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4Managing Catastrophe-bound Industrial Pollution with Game-theoretic Algorithm: the St Petersburg
Initiative 1
David Yeung2, Leon Petrosyan3
2 Center of Game Theory, Hong Kong Baptist University
E-mail address: [email protected]
3 Center of Game Theory, St. Petersburg State University,
Faculty of Applied Mathematics - Control Processes, Graduate School of Management E-mail address: [email protected]
Abstract. After several decades of rapid technological advancement and economic growth, alarming levels of pollutions and environmental degradation are emerging all over the world. Reports are portraying the situation as an industrial civilization on the verge of suicide, destroying its environmental conditions of existence with people being held as prisoners on a runaway catastrophe-bound train. Though cooperation in environmental control holds out the best promise of effective action, limited success has been observed. Existing multinational joint initiatives like the Kyoto Protocol can hardly be expected to offer a long-term solution because (i) the plans are limited to a confined set of controls like gas emissions and permits which is unlikely be able to offer an effective mean to reverse the accelerating trend of environmental deterioration, and (ii) there is no guarantee that participants will always be better off and, hence, be committed within the entire duration of the agreement.To create a cooperative solution a comprehensive set of environmental policy instruments including taxes, subsidies, technology choices, pollution abatement activities, pollution legislations and green technology R&D has to be taken into consideration.
1 This research was supported by Hong Kong Baptist University FRG Grant FRG/05-06/II-
22, FRG/06-07/II-39, Strategic Development Fund 03-17-224 and European Commission TOCSIN
Project RTD REG/I.5 (2006) D/553242.
The implementation of such a scheme would inevitably bring about different implications in cost and benefit to each of the participating nations. To construct a cooperative solution that every party would commit to from beginning to end, the proposed arrangement must guarantee that every participant will be better-off and the originally agreed upon arrangement remain effective at any time within the cooperative period for any feasible state brought about by prior optimal behavior. This is a “classic” game-theoretic problem. This paper applies the latest discoveries in cooperative game theory and mathematics by researchers at the Center of Game theory in St. Petersburg State University to suggest solutions and means to solve this deadlock problem of global environmental management. Such an approach would yield an effective policy menu to tackle one of the gravest problems facing the global market economy.
Keywords: Differential games, cooperative solution, subgame consistency,
Industrial pollution.
Introduction
After decades of rapid technological advancement and economic growth, alarming levels of pollutions and environmental degradation are emerging globally. Due to the geographical diffusion of pollutants, unilateral response of one nation or region is often ineffective. Reports are portraying the situation as an industrial civilization on the verge of suicide, destroying its environmental conditions of existence with people being held as prisoners on a runaway catastrophe-bound train. Though global cooperation in environmental control holds out the best promise of effective action, limited success has been observed. This is the result of many hurdles, ranging from commitment and sharing of costs to disparities in future developments. It is hard to be convinced that multinational joint initiatives like the Kyoto Protocol or pollution permit trading can offer a long-term solution because (i) the plans are limited to a confined set of controls like gas emissions and permits which is unlikely be able to offer an effective mean to reverse the accelerating trend of environmental deterioration, and (ii) there is no guarantee that participants will always be better off and, hence, be committed within the entire duration of the agreement.
To construct a theoretical framework capturing the essence of a transboundary industrial pollution paradigm a differential game approach is adopted. Differential games provide an effective tool to study pollution control problems and to analyze the interactions between the participants’ strategic behaviors and dynamic evolution of pollution. Applications of noncooperative differential games in environmental studies can be found in [Yeung, 1992], [Dockner and Long, 1993], [Tahvonen, 1994], [Stimming, 1999], [Feenstra et al 2001] and [Dockner and Leitmann, 2001]. Cooperative differential games in environmental control are presented by [Dockner and Long, 1993], [Jorgensen and Zaccour, 2001], [Fredj et al., (2004)], [Breton et al., 2005 and 2006], and [Petrosyan and Zaccour, 2003]. To incorporate the widely observed uncertainty phenomenon in pollution accumulation a stochastic differential
game framework is adopted. Cooperative stochastic differential games with resource and environmental management contents include [Haurie et al., 1994], [Yeung and Petrosyan, 2004] and [Yeung, 2007].
