑡)} characterized in Theorem 7.

Theorem 8. The Pareto optimal solution of the cooperative stochastic differential game Ψ(𝑠₀,∞−0) is given by 𝑒_𝑖=𝛼_𝑖+𝜇𝑊^′(𝑠)=𝛼_𝑖+𝜇(𝑎𝑠+𝑏),𝑖=1,2, (25) where 𝑎=𝑟+2𝜀−𝜎²−𝑟+2𝜀−𝜎²²+8𝜇²𝛽₁+𝛽₂4𝜇²𝜇𝛼<0,𝑏=¹+𝛼₂𝑎𝑟+𝜀−2𝜇²𝑎<0. (26)

Proof. Differentiating (24) with respect to 𝑒_𝑖 and using extreme value conditions, one can get 𝑒_𝑖=𝛼_𝑖+𝜇𝑊^′(𝑠),𝑖=1,2. (27) Using 𝑒_𝑖=𝛼_𝑖+𝜇𝑊^′(𝑠), 𝑖=1,2 into (24), one can obtain (𝑠)−𝜎𝑟𝑊²𝑠²2𝑊^′′(𝑠)=12𝛼₁+𝜇𝑊^′(𝑠)²+12𝛼₂+𝜇𝑊^′(𝑠)²−12𝛽₁+𝛽₂𝑠²−𝜀𝑠𝑊^′(𝑠). (28) From the above equation, it is rational to assume 𝑊(𝑠) is a quadratic polynomial.
Let 𝑊(𝑠)=(1/2)𝑎𝑠²+𝑏𝑠+𝑐, one can get 2𝜇²𝑎²−(𝑟+2𝜀−𝜎²)𝑎−(𝛽₁+𝛽₂)=0.
As [40], we have 𝑎=𝑟+2𝜀−𝜎²−𝑟+2𝜀−𝜎²²+8𝜇²𝛽₁+𝛽₂4𝜇²𝜇𝛼<0,𝑏=¹+𝛼₂𝑎𝑟+𝜀−2𝜇²𝑎𝛼<0,𝑐=¹+𝜇𝑏²+𝛼₂+𝜇𝑏²2𝑟>0. (29) Hence, 𝑒_𝑖=𝛼_𝑖+𝜇𝑊^′(𝑠)=𝛼_𝑖+𝜇(𝑎𝑠+𝑏), 𝑖=1,2.

Remark 9. (1) The Pareto optimal solution {𝑒₁,𝑒₂} possesses the Markov property.
(2) Theorem 8 shows that Pareto optimal solutions of the subgame and the original game are the same at the same time and conditions.
The cooperative value function is given by 𝑊^𝐶(𝑠)=(1/2)𝑎𝑠²+𝑏𝑠+𝑐, where 𝑎=𝑟+2𝜀−𝜎²−𝑟+2𝜀−𝜎²²+8𝜇²𝛽₁+𝛽₂4𝜇²𝜇𝛼<0,𝑏=¹+𝛼₂𝑎𝑟+𝜀−2𝜇²𝑎𝛼<0,𝑐=¹+𝜇𝑏²+𝛼₂+𝜇𝑏²2𝑟>0. (30)

3.2.2. The Expected Stock and Its Limitation

Theorem 10. The expectation stock and its limit in the Pareto optimal solution of the cooperative stochastic differential game Ψ(𝑠₀,∞−0) satisfy 𝐸𝑠^𝐶(𝑡)=𝑠₀𝑒^{(2𝜇2𝑎−𝜀)𝑡}+𝜇𝛼₁+𝜇𝛼₂+2𝜇²𝑎𝜀−𝜇²𝑎1−𝑒^{(2𝜇2𝑎−𝜀)𝑡},lim_𝑡→∞𝐸𝑠^𝐶(𝑡)=𝜇𝛼₁+𝜇𝛼₂+2𝜇²𝑏𝜀−2𝜇²𝑎. (31)

