An environmental game with coupling constraints

Environmental Modeling and Assessment (2005) 10:153–158 DOI 10.1007/s10666-005-5254-8  Springer 2005 An environmental game with coupling constraints Mabel Tidball a and Georges Zaccour b a INTRA, LAMETA, Montpellier, France b GERAD & HEC, Montreal, Canada Received 16 September 2004; accepted 9 April 2005 We consider a game where players face environmental constraints. We derive and compare noncooperative, cooperative and umbrella scenarios. In the latter, the players couple their environmental constraints and implement Rosen’s normalized equilibrium. It is shown that the cooperative outcome can be generated as a normalized equilibrium and that the results obtained in the literature do not necessarily generalize to this constrained setting. 1. Introduction An established result in environmental economics is that when players (countries, regions, etc.) cooperate, then each player pollutes less than under a noncooperative mode of play and total cooperative welfare is higher than the sum of individual noncooperative welfares. This result has been obtained for unconstrained environmental optimization problems, i.e., each player can choose freely her (non-negative) emissions level. International environmental agreements, e.g., the Kyoto Protocol, require that each signatory country limits its emissions to an exogenously denned level. Recently, Breton et al. [1] studied this constrained two-player problem under different scenarios with the aim of assessing the impact of foreign investment in environmental projects on players’ welfares. In a first scenario, countries face an environmental constraint but do not invest abroad in environmental projects while in the second one this option is available. The difference in welfares between the two scenarios provided a measure of the benefit of foreign investments (or joint implementation). In a dynamic extension [2], it is shown that the environmental constraint is not necessarily welfare deteriorating. We reconsider the problem of environmental policy coordination between countries where each one of them faces an environmental constraint. We shall analyze and compare the following three scenarios which may be seen as alternative options for the players to cope with their environmental constraints: • Noncooperative scenario: each player optimizes her welfare under her own environmental constraint. The players interact through the damage cost, which is a function of total emissions, and seek a Nash equilibrium. • Cooperative scenario: the players decide to jointly optimize their welfares under a common environmental constraint. • Umbrella scenario: the players remain independent entities and optimize individually their welfares but under a joint environmental constraint. The solution concept used here is the normalized equilibrium proposed by Rosen [7]. The first two scenarios are standard in this area and could be seen as benchmarks to the last one. With the umbrella scenario, we wish to study the case where countries, e.g., European Community, agree to couple their individual environmental constraints while still pursuing individual optimization objectives. Doing so provides these countries with a greater latitude in satisfying their environmental targets by, e.g., abating more in cheaper locations. This class of noncooperative games with coupled constraints has been dealt with originally by Rosen [7] who introduced the concept of normalized equilibrium. To compute this equilibrium, one appends to each coupled constraint a normalized multiplier defined as a common Lagrange multiplier divided by a weight (a positive number) which is specific to each player. For any given vector of weights, one obtains a normalized equilibrium (assuming existence is at hand). For applications of normalized equilibrium to environmental economics games, see, e.g., Haurie and Zaccour [4], Haurie and Krawczyk [3], Krawczyk [5], Krawczyk and Uryasev [6]. An open question in Rosen’s formalism is how to choose the weights. In the above references, these weights are typically interpreted as taxes, set by for instance a regulator, in order to insure the satisfaction of the coupled constraint. Our objectives here are different. We wish indeed to address the following questions: • How normalized and Nash equilibria compare in terms of total emissions and welfares? • Does there exist a vector of weights which leads to the cooperative outcome? 