To formulate the foundation for an effective policy menu to tackle one of the gravest problems facing the global market economy this analysis proposes a new cooperative initiative involving a comprehensive set of environmental policy instruments including taxes, subsidies, technology choices, pollution abatement activities, pollution legislations and green technology R&D. The implementation of such a scheme would inevitably bring about different implications in cost and benefit to each of the participating nations. To construct a cooperative solution that every party would commit to from beginning to end, the proposed arrangement must guarantee that every participant will be better-off and the originally agreed upon arrangement remain effective at any time within the cooperative period for any feasible state brought about by prior optimal behavior. This condition is known as a subgame consistency (see [Yeung and Petrosyan, 2004]) which guarantees that no participants would possess incentives to deviate from the previously adopted optimal cooperative behav-ior.To resolve this “classic” game-theoretic problem we further develop and apply the latest discoveries in cooperative game theory and mathematics by researchers at the Center of Game theory in St Petersburg State University (see [Yeung and Petrosyan, 2004, 2005, 2006a and 2006b], and [Yeung et.al., 2006], [Petrosyan and Yeung, 2007], and [Yeung, 2007]) to suggest solutions and means to solve this deadlock problem of global environmental management.
Last but not least, the analysis provides a payment distribution mechanism which governs each nation’s share of benefits and costs under the cooperative scheme so that the proposed subgame consistent solution can be realized. The analysis is also expected to open up a forum for policy research on global cooperative initiatives in environmental management.
The paper is organized as follows. Section 1 provides an analytical framework to study transboundary industrial pollution management. Noncooperative outcomes are characterized in Section 2. Cooperative arrangements, group optimal actions, and individually rational and subgame-consistent imputations are examined in Section 3. A payment distribution mechanism bringing about the proposed subgame-consistent solution is derived in Section 4. Policy Implications are discussed in Section 5 and concluding remarks are given in Section 6.
1. An Analytical Framework
In this section we present an analytical framework to study transboundary industrial pollution management.
1.1. The Industrial Sector
Consider a global economy which is comprised of n nations. At time instant s the demand system of the outputs of the nations is
Pi(s) = f[qi(s),q2(s), • • • ,qn(s), s], i & N = {1, 2, • --n},
(1)
where Pi(s) is the price vector of the output vector of nation i and qj(s) is the output of nation j. The demand system (1) shows that the world economy is a form of generalized differentiated products oligopoly.
Industrial profits of nation i at time s can be expressed as:
/i[qi(s),q2(s), ••• ,q„(s),s]qi(s) - ci[qi(s),vi (s)], for i £ N, (2)
where vi(s) is the set of environmental policy instruments of government i and ci[qi(s),vi(s)] is the cost of producing qi(s) under policy vi(s).
As mentioned before vi(s) is nation i’s comprehensive set of policy instruments including taxes, subsidies, technology choices, pollution abatement activities, pollution legislations and green technology R&D. Profit maximization by the industrial sectors yields:
/i[qi(s) q2(s) •••, qn(s), s] + /lqi[qi(s) q2(s) • • • qn(s), s]qi(s) -
- Cqi [qi(s),vi(s)] = 0, for i £ N. (3)
Condition (3) is a system of implicit functions in q(s) = [q1(s),q2(s), ••• qn(s)] with government policies v(s) = [v1(s), v2(s), • • • vn(s)] being regarded as parameters. The existence of a market equilibrium reflects the satisfaction of the Implicit Function Theorem in (3), and nation i's instantaneous market equilibrium output can be expressed as:
q*(s) = qi[vl(s),v2(s), ••• ,vn(s),s] = ql[v(s),s], for i £ N. (4)
One can readily observe from (4) that each nation’s output decision depends on government environmental policies.