Proof. Substituting 𝑒₁(𝑡),𝑒₂(𝑡) to (1) leads to 𝑑𝑠^𝐶(𝑡)=𝜇𝛼₁+𝜇𝑎𝑠^𝐶(𝑡)+𝑏+𝛼₂+𝜇𝑎𝑠^𝐶(𝑡)+𝑏−𝜀𝑠^𝐶(𝑡)𝑑𝑡+𝜎𝑠^𝐶(𝑡)(𝑡)=𝑑𝐵𝜇𝛼₁+2𝜇²𝑏+𝜇𝛼₂+2𝜇²𝑎−𝜀𝑠^𝐶(𝑡)𝑑𝑡+𝜎𝑠^𝐶(𝑡)𝑑𝐵(𝑡). (32) Solving the above stochastic differential equation leads to 𝑠^𝐶(𝑡)=𝑠₀𝑒^{(2𝜇2𝑎−𝜀−(1/2)𝜎2)𝑡+𝜎𝐵(𝑡)}+^𝑡₀𝜇𝛼₁+𝜇𝛼₂+2𝜇²𝑏×𝑒^{(2𝜇2𝑎−𝜀−(1/2)𝜎2)(𝑡−𝜏)+𝜎(𝐵(𝑡)−𝐵(𝜏))}𝑑𝜏. (33) Taking the expectation of the above equation leads to 𝐸𝑠^𝐶(𝑡)=𝐸𝑠₀𝑒^{(2𝜇2𝑎−𝜀−(1/2)𝜎2)𝑡+𝜎𝐵(𝑡)}+𝐸^𝑡₀𝜇𝛼₁+𝜇𝛼₂+2𝜇²𝑏×𝑒^{(2𝜇2𝑎−𝜀−(1/2)𝜎2)(𝑡−𝜏)+𝜎(𝐵(𝑡)−𝐵(𝜏))}𝑑𝜏=𝑠₀𝑒^{(2𝜇2𝑎−𝜀)𝑡}+𝜇𝛼₁+𝜇𝛼₂+2𝜇²𝑎𝜀−𝜇²𝑎1−𝑒^{(2𝜇2𝑎−𝜀)𝑡}. (34) Since 𝑎<0 and 2𝜇²𝑎−𝜀<0 hold, one can get lim_𝑡→∞𝐸𝑠^𝐶(𝑡)=(𝜇𝛼₁+𝜇𝛼₂+2𝜇²𝑏)/(𝜀−2𝜇²𝑎).

4. Numerical Simulations

4.1. Noncooperative Game

Let 𝑟=0.05, 𝜎=0.2,   𝜀=0.3, 𝑠₀=0.5, 𝛽₁=0.2, 𝛽₂=0.25, 𝛼₁=2, 𝛼₂=3, and 𝜇=0.8. For player 2, its noncooperative feedback Nash equilibrium 𝑒^𝑁₂, as shown in Figures 1, 2, and 3, increases with 𝜀 but decreases with either 𝑠 or 𝜎 or 𝛽₂. To be specific, the optimal quantity of GHG emission of player 2 is proportional to the natural absorption rate and inversely proportional to the stock of GHGs and constant affected by stochastic factors and cost parameter of player 2. Its limitation of GHG stock lim_𝑡→∞𝐸𝑠^𝑁(𝑡)=5.1297 is shown in Figure 4. For the sake of simplicity, in this section we denote 𝐸𝑉₂(𝑠) as 𝑉₂(𝑠). Its expected profit 𝐸𝑉₂(𝑠) increases with 𝜀 but decreases with either 𝜎 or 𝛽₂ and changes little with 𝑠 shown in Figures 5, 6, and 7. More specifically, the expected profit is proportional to the natural absorption rate, is inversely proportional to constant affected by stochastic factors and cost parameter of player 2, and has little to do with the stock of GHGs.

Figure 1 

𝑒𝑁2 versus 𝑠 and 𝜀.

Figure 2 

𝑒𝑁2 versus 𝑠 and 𝜎.

Figure 3 

𝑒𝑁2 versus 𝑠 and 𝛽2.

Figure 4 

𝐸𝑠𝑁(𝑡) and lim𝑡→∞𝐸𝑠𝑁(𝑡) versus 𝑡.

Figure 5 

𝑉2(𝑠) versus 𝑠 and 𝜀.

Figure 6 

𝑉2(𝑠) versus 𝑠 and 𝜎.

Figure 7 

𝑉2(𝑠) versus 𝑠 and 𝛽2.