154 M. Tidball, G. Zaccour / An environmental game with coupling constraints • Does the result stating that total emissions are lower under cooperation than under noncooperation remain always true when emissions are constrained? The rest of the paper is organized as follows. Section 2 introduces the model for two-player game and defines formally the scenarios. Section 3 characterizes and compares the solutions of the different scenarios. Section 4 extends the results to n players. Section 5 concludes. • In the umbrella scenario, player i, i = 1, 2, optimizes her welfare under a coupled constraint, i.e., e1 + e2
E1 + E2 . (4) max fi (ei ) − di (e1 + e2 ), ei It is easy to see that this scenario admits multiple equilibria. In order to obtain uniqueness, we shall look here for a normalized equilibrium à la Rosen and results will be superscripted by R. 3. Solutions and comparison 2. The model 3.1. Characterization of solutions We consider a two-player pollution control game. Each player aims at the maximization of her total welfare given as the difference between the net revenue from production of goods and services and the damage cost due to pollution. We assume that pollution is a proportional by-product of production and therefore player i’s revenues can be expressed as a function of emissions ei , i = 1, 2. Denote by fi (ei ) the nonnegative, twice-differentiable, concave and increasing net revenue function (gross revenue minus production cost). Damage cost depends on both players’ emissions. Denote this convex twice-differentiable increasing cost by di (e1 + e2 ). Player i’s welfare function is given by: wi (e1 , e2 ) = fi (ei ) − di (e1 + e2 ). (1) Assume that each country faces an environmental constraint of the form ei
E i , where Ei is an exogenous given upper bound on emissions. For instance, it could be the level agreed upon in an international treaty. The three scenarios sketched in the introduction are now defined formally. • In the noncooperative scenario, player i, i = 1, 2, optimization problem is max fi (ei ) − di (e1 + e2 ), ei ei
E i . (2) The solution concept adopted here is Nash equilibrium and results will be superscripted by N. The assumption made on the welfare functions implies the existence of Nash equilibria. We suppose that it is unique. • In the cooperative scenario, we assume that the two players agree to jointly maximize their welfares, i.e., max e1 ,e2 2
  fi (ei )−di (e1 +e2 ) , e1 +e2
E1 +E2 . (3) i=1 Optimal values will be superscripted by C. The assumption made on the welfare functions implies that the solution to this optimization problem is unique. We characterize and compare in this section the solutions for the three scenarios. Introduce, in R+ , the following sets   R1 = (e1 , e2 ): e1 < E1 , e2 < E2 ,   R2 = (e1 , e2 ): e1 < E1 , e2  E2 , e1 + e2 < E1 + E2 ,   R2′ = (e1 , e2 ): e1  E1 , e2 < E2 , e1 + e2 < E1 + E2 ,   R3 = (e1 , e2 ): e1 + e2 = E1 + E2 . Note that regions R2 and R2′ are symmetric and the results are analogous. We therefore focus on region R2 . First-order necessary conditions are for i = 1, 2 as follows: • Nash equilibrium:   N = fi (ei ) − di (e1 + e2 ) + λN LN i (Ei − ei ) i ei , λi fi′ (ei ) − di′ (e1 + e2 ) − λN i = 0, ei
E i , λN i  0, λN i (Ei − ei ) = 0, N N where LN i (ei , λi ) is player i’s Lagrangian and λi the Lagrange multiplier. • Normalized equilibrium:   R LR i ei , λ , ri = fi (ei ) − di (e1 + e2 ) λR (E1 + E2 − e1 − e2 ), ri λR fi′ (ei ) − di′ (e1 + e2 ) − = 0, ri e1 + e2
E1 + E2 , λR  0, λR (E1 + E2 − e1 − e2 ) = 0, + R R where LR i (ei , λ , ri ) is player i’s Lagrangian, λ the Lagrange multiplier and ri a positive weight assigned to player i. Note again that we have one Lagrange multiplier for all players. Uniqueness of a normalized equilibrium is guaranted by the following condition: let e = (e1 , e2 ), r = (r1 , r2 ) and g(e, r) = (r1 ∇1 w1 (e), r2 ∇2 w2 (e)), where ∇i w(e) is the gradient with respect to ei of w. Given r = (r1 , r2 ) such that ri > 0 and r1 + r2 = 1 for all e = (e1 , e2 ) and for all e = (e1 , e2 ) such that e1 + e2