1.2. Accumulation Dynamics of Pollutants
Industrial production emits pollutants into the environment, and the amount of pollution created by different nations’ outputs may be different. The pollutant will then add to the stock of existing pollution. Each government adopts its own pollution abatement policy to reduce the pollution stock. Let x(s) C Rm denote the level of pollution at time s, the dynamics of pollution stock is governed by the stochastic differential equation:
dx(s) =
n
53 a vdj
j=i
a fa(s),vj(s)] bj (s),x(s)] _ ¿[x(sMs))
j=i
ds +
+ a[x(s)]dz(s),
x(t0 ) = xt0,
where
a is a noise parameter and z(s) is a Wiener process,
aj [qj (s), Vj (s)] is the amount of pollution created by qj (s) amount of output produced under policy Vj(s),
Uj(s) is the pollution abatement effort of nation j,
bj[uj(s),x(s)] is the amount of pollution removed by Uj(s) unit of abatement effort
of nation j, and £[x(s)] is the natural rate of decay of the pollutants.
Moreover, S(x) is negatively related to x reflecting the phenomenon that the natural rate of decay declines as the level of pollution stock rises.
The stochastic nature of (5) reflects the uncertainty in the evolution of the pollution stock.
1.3. The Governments’ Objectives
The governments have to promote business interests and at the same time handle the financing of the costs brought about by pollution. In particular, each government maximizes the gains in the industrial sector plus tax revenue minus expenditures on pollution abatement and damages from pollution. The instantaneous objective of government i at time s can be expressed as:
where cP [vi (s)] is the cost of implementing the vector policy instrument vi (s), c“ [u j (s)] is the cost of employing ui amount of pollution abatement effort, and hi[x(s)] is the value of damage to country i from x(s) amount of pollution.
The governments’ planning horizon is [to,T]. It is possible that T may be very large. The discount rate is r. At time T, the terminal appraisal of pollution damage is gi[x(T)] where dg%/dx < 0. Each one of the n governments seeks to maximize the integral of its instantaneous objective (6) over the planning horizon subject to pollution dynamics (5) with controls on the level of abatement effort and output tax.
Substituting qi(s), for i e N, from (4) into (5) and (6) one obtains a stochastic differential game in which government i e N seeks to:
f[qi(s),q2(s), ■■■ ,q„(s),s]q¿(s) - ci[qi(s),wi (s)] - c[ [^(s)]
— c“Ms)] — hi[x(sУ], i G N,
(6)
max Et0 {
ft{q1v(s), s], q2[v(s), s], ■■■ , qn[v(s), s], s}q>(s), s]
(7)
subject to
n
n
dx(s) = I aj{q-i [v(s), s], Vj (s)} — bj [uj (s), x(s)] — J[x(s)]x(s) ds +
j=i
+ a[x(s)]dz(s), x(to)= xt0.
j=1
(8)
2. Noncooperative Outcomes Characterization
Since the payoffs of nations are measured in monetary terms, the game (7)-(8) is a transferable payoff game.