4.2. Cooperative Game Situation

Let 𝑟=0.05, 𝜎=0.2, 𝜀=0.3, 𝑠₀=0.5, 𝛽₁=0.2, 𝛽₂=0.25, 𝛼₁=2, 𝛼₂=3, and 𝜇=0.8. For player 1 and player 2, their Pareto optimal solutions 𝑒₁ and 𝑒₂, as shown in Figures 8, 9, 10, 11, 12, 13, 14, and 15, and their GHG emissions both increase with 𝜀 but decrease with 𝑠 or 𝜎 or 𝛽₁ or 𝛽₂. Practically speaking, the optimal quantities of GHG emissions of player 1 and player 2 are all proportional to the natural absorption rate and inversely proportional to the stock of GHGs and constant affected by stochastic factors and cost parameters of player 1 and player 2. The limitation of GHG stock lim_𝑡→∞𝐸𝑠^𝐶(𝑡)=3.2115 is shown in Figure 16. For the sake of simplicity, we denote 𝐸𝑊^𝐶(𝑠) as 𝑊^𝐶(𝑠) in this section. The whole expected cooperative profit 𝐸𝑊^𝐶(𝑠) decreases with either 𝑠 or 𝜀 or 𝜎 or 𝛽₁ or 𝛽₂ shown in Figures 17, 18, 19, and 20. In fact, the whole expected cooperative profit is inversely proportional to the natural absorption rate and the stock of GHGs and constant affected by stochastic factors and cost parameters of player 1 and player 2.

Figure 8 

𝑒1 in terms of 𝑠 and 𝜀.

Figure 9 

𝑒2 versus 𝑠 and 𝜀.

Figure 10 

𝑒1 versus 𝑠 and 𝜎.

Figure 11 

𝑒2 versus 𝑠 and 𝜎.

Figure 12 

𝑒1 versus 𝑠 and 𝛽2.

Figure 13 

𝑒2 versus 𝑠 and 𝛽2.

Figure 14 

𝑒2 versus 𝑠 and 𝛽1.

Figure 15 

𝑒1 versus 𝑠 and 𝛽1.

Figure 16 

𝐸𝑠𝐶(𝑡),lim𝑡→∞𝐸𝑠𝐶(𝑡) versus 𝑡.

Figure 17 

𝑊𝐶(𝑠) versus 𝑠 and 𝜀.

Figure 18 

𝑊𝐶(𝑠) versus 𝑠 and 𝜎.

Figure 19 

𝑊𝐶(𝑠) versus 𝑠 and 𝛽1.

Figure 20 

𝑊𝐶(𝑠) versus 𝑠 and 𝛽2.

5. The Distribution of Cooperative Dividend

We consider the long-term cooperation agreement related to the following crucial questions.(1)When is a cooperative agreement globally feasible? (2)In the parameter space where cooperation is achievable, how should the cooperative dividend be distributed among players?(3)What can ensure that players abide by the cooperative agreement over time?

5.1. Global Feasible Conditions for the Cooperation Agreement

A cooperative agreement is globally feasible if the total cooperative payoff must be greater than the total individual noncooperative payoff. The difference between the two is called the cooperative dividend, which is denoted as DC: DC𝑠₀=12𝑎−𝐴₂𝑠²₀+𝑏−𝐵₂𝑠₀+𝐶, (35) where 𝐶=𝑐−𝐶₂−(𝛼²₁/2𝑟)=(𝜇/2𝑟)(𝜇(2𝑏²−𝐵²₂)+2(𝛼¹+𝛼²)(𝑏−𝐵₂)).

Here, DC(𝑠₀) is a quadratic polynomial of 𝑠₀. Let DC(𝑠₀)=0, one can get 𝑠⁻₀=−𝑏−𝐵₂−𝑏−𝐵₂²−2𝑎−𝐴₂𝐶𝑎−𝐴₂,𝑠⁺₀=−𝑏−𝐵₂+𝑏−𝐵₂²−2𝑎−𝐴₂𝐶𝑎−𝐴₂ (36) for (𝑏−𝐵₂)²−2(𝑎−𝐴₂)𝐶≥0.

Remark 11. From Definition 1, the discount value of cooperative dividend DC along the optimal trajectory is equal to the cooperative dividend DC(𝑠₀) in the initial state.