E1 + E2 , ē1 + ē2
E1 + E2 (e − ē)g(ē, r) + (ē − e)g(e, r) > 0, 155 M. Tidball, G. Zaccour / An environmental game with coupling constraints then there exists a unique normalized equilibrium. We assume in the sequel that this condition is satisfied. • Cooperative solution: 2  
  fi (ei ) − di (e1 + e2 ) LC e i , λ C = i=1 + λC (E1 + E2 − e1 − e2 ), fi′ (ei ) − d1′ (e1 + e2 ) − d2′ (e1 + e2 ) − λC = 0, e1 + e2
E1 + E2 , λC  0, λC (E1 + E2 − e1 − e2 ) = 0, C C where LC i (ei , λ ) is the Lagrangian and λ the Lagrange multiplier. Remark 1. Note that if wi is independent of ej then the normalized Nash equilibrium when r = 1/2 and the cooperative solution coincide. Denote by AN ij (that we suppose greater or equal to zero) the interior Nash equilibrium value for player i and by BijN the best response of player i to the strategy ej  Ej . Denote N by BijR the emissions of player i when AN ij + Aj i  Ei + Ej . Finally, denote by AC ij the interior cooperative solution for player i and by BijC the cooperative solution value for player C i when AC ij + Aj i  Ei + Ej . The following propositions characterize the solutions for the three scenarios. Proposition 1. A Nash equilibrium is given by  N N N A , Aj i if AN  ij < Ei and Aj i < Ej ,     ij N N N   N N B , Ej if Aij < Ei and Aj i  Ej , ei , ej =  ij N  N  Ei , Bj i if AN  ij  Ei and Aj i < Ej ,   N (Ei , Ej ) if AN ij  Ei andAj i  Ej . (5) Proof. Straightforward from first-order equilibrium conditions.  Proposition 2. A normalized equilibrium is given by  N N N Aij , Aj i if AN  R R ij + Aj i < Ei + Ej , ei , ej =  R R  N Bij , Bj i if AN ij + Aj i  Ei + Ej . Proof. Straightforward from first-order normalized equilibrium conditions.  Proposition 3. A cooperative solution is given by  C C C Aij , Aj i if AC  C C ij + Aj i < Ei + Ej , ei , ej =  C C  C Bij , Bj i if AC ij + Aj i  Ei + Ej . Proof. Straightforward from first-order optimality conditions for the cooperative game.  3.2. Comparison The next two propositions compare Nash and Rosen equilibria and thus address one of our questions. They state that, for as long as the solutions are in regions R1 or R2 (R2′ ) choosing Nash equilibrium is better from both the environment (total emissions) and economics (welfares) perspectives for player 1 (for player 2). Proposition 4. In region R1 Nash and Rosen equilibria coincide. Proof. Follows immediately from propositions 1 and 2.  Proposition 5. In region R2 we have that e1N + e2N
e1R + e2R , w1 (e1N , e2N )  w1 (e1R , e2R ). Proof. In R2 we have R N AN 21 = e2  E2 = e2 , R AN 12 = e1 < E1 . First-order conditions show that Nash equilibrium is given by: f1′ (e1 ) = d1′ (e1 + e2 ), e2 = E 2 , and a normalized equilibrium by: fi′ (ei ) = di′ (e1 + e2 ), i = 1, 2. Let us call e1 = g(e2 ) the solution of f1′ (e1 ) = d1′ (e1 + e2 ). It is easy to see that g ′ (e2 ) = d1′′ (e1 + e2 ) < 0. f1′′ (e1 ) − d1′′ (e1 + e2 ) (6) As e2N
e2R , (6) implies that e1N  e1R . Moreover     e1N +e2N = (d1′ )−1 f1′ (e1N ) , e1R +e2R = (d2′ )−1 f1′ (e1R ) . As by assumption f1 , d1 and d1′ are increasing functions and f1′ is a decreasing function, we obtain the results.  The next proposition answers our second question by showing that if one defines the ri as proportions, then one can choose them as a mean to reach the cooperative solution, at least in region R3 . As mentioned previously, the literature has interpreted these numbers as a taxing device to insure, under still a non-cooperative mode of play, satisfaction of the coupled constraint. Our result provides an alternative perspective which is to design these numbers as a way to achieve cooperation as an equilibrium. Proposition 6. In R3 there exists ri , i = 1, 2, r1 + r2 = 1, such that eiC = eiR , i = 1, 2. Proof. In region R3 the cooperative solution is given by: fi′ (ei ) − d1′ (e1 + e2 ) − d2′ (e1 + e2 ) = λC , e1 + e2 = E 1 + E 2 , i = 1, 2, 156 M. Tidball, G. Zaccour / An environmental game with coupling constraints and the normalized equilibrium by i = 1, 2, The following proposition answers our last question, namely whether total emissions are always