Under a noncooperative framework, a feedback Nash equilibrium solution (if it exists) can be characterized as (see [Basar and Olsder, 1995] and [Yeung and Petrosyan, 2006b]):
Definition 1. A set of feedback strategies {u*(t) = ¡ii(t,x), v*(t) = $i(t,x), for i e N} provides a Nash equilibrium solution to the game (7)-(8) if there exist suitably smooth functions : V(to )i(t, x) : [to, T] xR ^ R, i e N, satisfying the following partial differential equations:
-VtMi(t,x) -\Y^akiWV&%x) = k,j
= max{ f{q [vi,$=i (t,x),t],q [vi,$=i(t,x),t], ■■■ ,
Vi,Ui L
qn[vi,0=i(t,x),t]} x qi[vi,0=i(t,x),t] - c{qi[vi,0=i(t, x),t],Vi} -
n
- cP [Vi] - ca[ui]- hi(x) e-r(t-to) + VXto)i^Y^ aj W [vi,$=i (t,x),t],Vj }-
j=1 n
- bi(ui,x) - bj [ij (t,x),x]- 5(x)x },
j=1,j=i
V(to)i(T, x) = gi[x]e-r(T-to), (9)
where
$=i(t,x) = [$1(t,x),$2 (t,x), ■■■ ,4>i-i (t,x),$i+1 (t,x), ■■■ ,$n(t,x)]. (10)
In a prevailing Nash equilibrium the function V(to)i(t, x) is then the integral:
Eto{Jt f{ql[4>(s,x(s)),s],q2 [$(s,x(s)),sL ■■■ ,qn [$(s,x(s)),s],s} x
x qi[$(s,x(s)), s] - c{qi[$(s,x(s)), s],$i(s,x(s))} - cp [$i(s,x(s))] -
- ca[ii(s,x(s))] - hi[x(s)]
e-r(s-to)ds +
+ gi[x(T)]e r(T to) x(t) = x,^ (11)
for i e N.
The game equilibrium dynamics then becomes:
n n
dx(s) = [53 aj {qi[$(s,x(s)),s],$j (s,x(s))}-$3 bj [ij(s,x(s)),x(s)] -j=i j=i
-J[x(s)]x(s) ds + a[x(s)]dz(s), x(to) = xto. (12)
Remark 1. One can readily verify that V(t )i(t,xt) = V(to)i(t,xt)er(T to), for t e e [t0, T], is the value function to player i at time t e [t, T] when the state x(t) = xt in the game (7)-(8) which starts at time t.
3. Cooperative Arrangement
Now consider the case when all the nations want to cooperate and agree to act so that an international optimum could be achieved. For the cooperative scheme to be upheld throughout the game horizon both group rationality and individual rationality are required to be satisfied at any time.
Group optimality ensures that all potential gains from cooperation are captured. Failure to fulfill group optimality leads to condition where the participants prefer to deviate from the agreed upon solution plan in order to extract the unexploited gains. Individual rationality is required to hold so that the payoff allocated to a nation under cooperation will be no less than its noncooperative payoff. Failure to guarantee individual rationality leads to condition where the concerned participants would reject the agreed upon solution plan and play noncooperatively.
Finally, as mentioned in Introduction, to ensure that the cooperative solution is dynamically consistent, the agreement must be subgame-consistent. In the absence of a punishment scheme, the cooperative plan will dissolve if any of the nations deviates from the agreed-upon plan.
3.1. Group Optimality and Cooperative State Trajectory
Consider the cooperative stochastic differential games with payoff structure (7) and dynamics (8). To secure group optimality the participating nations seek to maximize their joint expected payoff by solving the following stochastic control problem:
subject to (8).
Invoking Fleming’s (1969) technique in stochastic control a set of controls
constitutes an optimal solution to the stochastic control problem (13) and (8) if there exists continuously differentiable function:
n
— hi[x(s)] e-r(t-to)ds + ^ gi[x(T)]e-r(T-t0)}.