Theorem 12. Let 𝛽⁺₂=𝛽₁+((𝑟+2𝜀−𝜎²)/𝜇)𝛽₁.
If 𝛽₂>𝛽⁺₂, then DC𝑠₀>0,𝑖𝑓𝐶≥0,∀𝑠₀≥0,<0,𝑖𝑓𝐶<0,𝑠₀∈0,𝑠⁺₀,>0,𝑖𝑓𝐶<0,𝑠₀>𝑠⁺₀. (37) If 𝛽₂∈[𝛽₁,𝛽⁺₂] and 𝑏−𝐵₂>0, then DC𝑠₀>0,𝑖𝑓𝐶≥0,𝑠₀∈0,𝑠⁻₀,>0,𝑖𝑓𝐶<0,𝑠₀∈𝑠⁺₀,𝑠⁻₀. (38) If 𝛽₂∈[𝛽₁,𝛽⁺₂] and 𝑏−𝐵₂<0, then DC(𝑠₀)<0 for all 𝑠₀>0.

Proof. It is easy to prove it by analyzing the quadratic function DC(𝑠₀)=0 of 𝑠₀.

Remark 13. The above theorem states that the cooperative agreement is achievable if DC(𝑠₀)>0.

5.2. The Distribution Method

Assuming that the cooperation is globally feasible, then it is an important problem for us to tackle and how to allocate DC(𝑠₀) between two players. From cooperative game or bargaining theory, one can find many solutions to the problems, such as core, Shapley value, Nucleolus, and Nash bargaining solution (NBS).

The NBS method will be applied into this paper. Let (𝑊^𝑁₁,𝑊^𝑁₂) denote the Nash equilibrium, and let (𝑊^NBS₁,𝑊^NBS₂) denote the Nash bargaining solution. One can get 𝑊^NBS₁=𝑊^𝑁₁+DC𝑠₀2=𝑊^𝑁₁+12𝑊^𝐶₁+𝑊^𝐶₂−𝑊^𝑁₁+𝑊^𝑁₂,𝑊^NBS₂=𝑊^𝑁₂+DC𝑠₀2=𝑊^𝑁₂+12𝑊^𝐶₁+𝑊^𝐶₂−𝑊^𝑁₁+𝑊^𝑁₂, (39) here 𝑊^𝑁₁=max_𝑒1𝐸^∞₀𝑒^−𝑟𝑡𝑅₁𝑒^𝑁₁(𝑡),𝑊𝑑𝑡_𝑁2=max_𝑒2𝐸^∞₀𝑒^−𝑟𝑡𝑅₂𝑒^𝑁₂(𝑡)−𝐷₂𝑠^𝑁(𝑡),𝑊𝑑𝑡_𝐶𝑖=max_𝑒𝑖𝐸^∞₀𝑒^−𝑟𝑡𝑅_𝑖𝑒^𝑁_𝑖(𝑡)−𝐷_𝑖𝑠^𝑁(𝑡)𝑑𝑡,𝑖=1,2. (40)(41)

Remark 14. (1) For non-cooperation stochastic differential game Γ(𝑠_𝜏,∞−𝜏), the current value of expected payoff is equal to its subgame value function for the corresponding discount in the state variable 𝑠(𝑡)=𝑠_𝑡 at the interval [𝑡,∞).
(2) For cooperation stochastic differential game Ψ(𝑠_𝜏,∞−𝜏), the current value of expected payoff is equal to its subgame value function for the corresponding discount in the state variable 𝑠(𝑡)=𝑠_𝑡 at the interval [𝑡,∞).

5.3. Abide by the Cooperative Agreement

It is crucial to build a credible basis for abiding strictly by the cooperative agreement in a long time. Inspired by [14, 35, 39, 41–43], we try to design a transfer payment mechanism for subgame consistent agreement based on the Nash bargaining solution.

Denote 𝑊^NBS_𝑖(𝑠^𝐶(𝑡)),𝑊^𝑁_𝑖(𝑠^𝐶(𝑡)) as cooperation and noncooperative distribution, respectively.

