Games with adaptation and mitigation

Chritonenko, Natalija V.; Hritonenko, Victoria; Jacenko, Jurij P. Article Games with adaptation and mitigation Games Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Chritonenko, Natalija V.; Hritonenko, Victoria; Jacenko, Jurij P. (2020) : Games with adaptation and mitigation, Games, ISSN 2073-4336, MDPI, Basel, Vol. 11, Iss. 4, pp. 1-16, https://doi.org/10.3390/g11040060 This Version is available at: https://hdl.handle.net/10419/257478 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ games Article Games with Adaptation and Mitigation Natali Hritonenko 1, Victoria Hritonenko 2and Yuri Yatsenko 3,* 1Department of Mathematics, Prairie View A&M University, Prairie View, TX 77446, USA; [email protected] 2Department of Sciences, Mathematics and Biotechnology, University of California Extension, Berkeley, CA 94704, USA; [email protected] 3Dunham College of Business, Houston Baptist University, Houston, TX 77074, USA *Correspondence: [email protected] Received: 22 October 2020; Accepted: 4 December 2020; Published: 7 December 2020   Abstract: We formulate and study a nonlinear game of nsymmetric countries that produce, pollute, and spend part of their revenue on pollution mitigation and environmental adaptation. The optimal emission, adaptation, and mitigation investments are analyzed in both Nash equilibrium and cooperative cases. Modeling assumptions and outcomes are compared to other publications in this fast-developing area of environmental economics. In particular, our analysis implies that: (a) mitigation is more effective than adaptation in a crowded multi-country world; (b) mitigation increases the effectiveness of adaptation; (c) the optimal ratio between mitigation and adaptation investments in the competitive case is larger for more productive countries and is smaller when more countries are involved in the game. Keywords: economic-environmental model; environmental adaptation; pollution mitigation; Nash equilibrium 1. Introduction Pollution mitigation and adaptation are two major policies commonly used by governments to reduce the environmental damage. Spending on mitigation and adaptation is enormous. Indeed, the total global economic cost of mitigating climate change is estimated to be € 200–350 billion per year by 2030 [ 1 ]. Accurate assessment of effective environmental policies and actions has been a subject of intensive research for the last forty years. It includes analytic models [ 2 – 5 ] and computer simulation methods, known as the integrated assessment models [ 6 – 8 ]. This paper contributes to analytic modeling of adaptation and mitigation activities in the competitive world. Analytic models lead to better understanding of observed environmental changes and predicting consequences of human impact on the environment. They also stimulate the improvement of computer simulation models. However, a systematic analytic theory of adaptation and mitigation is to be developed. A short survey below demonstrates a growing number of related analytic models with scattered underlying assumptions and fragile links among various models and their outcomes. The modeling tools often reflect the analytic expertise of their authors. The adaptation–mitigation models can be deterministic or stochastic, static, or dynamic, in continuous or discrete time (including twoor three-stage versions). Because of analytic complexity, all models make simplifying assumptions about production, pollution, mitigation, and adaptation [ 5 , 6 , 9 – 12 ]. Corresponding optimization problems can involve one or several objectives, and one or many players [ 13 ]. Some adaptation–mitigation models consider one country and neglect the international dimension of environmental protection [ 5 , 12 , 14 – 17 ]. The optimization models with several players reflect the international context of the environmental protection and lead to static or dynamic games. Multi-country models usually restrict their analysis to a symmetric case of identical countries [18–24]. Games 2020,11, 60; doi:10.3390/g11040060 www.mdpi.com/journal/games Games 2020,11, 60 2 of 16 Here, we focus on a rigorous analysis of optimal adaptation and mitigation. To reduce