(13)
{[v*i(t),u*(t)] = [^i(t,x),Wi(t,x)], i G N}
W(to)(t, x) : [t0, T] x R ^ R,
i e N, satisfying the following partial differential equations:
-wfVx) - =
k,j
n
= max { V'/i[q1(v,t),q2(v,t), ■■■ ,qn (v,t),t]ql (v,t)-
V1,V2,--- ,Vn;Ui,U2,--- ,Un K
i=1
-ci[qi(v,t),vi] - cP(vi) - ca(vi) - hi(x)]e-r(t-to) +
n n
+WX (t,x)^^2 aj [qj (v,t),Vj ] - ^2 bj (uj, x) - S(x)x |, j=1 j=1
and
n
W(to)(T,x) = J2 gi(x)e-r(T-to). (14)
i=1
Hence, the players will adopt the cooperative control {[^i(t, x), mi(t, x)], for i e N and t e [t0, T]}. The value function W(to)(t, x) is then the integral:
rT n
UT n
x(s)) «], q2 [^(sx(s)), sL
, qn[^(s, x(s)), s], s}qt[^(s, x(s)), s] - ci{qi[^(s, x(s)), s], fa(s, x(s))} -
- cp[^i(s,x(s))] - c“N(s,x(s))] - Mx(s)]
e-r(s-to)ds +
n
+ gi[x(T)]e-r(T-to) x(t) = x^ for i e N. (15)
i=1
The optimal trajectory under cooperation becomes
nn
dx(s) = E aj {qj №(s,x(s)),s],^j(s,x(s))} -$3 bj [mj(s,x(s)),x(s)] -j=1 j=1
- S[x(s)]x(s)]ds + a[x(s)]dz(s), x(to) = xto. (16)
The solution to (16) can be expressed as:
/t n
{^2 aj{qj №(s,x* (s)),s,^j (s,x*(s))} -
o j=1
n /* t
-^^ bj [ttj (s,x* (s)),x* (s)]-S[x*(s)]x* (s)]}ds + ax* (s)dz(s). (17)
to
j=1
We use X* to denote the set of realizable values of x*(t) at time t generated by (17). The term x* is used to denote an element in the set X*.
The cooperative control for the game Tc(x0, T -10) over the time interval [t0,T] can be expressed more precisely as:
Note that for group optimality to be achievable, the cooperative controls (18) must be exercised throughout time interval [t0,T].
Remark 2. One can readily verify that W(t)(t,x*) = W(to)(t, x*)er(T-to), for t e [t0,T], is the value function at time t e [t,T] of the stochastic control problem (8) and (13) which starts at time t with x(t) = x* e X*.
3.2. Individually Rational and Subgame-consistent Imputation
An agreed upon optimality principle must be sought to allocate the cooperative payoff. In a dynamic framework individual rationality has to be maintained at every instant of time within the cooperative duration [t0, T] given any feasible state generated by the cooperative trajectory (21). For t e [t0,T], let £(t )i(т, x*) denote the solution imputation (payoff under cooperation) over the period [t, T] to player i e N given that the state is x* e X*. Individual rationality along the cooperative trajectory requires:
Since nations are asymmetric and the number of nations may be large, a reasonable solution optimality principle for gain distribution is to share the expected gain from cooperation proportional to the nations’ relative sizes of expected noncooperative payoffs. As mentioned before, a very stringent condition - subgame consistency -is required for a credible cooperative solution under a dynamic stochastic framework. In particular, the solution optimality principle must be maintained in any subgame which starts at a later time with any feasible state brought about by prior optimal behaviors so that no player has incentives to deviate from the previously adopted optimal behavior throughout the game.
In order to satisfy the property of subgame consistency, the optimality principle of sharing the expected gain proportional to the nations’ relative sizes of expected noncooperative payoffs has to remain in effect throughout the cooperation period. Hence, the solution imputation scheme {£(t)i(T,x*); for i e N} has to satisfy:
Condition 1.
fa(t,xi (t)) and wi(t,xi (t)) for t G [t0,T] and i G N.
(18)
for i G N, xi G Xi and t G [t0, T].
One can easily verify that the imputation scheme in Condition 1 satisfies both group optimality and individual rationality. Crucial to the analysis is the formulation of a payment distribution mechanism that would lead to the realization of Condition 1. This will be done in the next Section.