higher under noncooperation than under cooperation. To obtain that both solutions coincide we require: Proposition 7. Total Nash and cooperative emissions compare as follows: fi′ (ei ) − di′ (e1 + e2 ) = λR ri , e1 + e 2 = E 1 + E 2 . fi′  C   λR   ei − di′ e1C + e2C = = λC + dj′ e1C + e2C , ri i = 1, 2, (7) which leads to    λR = ri λC + dj′ e1C + e2C > 0, Proof. (i) Nash equilibrium is the solution of i, j = 1, 2, i = j. + di′ (e1C + e2C ) 2λC + d1′ (e1C + e2C ) + d2′ (e1C λC + e2C ) . (8) Remark 2. The result in the above proposition has been obtained under joint optimization. To see how this generalizes to the case where the players optimize a weighted sum of their individual objectives, let αi  0 be the “political” weight of player i with α1 + α2 = 1. The optimization problem becomes max e1 ,e2 i=1   αi fi (ei )−di (e1 +e2 ) , e1 +e2
E1 +E2 . (9) We still denote by eiC the solution of this problem. To see the link between the normalized equilibrium weights and the political ones, note that the relations in (7) become λC + αj dj′ (e1C + e2C )  λR    fi′ eiC − di′ e1C + e2C = = , ri αi i = 1, 2, which implies that r1 = α1 λC + α1 d1′ (e1C + e2C ) λC + (1 − 2α1 α2 )d1′ (e1C + e2C ) . ek ≡ hi 2
ek k=1 k=1 and cooperative solution fi′ (ei ) = 2
i=1 The definition of ri in (8) states that each player’s weight depends on both players’ damage costs and the shadow price of the constraint. Note that if the damage costs are equal for the two players, then r1 = r2 = 1/2, irrespective of the individual welfare functions. 2
fi′ (ei ) = di′  2
2 C k=1 ek ;  (ii) In R2 , total noncooperative emissions are not necessarily larger than total cooperative emissions. The above equations and the assumption that r1 + r2 = 1 lead to ri = 2 N k=1 ek (i) In R1 , di′ 2
ek ≡ h̄ 2
ek . k=1 k=1 Note that hi (x)
h̄(x) for all x, hi , h̄, (hi )−1 , (h̄)−1 are increasing functions and (hi )−1 (y)  (h̄)−1 (y) for all y. Moreover: 2
k=1   ekN = (hi )−1 fi′ (eiN ) , 2
k=1   ekC = (h̄)−1 fi′ (eiC ) . If for all i, eiN  eiC , we have proven the proposition; if not, suppose there exists l such that elN < elC . We have that y = fl′ (elN ) > fl′ (elC ) = ȳ and 2
k=1 ekN − 2
ekC = (hl )−1 (y) − (h̄)−1 (ȳ) k=1 > (hl )−1 (ȳ) − (h̄)−1 (ȳ) > 0. (ii) To show the result, we provide an example where total emissions under Nash are lower than under cooperation. Assume the following functional forms for revenue and damage cost di fi (ei ) = ai ln(ei ), di (e1 + e2 ) = . 2(e1 + e2 )2 Clearly, these functions satisfy the assumptions stated in section 2. For a1 = 1.3, a2 = 3, d1 = d2 = 1/30 we have that N C C AN 12 = 3.433, A21 = 7.924, e1 = 2.428, e2 = 5.603. If E1 = 5.5 and E2 = 3, then e1N = 4.922, e2N = 3. Then e1N + e2N − (e1C + e2C ) < 0 and hence the result.  Taking into account that α2 = 1 − α1 , we have that α1 = α2 = 1/2 ⇐⇒ r1 = r2 = 1/2, α1 ∈ (1/2, 1] ⇐⇒ r1 > α1 , α1 ∈ [0, 1/2) ⇐⇒ r1 < α1 . The abowe result says that the player having the higher political weight, has to have an even higher weight in the normalized equilibrium in order to obtain cooperation. Being at first glance surprising, this last result can be explained as follows1 : Consider a cooperative case with one constraint E−e1 −e2 = 0, where E is a given constant. Suppose that in the cooperative solution, given by e = (e1C , e2C ), player 1 pollutes much more than player 2(e1C ≫ e2C ). Now choose E1 and E2 such that E1 + E2 = E and impose on 1 We are grateful to a Reviewer for providing this interpretation. 157 M. Tidball, G. Zaccour / An environmental game with coupling constraints country i, i = 1, 2, the constraint Ei − ei = 0. If E1 is (much) lower than e1C , then this country must reduce its emissions. Player 2 would then increase its emissions but not necessarily, being e.g. less productive, in the amount of the reduction of emissions by player 1. Hence, total Nash emissions would be lower than total cooperative ones. n
i=1 ei = n
Ei . i=1 To obtain that both solutions coincide we require that for all i = 1, . . . , n: λR = ri λC +