modeling complexity, some analytic games with adaptation do not involve a separate mitigation variable and use the emission reduction caused by environmental damage as a proxy for mitigation effort [ 10 , 12 , 16 , 22 , 25 ]. Such concept of mitigation is costless, so, the related models cannot compare adaptation and mitigation investments. Two-stage dynamic game [ 9 ] analyzes both adaptation and mitigation but ignores endogenous production. A two-country static game of [ 10 ] considers endogenous production, pollution, and adaptation, but oversimplifies mitigation as a reduced emission. The multi-country model of [ 22 ] significantly generalizes the game [ 10 ] by adding cross-country differences in adaptation costs but it also does not include a separate mitigation cost. Papers [ 12 , 16 ] consider dynamic optimization with uncertain damage, though over-simply mitigation. The models [ 5 , 12 , 15 , 16 , 26 ] rigorously analyze mitigation and adaptation actions in one country ignoring international aspect of the problem. A static game of [ 27 ] and a dynamic game of [ 28 ] include separate endogenous mitigation variable, nevertheless, do not consider adaptation. Paper [ 11 ] analyzes a static game of ncountries with pollution and adaptation but does not explicitly include mitigation. A separate group of studies focuses on the formation and stability of possible coalitions in environmental protection, see [ 11 , 18 , 23 , 24 ] and the references therein. The impact of strategic commitment in a model with nsymmetric countries, including adaptation and mitigation, is evaluated in [ 11 , 18 ]. Two-stage coalition formation model of [ 23 ] employs a general static game of nidentical countries with separate endogenous mitigation and adaptation variables. The authors show that adaption can lead to larger stable coalitions and higher global welfare (compared to the only mitigation case), but they do not estimate related adaptation and mitigation investments. Different modeling assumptions about payoff functions, pollution disutility/damage, and mitigation and adaptation effectiveness are used in [ 29 – 38 ] to explore economic, agricultural, welfare, political, and regional aspects of strategic interactions among pollution, mitigation, and adaptation. The novelty of the present paper relative to the existing literature is to analyze and compare a country’s strategic investments in mitigation and adaptation in the competitive world. Estimating the optimal mix of adaptation and mitigation efforts has tremendous policy implications [ 5 , 20 ]. We introduce a multi–country model with separate mitigation and adaptation investment controls and systematically analyze its competitive and cooperative cases, focusing on analytic solutions for the optimal emission, adaptation, and mitigation. The constructed nonlinear model follows natural economic assumptions and is not restricted to linear-quadratic cases. In general, it is not easy to find analytic solutions to non-quadratic games. For clarity, we employ the static game framework to obtain closed-form solutions, useful for policy analysis. Dynamic models of [ 20 , 39 ] with endogenous production, emission, mitigation, and adaptation are conceptually close to the present paper though they differ in modeling assumptions and are restricted to two regions. The paper is organized as follows. Section 2formulates optimization problems for competitive and cooperative scenarios. Section 3provides a comparative analysis of the competitive (Nash equilibrium) and cooperative solutions, focusing on their dependence on the number of countries and country’s productivity. Section 4discusses obtained outcomes and their policy implications and concludes. 2. Models and Methods: Games with Mitigation and Adaptation This section provides formal statements and interpretation of mathematical problems under study. 