4. Payment Distribution Mechanism
Following Yeung and Petrosyan (2004 and 2006b), we formulate a payment distribution scheme over time so that the agreed upon imputation Condition 1 can be realized for any time instant t e [t0, T] with the state being x* e X*. Let the vectors
B(s,x**) = [Bi(s,x*),B2 (s,x* ), ■■■ ,Bn(s,x* )]
denote the instantaneous payment to the n nations at time instant s when the state is x* e X*. A terminal value of gi[x*r] is realized by nation i at time T.
To satisfy Condition 1 it is required that
-r(T-T )
x(t ) = x* },
= ET{J Bi(s, x*(s)e-r(s-T^ds + gi[x*r]e for i e N,x* e X* and t e [t0, T].
To facilitate further exposition, we use the term £(t)i(t,x*) which equals to:
(21)
f t
Et { Bi (s,x*(s))e-r(s-T ) ds + gi[x*T ]e
i T/f-rWi VV
-r(T-T )
En=i v(T )i(t,x*)
j V(t)j (t,x* )
x(t) = x*t} =
W (t)(t,x* )e-r(t-T )
e(t)i (t,xtt )e-
r(t-T )
for x* G X* and t G [t,T ].
(22)
to denote the expected present value (with initial time set at t) of nation i's cooperative payoff over the time interval [t, T].
Theorem 1. A distribution scheme with a terminal payment —gz[xj, — x1] at time T and an instantaneous payment at time t e [t0,T]:
Bî(t, x*
k,j
n X)i(t,x* ) 1 X fm aj {qj №(t,x*t ),T],^j (t,x* )}-
z t=T J L
j=1
— bj [wj(T,x*),x*] — S(x*)x* , for i G N yields Condition 1.
j=i
x
x
t=T.
Proof.
Since £(t)l(t, x*) is continuously differentiable in t and x* , using (22) and Remarks
1 and 2 one can obtain:
pT+Ai
/r+ At
Bi(s,xt (s))e-r(s-T^ds x(t) = x* }
= Et{£(t)i(T, x* ) - e-rAtÇ(T+At)i(T + At, x* +At) x(T) = x;} =
= Et{£(t)i(T, x;) - £(t)i(T + At, x;+At) |x(r) = x;}, for i G W and t G [to, T],
(24)
where
AxT = [X} WWT x*T ), t],^3 (т, x*T )}-X^ bj [m3 (т, x*; ),x; - S(x*r )x-
j=l
j=l
+a(x* )AzT + o(At),
AzT = z(t + At) — z(t), and ET [o(At)]/At ^ 0 as At ^ 0. With At ^ 0, condition (24) can be expressed as:
At +
Et {Bí(t, x; )At + o(At)} = Et {- éT ]i(t, x T )
At
&4)t(t,x*T ) j X \Y1 aj {qj №(t,x*t ),T],^j (t,x*; )}-
t í=tJ L
j=l
“X16j[II7j(T’X*)’X*]_(5(X*)X* At“ôXlCrfcj(X*) £xix*{t,x*t)
j=l
k,j
At
é¿ (t,x*t) a(x)AzT - o(At)}
(25)
Taking expectation and dividing (25) throughout by At, with At ^ 0, yields (23). Hence, Theorem 21 follows.
Finally, explicit illustrative examples of the theoretical framework can be found in Yeung (2007) and Yeung and Petrosyan (2008).
5. Policy Implications
Facing with increasing demand for a sustainable solution the international community has responded to the deteriorating problem of global pollution. Over a decade ago, most countries joined an international treaty - the United Nations Framework Convention on Climate Change (UNFCCC) - to consider what can be done to reduce global warming and to cope with whatever temperature increases are inevitable.
t=T-
t=T-
Recently, a number of nations have approved an addition to the treaty: the Kyoto Protocol, which has more powerful and legally binding measures. In brief, the Kyoto Protocol is an international agreement, which builds on the United Nations Framework Convention on Climate Change, and sets legally binding targets and timetables for cutting the greenhouse-gas emissions of industrialized countries. Conditions for entry into effect are that some UNFCCC parties cut greenhouse-gas emissions of at least 5% from 1990 levels in the commitment period 2008-2012. As for December 2006, 169 countries and other governmental entities ratified the agreement. Notable exceptions include the United States. Other countries, like India and China, which have ratified the protocol, are not required to reduce carbon emissions under the present agreement despite their relatively large industrial production activities.