dl′ l=i 4. Generalization to n players We generalize in this section the results to the n-player case. We continue to assume that in each scenario the solution is unique. Note that characterizing the three solutions using the notation of propositions 1–3 is not necessary to derive the results. The generalizations of propositions 4 and 7 are straightforward. Proposition 5 stated that if Rosen and Nash equilibria are in region R2 then total Nash emissions are less or equal to Rosen’s counterpart and the player whose control is interior makes a higher payoff in Nash than in Rosen equilibrium. The following proposition generalizes the result. n
k=1 ek , n
ri = 1. i=1 This is a system of n equations with n unknowns. This system has a solution because its determinant is different from zero since λC + l=i dl′ ( nk=1 ek ) > 0.  5. Concluding remarks Without assuming special functional forms for revenue and damage cost, we showed in this paper that when the players face environmental constraints, then • Nash equilibrium may be better than Rosen’s normalized equilibrium; Proposition 8. Let ei < Ei for all i = 1, . . . , m, ei  n Ei for all i = m + 1, . . . , n, and ni=1 ei < i=1 Ei , n n N N R R e . Moreover, if e  e then i=1 ei
i=1 i i∗ i∗ then wi∗ (e1N , . . . , enN )  wi∗ (e1R , . . . , enR ). • Cooperative solution may be attained by a suitable choice of the weights of the normalized equilibrium; Proof. In this region, we have that for all i = m+1, . . . , n, eiN < eiR . If for all i = 1, . . . , m we have eiN
eiR we obtain ni=1 eiN
ni=1 eiR . If it is not the case, there exists N > eR and by first order conditions i∗ such that ei∗ i∗  ′  N  N ′ −1 e1 + · · · + enN = (di∗ ) fi∗ ei∗ ,    ′ −1 ′ R ) fi∗ ei∗ . e1R + · · · + enR = (di∗ The comparative analysis rests on the assumption that each scenario admits a unique solution. The required conditions to have uniqueness are rather popular in the economics environmental literature. From a game theoretic perspective, multiple equilibria imply a selection problem and it is not clear how a comparative analysis could then be conducted. This is a challenging topic which deserves an investigation. Another extension would be the analysis of a dynamic game. ′ The result follows from the assumptions that fi∗ , di∗ and di∗ ′ are increasing functions and fi∗ is a decreasing function.  The following proposition generalizes the result in proposition 6. n Proposition 9. In the region where ni=1 ei = i=1 Ei there exists ri , i = 1, . . . , n, r1 + · · · + rn = 1, such that eiC = eiR , i = 1, . . . , n. Proof. In this region the cooperative solution is given by: fi′ (ei ) − n
di′ i=1 n
i=1 ei = n
n
= λC , ek i = 1, . . . , n, k=1 Ei , i=1 and the normalized equilibrium by fi′ (ei ) − di′ n
k=1 ek = λR , ri i = 1, . . . , n, • Noncooperative Nash emissions need not necessarily be higher than their cooperative counterparts. Acknowledgements We wish to thank the two anonymous reviewers for their helpful comments. Research completed when the first author was visiting GERAD, Montréal. Research supported by NSERC, Canada. References [1] M. Breton, G. Zaccour and M. Zahaf, A game-theoretic formulation of joint implementation of environmental projects, European Journal of Operational Research 168(1) (2005) 221–239. [2] M. Breton, G. Zaccour and M. Zahaf, A differential game of joint implementation of environmental projects, to appear in Automatica, Cahier du GERAD G-2004-10, Montreal (2004). [3] A. Haurie and J.B. Krawczyk, Optimal charges on river effluent from lumped and distributed sources, Environmental Modeling and Assessment 2(3) (1997) 93–106. [4] A. Haurie and G. Zaccour, Differential game models of global environmental management, Annals of the International Society of Dynamic Games 2 (1995) 3–24. 158 M. Tidball, G. Zaccour / An environmental game with coupling constraints [5] J.B. Krawczyk, An open-loop Nash equilibrium in an environmental game with coupled constraints, in: Proceedings of the 2000 Symposium of the International Society of Dynamic Games, Adelaide, South Australia (2000) pp. 325–339. [6] J.B. Krawczyk and S. Uryasev, Relaxation algorithms to find Nash View publication stats equilibria with economic applications, Environmental Modeling and Assessment 5 (2000) 63–73. [7] J.B. Rosen, Existence and uniqueness of equilibrium points for concave N-person games, Econometrica 33 (1965) 520–534.