2.1. Modeling Framework Let us consider ncountries, each of which produces an economic output q i , emits pollution x i and reduces it using a mitigation investment y i . Following other environmental games [ 18 , 25 , 27 , 28 ], we express the output qiin terms of the pollution xias qi=Aixiyik,i=1, . . . ,n, (1) Games 2020,11, 60 3 of 16 where the parameter A i describes the country’s productivity (more exactly, environmental cleanness of production), while k, 0 <k<1, represents the marginal efficiency of the mitigation investment y i . The variables q i ,x i , and y i are per capita. The mitigation actions are less effective at a smaller kand completely useless at k=0. The total pollution X= n P i=1 xi from all countries causes the environmental damage Ωi =B i X 2 to the country i[2,6,40], which can be reduced by the country’s adaptation spending zias Ωi=Bi 1 1+aizi +Di!       n X i=1 xi       2 ,i=1, . . . ,n. (2) The parameter B i >0 describes the country vulnerability to environmental damage (in monetary units), a i >0 is the efficiency of adaptation, and D i >0 is the residual non-avoidable damage in the country. The adaptation is not possible at a i =0. Concave effectiveness of mitigation y i in Equation (1) and adaptation ziin Equation (2) is in line with the majority of related studies [3–6,19,20,25,41]. The individual consumption is the difference c i =q i− y i− z i between the output and mitigation and adaptation investments. The objective of a country iis to maximize the individual welfare, measured by the difference between the consumption utility Ci1−η and the monetarized disutility (2) of environmental damages: F(qi,xi,yi,zi)=(qi−yi−zi)1−η−Bi 1 1+aizi +Di!        n X j=1 xj        2 , 0 <η<1. (3) The objective function (3) uses the standard isoelastic (also known as CRRA) utility function Ci1−η with the risk aversion parameter 0< η <1 [ 2 , 24 ] rather than a quadratic payoff. In doing so, we keep our model in line with the mainstream economic theory. The choice of the benefit function should be theoretically and empirically grounded. Quadratic payofffunctions are favorite in the game theory because they allow for finding analytic solutions in many cases [ 11 , 18 , 27 , 36 ]. However, the quadratic utility leads to increasing absolute risk aversion (which never happens in reality) and has never been seriously considered by economists. The isoelastic utility possess a tremendous potential to increase the quality of many economic-environmental models, including games. An empiric justification of the isoelastic utility for the multi-country world was recently provided in [ 24 ] on a dataset about 264 countries, where Ci1−η with η≈ 0.875 appears to be statistically significant. Because 0 < η <1, we use Ci1−ηin Equation (3) rather than its more general version Ci1−η/(1 −η). The model (1)–(3) provides a simple framework for the current policy debate about environmental policies. Its three control variables describe output/pollution intensity, mitigation effort, and adaptation effort. For convenience, Table 1contains descriptions of all variables and parameters of the model. 2.2. Competitive and Cooperative Games We will analyze two cases, competitive and cooperative. In the competitive case, all countries compete and each country i, 1 ≤ i ≤ n, maximizes its own payoffby taking strategies of other countries as given. The competitive game with payoff(3) is described as max xi,yi,zi FN(xi,yi,zi)=Aixiyik−yi−zi1−η−Bi 1 1+aizi +Di!        n X j=1 xj        2 ,i=1, . . . ,n. (4) A solution (x i ,y i ,z i ), x i≥ 0, y i≥ 0, and z i≥ 0, i=1, . . . ,n, of the nonlinear static game (4), if it exists, represents the Nash equilibrium [13,19,20,25,27]. Games 2020,11, 60 4 of 16 Table 1. The list of used parameters and variables. Notation Description n,n≥1 the number of countries xi,I=1, 2, . . . ,npollution intensity of the country i cithe consumption in the country i qithe production output of the country i Aia productivity factor yi,I=1, 2, . . . ,nmitigation investment in the country i k,0<k<1 the efficiency of mitigation investment zi,I=1, 2, . . . ,nadaptation investment in the country i ai,ai>0 the efficiency of adaptation investment η,0<η<1 the risk aversion parameter of utility function Fthe payofffunction Bi,Bi>0 vulnerability to environmental damage Di,Di>0 the non-avoidable damage in the country (xN,yN,zN) the Nash equilibrium solution of the game (4) (xC,yC,zC) the solution of the cooperative problem (5) σan auxiliary parameter defined by Equation (27) van auxiliary