As mentioned before placing a constraint just on certain types of pollution emissions cannot offer a long-term solution, because the plans are limited to a confined set of controls like gas emissions and permits which is unlikely be able to offer an effective mean to reverse the accelerating trend of environmental deterioration, and there is no guarantee that participants will always be better off and, hence, be committed within the entire duration of the agreement. Guided by the analysis shown above, a grand coalition of all nations should be formed to pursue a comprehensive cooperative scheme of industrial pollution abatement. In particular, the entire set of policy instruments available - including environmental taxes and charges, adoption of environment-friendly production technology, subsidy to the replacement of polluting techniques, joint research and development in clean technology, restoration and preservation of the natural ecosystem, and legislations to outlaw environmentally unacceptable practices - will be used achieve an optimal cooperative outcome. A payment distribution mechanism has to be formulated so that cooperative gains will be shared according to the proportions of the nations relative sizes of expected noncooperative payoffs throughout the planning horizon. In sum, appropriate policy coordination will lead to the enhancement of economic performance and the realization of a cleaner environment.
This analysis opens up a novel policy forum for the international community. A particularly relevant instance would be the formation of a United Nations Agency to coordinate international cooperative actions on pollution and climate change. The Agency is proposed to be comprised of three divisions. An executive branch would be established to coordinate adoption and development of clean technology, pollution abatement activities, use of materials, waste disposal, mode of resource extraction and cooperation in environmental R&D. A financial branch (or FUND) would be set up to handle pollution charges, clean technology subsidies and allocate payoff distributions so that the agreed upon optimality principle will be realized throughout the cooperative period. Lastly, a legislative body would be in place to enact regulations on the use of dirty technologies, toxic disposal, pollutant emissions, activities damaging the environment and violation of the cooperative agreement.
Finally, a large scale scheme is in order for research in mechanism design theory initiated by Hurwicz (1973) and refined and applied Myerson (1989) and Maskin
(1999). In particular, mechanism designs for conventional markets in the face of impacts from a comprehensive set of environmental policy instruments including taxes, subsidies, technology choices, pollution abatement activities, pollution legislations and green technology R&D have to be considered. In addition, mechanism designs for inter-government transfers, institution formation, like-market and beyond-conventional market arrangements have also to be investigated.
6. Conclusions
After several decades of rapid technological advancement and economic growth, alarming levels of pollutions and environmental degradation are emerging all over the world. Though cooperation in environmental control holds out the best promise of effective action, limited success has been observed. Existing multinational joint initiatives like the Kyoto Protocol or pollution permit trading can hardly be expected to offer a long-term solution because there is no guarantee that participants will always be better off within the entire duration of the agreement.
A practicable cooperative scheme which guarantees that every participant will be better-off and the agreed upon arrangement will remain optimal along the cooperative path is characterized. Such an approach would yield an effective policy menu to tackle one of the gravest problems facing the global market economy. Finally this analysis opens up a novel policy forum for the international community.
References
Basar T., Olsder G.J. 1995. Dynamic Noncooperative Game Theory, 2nd Edition. Academic Press, London, England.
Breton M., Zaccour G., Zahaf M. 2005. A Differential Game of Joint Implementation of Environmental Projects. Automatica, 41: 1737-1749.
Breton M., Zaccour G., Zahaf M. 2006. A Game-Theoretic Formulation of Joint Implementation of Environmental Projects. European Journal of Operational Research, 168: 221-239.
Dockner E. J., Leitmann G. 2001. Coordinate Transformation and Derivation of Open-Loop Nash Equilibria. Journal of Economic Dynamics and Control, 110: 1-15.