Definition 15 (see [29]). (𝑊^NBS₁(𝑠^𝐶(𝑡)),𝑊^NBS₂(𝑠^𝐶(𝑡))) is an effective distribution if it satisfies the following:(1)(𝑊^NBS₁(𝑠^𝐶(𝑡)),𝑊^NBS₂(𝑠^𝐶(𝑡))) is a Pareto optimal distribution,(2)𝑊^NBS_𝑖(𝑠^𝐶(𝑡))≥𝑊^𝑁_𝑖(𝑠^𝐶(𝑡)), 𝑖=1,2,where 𝑠^𝐶(𝑡) is state trajectory of the cooperative state trajectory.

Definition 16 (see [29]). (𝑊^NBS₁(𝑠^𝐶(𝑡)),𝑊^NBS₂(𝑠^𝐶(𝑡))) is a subgame consistent distribution if it satisfies(1)(𝑊^NBS₁(𝑠^𝐶(𝑡)),𝑊^NBS₂(𝑠^𝐶(𝑡))) is a Pareto optimal distribution,(2)𝑊^NBS_𝑖(𝑠^𝐶(𝑡))≥𝑊^𝑁_𝑖(𝑠^𝐶(𝑡)), 𝑖=1,2 for 𝑡∈[𝜏,∞),where 𝑠^𝐶(𝑡) is state trajectory of the cooperation state trajectory.

Denote 𝜉_𝑖(𝑡), 𝑖=1,2 instantaneous payoff-distribution procedure of the subgame, which satisfies ^∞₀𝑒^−𝑟𝑡𝜉_𝑖(𝑡)𝑑𝑡=𝑊^NBS_𝑖𝑠₀𝑊,𝑖=1,2._NBS𝑖𝑠₀=^𝑡₀𝑒^−𝑟𝜏𝜉_𝑖(𝜏)𝑑𝜏+𝑒^−𝑟𝑡𝑊^NBS_𝑖𝑠^𝐶(𝑡),𝑖=1,2, (42)(43) where 𝑊^NBS_𝑖𝑠^𝐶(𝑡)=𝑊^𝑁_𝑖𝑠^𝐶(𝑡+1)2²_𝑖=1𝑊^𝐶_𝑖𝑠^𝐶(𝑡)−𝑊^𝑁_𝑖𝑠^𝐶(𝑡),𝑖=1,2. (44)

For the sake of simplicity, let 𝑠^𝐶(𝑡) denote 𝐸𝑠^𝐶(𝑡), and let Γ(𝑠^𝐶(𝑡)) denote a subgame in state 𝑠^𝐶_𝑡=𝑠^𝐶(𝑡). If a cooperative agreement is feasible in the subgame, player 𝑖, 𝑖=1,2 can get a Nash bargaining solution from (43).

Differential (43) with respect to 𝑡, one has 𝜉_𝑖(𝑡)=𝑟𝑊^NBS_𝑖𝑠^𝐶(𝑡)−𝑑𝑊𝑑𝑡^NBS𝑖𝑠^𝐶(𝑡),𝑖=1,2. (45)

Equation (45) shows that the instantaneous payoff of player 𝑖 in the time 𝑡 is equal to its interest payoff minus the ratio of Nash bargaining solution.

Remark 17. If (45) is multiplied by discount factor 𝑒^−𝑟𝑡, by integrating both sides, one can get ^∞₀𝑒^−𝑟𝑡𝜉_𝑖(𝑡)=𝑑𝑡_∞0𝑟𝑒^−𝑟𝑡𝑊^NBS_𝑖𝑠^𝐶(𝑡)−𝑒^−𝑟𝑡𝑑𝑊𝑑𝑡^NBS𝑖𝑠^𝐶(𝑡)𝑑𝑡=−𝑒^−𝑟𝑡𝑊^NBS_𝑖𝑠^𝐶(𝑡)^∞₀=𝑊^NBS_𝑖𝑠^𝐶(0)=𝑊^NBS_𝑖𝑠₀𝑊,𝑖=1,2,_NBS𝑖𝑠^𝐶(𝑡)−𝑊^𝑁_𝑖𝑠^𝐶(𝑡)=12²_𝑖=1𝑊^𝐶_𝑖𝑠^𝐶(𝑡)−𝑊^𝑁_𝑖𝑠^𝐶(𝑡)=12DC𝑠^𝐶(𝑡),𝑖=1,2, (46) where DC𝑠^𝐶(𝑡)=²_𝑖=1𝑊^𝐶_𝑖𝑠^𝐶(𝑡)−𝑊^𝑁_𝑖𝑠^𝐶(𝑡=1)2𝑎−𝐴₂𝑠^𝐶(𝑡)²+𝑏−𝐵₂𝑠^𝐶(𝑡)+𝐶≥0. (47) Hence, 𝑊^NBS_𝑖(𝑠^𝐶(𝑡))−𝑊^𝑁_𝑖(𝑠^𝐶(𝑡)), 𝑖=1,2 has the same sign as (1/2)DC(𝑠^𝐶(𝑡)).
In fact, if DC(𝑠₀)<0, the cooperative game will not be played. So the next section will discuss the conditions of DC(𝑠₀)≥0.

