^{1,2}

^{1,3}

^{1}

^{2}

^{3}

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.

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 CO_{2} 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. [

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 [

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 [

The remainder of this paper is organized as follows. In Section

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 [

Player

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

Consider

Without consensus agreement between both players, they will, alone, maximize their payoffs as follows:

We use

The rational economic explanation for the noncooperative game is that (

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 [

We use

In order to reduce the difficulty of mathematical handling, we assume that

When the state variable

The strategy

If above conditions are satisfied, its feedback Nash equilibrium is

For player 1,

Substituting

As [

(1) The first-order extreme value conditions show that player 2 determines its GHG emissions whose marginal profit

(2) From Theorem

(3) Theorem

The individual gross payoff in the noncooperative game can be written as

From Definition

For player

The expectation stock and its limit in noncooperative game feedback Nash equilibrium satisfy

Substituting

The strategy

Hence, the players will adopt the cooperative control

The Pareto optimal solution of the cooperative stochastic differential game

Differentiating (

Let

As [

(1) The Pareto optimal solution

(2) Theorem

The cooperative value function is given by

The expectation stock and its limit in the Pareto optimal solution of the cooperative stochastic differential game

Substituting

Let

Let

We consider the long-term cooperation agreement related to the following crucial questions.

When is a cooperative agreement globally feasible?

In the parameter space where cooperation is achievable, how should the cooperative dividend be distributed among players?

What can ensure that players abide by the cooperative agreement over time?

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:

Here, DC

From Definition

Let

If

It is easy to prove it by analyzing the quadratic function DC

The above theorem states that the cooperative agreement is achievable if

Assuming that the cooperation is globally feasible, then it is an important problem for us to tackle and how to allocate DC

The NBS method will be applied into this paper. Let

(1) For non-cooperation stochastic differential game

(2) For cooperation stochastic differential game

It is crucial to build a credible basis for abiding strictly by the cooperative agreement in a long time. Inspired by [

Denote

Denote

For the sake of simplicity, let

Differential (

Equation (

If (

In fact, if

If the cooperative solution is interior, and

If for all

Next we prove

From (

According to

It is proved.

From (

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.

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).

_{2}emission quotas by acceptability