Dockner E. J., Long N. V. 1993. International Pollution Control: Cooperative Versus Noncooperative Strategies. Journal of Environmental Economics and Management, 25: 13-29.
Feenstra T., Kort P. M., De Zeeuw A. 2001. Environmental Policy Instruments in an International Duopoly with Feedback Investment Strategies. Journal of Economics Dynamics and Control. 25: 1665-1687.
Fleming W. H. 1969. Optimal Continuous-Parameter Stochastic Control. SIAM Review, 11: 470-509.
Fredj K., Martin-Herran G., Zaccour G. 2004. Slowing Deforestation Pace through Subsidies: A Differential Game. Automatica, 40: 301-309.
Haurie A., Krawczyk J. B., Roche M. 1994. Monitoring cooperative equilibria in a stochastic differential game. Journal of Optimization Theory and Applications, 81: 73-95.
Hurwicz L. 1973. The design of mechanisms for resource allocation. American Economic Review. Papers and Proceedings, 63: 1-30.
Jorgensen S., Zaccour G. 2001. Time Consistent Side Payments in a Dynamic Game of Downstream Pollution. Journal of Economic Dynamics and Control, 25: 1973-1987.
Maskin E. 1999.Nash Equilibrium and Welfare Optimality. Review of Economic Studies, 66: 23-38.
Myerson R. 1989. Mechanisms design. The New Palgrave: Allocation, Information and Markets in: J. Eatwell, M. Milgate and P. Newman (eds.). Norton, New York.
Petrosyan L., Yeung D.W.K. 2007. Subgame-consistent Cooperative Solutions in Randomly-furcating Stochastic Differential Games. International Journal of Mathematical and Computer Modelling (Special Issue on Lyapunov’s Methods in Stability and Control), 45: 1294-1307.
Petrosyan L., Zaccour G. 2003. Time-Consistent Shapley Value Allocation of Pollution Cost Reduction. Journal of Economic Dynamics and Control, 27: 381-398.
Singh N., Vives X. 1984. Price and Quantity Competition in a Differentiated Duopoly. Rand Journal of Economics, 15: 546-554.
Stimming M. 1999. Capital Accumulation Subject to Pollution Control: Open-loop versus Feedback Investment Strategies. Annals of Operations Research, 88: 309336.
Tahvonen O. 1994. Carbon Dioxide Abatement as a Differential Game. European Journal of Political Economy, 10: 685-705.
Yeung D.W.K. 1992. A Differential Game of Industrial Pollution Management. Annals of Operations Research, 37: 297-311.
Yeung D.W.K. 2007. Dynamically Consistent Cooperative Solution in a Differential Game of Transboundary Industrial Pollution. Journal of Optimization Theory and Applications, 135 (1).
Yeung D.W.K., Petrosyan L. 2004. Subgame Consistent Cooperative Solutions in Stochastic Differential Games. Journal of Optimization Theory and Applications, 120: 651-666.
Yeung D.W.K., Petrosyan L. 2005. Subgame consistent solution of a cooperative stochastic differential game with nontransferable payoffs. Journal of Optimization Theory and Applications: 701-724.
Yeung D.W.K., Petrosyan L. 2006a. Dynamically Stable Corporate Joint Ventures. Automatica, 42: 365-370.
Yeung D.W.K., Petrosyan L. 2006b. Cooperative Stochastic Differential Games. Springer-Verlag, New York.
Yeung D.W.K., Petrosyan L. 2008. A Cooperative Stochastic Differential Game of Transboundary Industrial Pollution. Automatica.
Yeung D.W.K., Petrosyan L., Yeung P.M. 2006. Subgame Consistent Solutions for a Class of Cooperative Stochastic Differential Games with Nontransferable Payoffs. Annals of the International Society of Dynamic Games, 9: 153-170.