5.4. Sufficient Conditions for DC(𝑠^𝐶(𝑡))≥0

Theorem 18. If the cooperative solution is interior, and DC(𝑠^𝐶(𝑡)) is increasing for DC(𝑠₀)>0, DC(𝑠^𝐶(𝑡))≥0 for all 𝑡∈[0,∞) holds.

Proof. If for all 𝑡∈[0,∞), then 𝑠^𝐶(𝑡)>𝑠₀, that is, the theorem is proved.
Next we prove 𝑠^𝐶(𝑡)>𝑠₀.
From (34), one can have 𝑠^𝐶(𝑡)−𝑠₀=𝑒^{(2𝜇2𝑎2−𝜀)𝑡}−1𝜇𝛼₁+𝜇𝛼₂+2𝜇²𝑏2𝜇²𝑎²−𝜀−𝑠₀. (48) Since the GHG cooperative emission is interior, and 𝑒^𝐶_𝑖=𝛼_𝑖+𝜇(𝑎𝑠+𝑏), 𝑖=1,2, one can get 𝛼_𝑖+𝜇𝑏>−𝜇𝑎𝑠>0, 𝑖=1,2, so (𝜇𝛼₁+𝜇𝛼₂+2𝜇²𝑏)/(2𝜇²𝑎²−𝜀)<0.
According to 𝑎<0 and 𝑒^{(2𝜇2𝑎2−𝜀)𝑡}−1<0, 𝑠^𝐶(𝑡)>𝑠₀ holds.
It is proved.

From (45), one can get the instantaneous payoff 𝜉_𝑖(𝑡), 𝑖=1,2 as follows: 𝜉₁(𝑡)=12𝑟𝐴₁+12𝑟12𝑎𝑠²−()𝜇𝑒+𝑏𝑠+𝑐𝑎𝑠+𝑏₁+𝑒₂−1−𝜀𝑠2𝑎𝜎²−12𝑟12𝐴₂𝑠²+𝐵₂𝑠+𝐶₂−𝐴₂𝑠+𝐵₂𝜇𝑒₁+𝑒₂−1−𝜀𝑠2𝐴₂𝜎²,𝜉₂(𝑡)=12𝑟12𝐴₂𝑠²+𝐵₂𝑠+𝐶₂−𝐴₂𝑠+𝐵₂𝜇𝑒₁+𝑒₂−1−𝜀𝑠2𝐴₂𝜎²+12𝑟12𝑎𝑠²−()×𝜇𝑒+𝑏𝑠+𝑐𝑎𝑠+𝑏₁+𝑒₂−1−𝜀𝑠2𝑎𝜎²−12𝑟𝐴₁. (49)

6. Conclusion

Taking the effect of uncertainty into consideration, we use stochastic differential game to build cooperative and noncooperative game model for global GHG emission between developing and developed countries. Then we calculate the feedback Nash equilibrium and the Pareto optimal solution, give the globally feasible and sustainable conditions of the cooperative agreement, and propose the distribution method of payoff. At last, we illustrate the above results by numerical simulations. These results show that developing and developed countries would rationally adopt the Nash equilibrium as their GHGs emissions Pareto optimal solution in the long term.

Acknowledgments

This work is supported partly by Excellent Young Scientist Foundation of Shandong Province (Grant no. BS2011SF018), National Social Science Foundation of China (Grant no. 12BJY103), Humanities and Social Sciences Foundation of the Ministry of Education of China (Grant no. 11YJCZH200), and National Natural Science Foundation of China (Grant no. 71272148).