variable (in Theorems 3 and 5) The cooperative case maximizes the total payoffof all countries in the ideal case of a global environmental agreement and is described by the following optimization problem: max xi,yi,zi,i=1,...nFc=n P i=1 Fi(xi,yi,zi) =n P i=1Aixiyik−yi−zi1−η−Bi1 1+aizi+Di      n P j=1 xj      2 (5) This problem has 3nunknown variables: xi≥0, yi≥0, and zi≥0, i=1, . . . ,n. Following standard assumptions of environmental games [ 18 – 24 , 27 , 28 ], we restrict ourselves to the symmetric case of nidentical countries: Ai=A,ai=a,Di=D,Bi=B,i=1, . . . ,n. (6) Similar models with several asymmetric countries have been analyzed in [25]. Under condition (6), the optimal pollution x i , mitigation y i , and adaptation z i in problems (4) and (5) are the same for all countries. We denote the solution of the competitive game (4) as (x N ,y N ,z N ) and the solution of the cooperative problem (5) as (x C ,y C ,z C ). A simple link between those solutions is presented below. Theorem 1. Let the nonlinear game (4) have a solution x N (n, B), y N (n, B), z N (n, B) for any n =1,2,3, . . . and any B >0. Then, the solution of the cooperative problem (5) is: xC=xN(1, Bw), yC=yN(1, Bw), zC=zN(1, Bw), (7) where Bw=Bn2. Games 2020,11, 60 5 of 16 Proof. Let us consider the cooperative optimization problem (5). Substituting Equation (6) and x i =x C , yi=yC,zi=zC,i=1, . . . ,nto Equation (5), we obtain max xi,yi,zi,i=1,...nFc=n·max xC,yC,zCAxCyCk−yC−zC1−η−Bn2x2 c1 1+azc +D i.e., the solution (xC,yC,zC) coincides with the solution of the one-country model max x1,y1,z1 F1=max x1,y1,z1"Ax1y1k−y1−z11−η−Bn2x2 1 1 1+az1 +D!# (8) with the modified parameter B=B w =Bn 2 . On the other side, the one-country model (8) is a special case of the game (4) at n=1. Therefore, the solution (x N ,y N ,z N ) to Equation (4) coincides with the solution (x1,y1,z1) to Equation (8). It justifies the formulas (7). The Theorem is proven.  Theorem 1 reduces a technical complexity of the forthcoming analysis and allows us to compare competitive and cooperative strategies with less effort. 3. Results: Comparative Analysis In this section, we investigate and compare analytic properties of the competitive game (4) and cooperative problem (5). In our analysis, we emphasize the dynamics of competitive and cooperative strategies when the number of countries is large. To demonstrate our technique, let us start with the simplest case. Special case k=0, a =0(no adaptation and no mitigation). Then, the game (4) becomes max xi FN(xi)=(Axi)1−η−B(1+D)       n X i=1 xi       2 for all i =1, . . . ,n,. . . (9) and does not include mitigation and adaptation controls yiand zi. The only control in Equation (9) is the pollution level x i that also defines the economic output (1). Differentiating (9) in x i , setting the derivative to zero, and using the symmetry assumption (6), we obtain the Nash equilibrium solution of the game (4) as xN="(1−η)A1−η 2B(1+D)n#1/(1+η) , (10) FN=1−n 2(1−η)A1−ηx1−η N=1−n 2(1−η)"(1−η)A2 2Bn(1+D)#(1−η)/(1+η) (11) By Theorem 1, the cooperative solution is xC="(1−η)A1−η 2B(1+D)n2#1/(1+η) , (12) Fc=1−η 2A1−ηxC(1−η)=1−η 2 (1−η)A2/(1−η) 2B(1+D)n2 (1−η)/(1+η) (13) It is easy to see that the pollution is higher: xN=n1/(1+η)·xC , and the payoffis smaller in the competitive game (4) than in the cooperative case: FN=2−n(1−η) 1+ηn(1−η)/(1+η)FC. (14) Games 2020,11, 60 6 of 16 By Equations (11) and (14), the cooperative payoffF C is always positive, but the competitive payoff F N >0 only when n(1 −η )<2. Thus, the concave utility η >0 is required for a positive Nash payoff at n>1 . A similar condition on model parameters appears in [ 18 ] to guarantee that each player’s decision is interior in equilibrium. Next, we explore the properties of competitive and cooperative strategies in models with mitigation, adaptation, and both controls. We compare competitive and cooperation strategies in the terms of pollution, adaptation, and mitigation. We also analyze how those strategies depend on the key model parameters, the number nof countries and their stage of development, represented by the production cleanness factor A. 3.1. Model with Mitigation The competitive game (4) with mitigation is presented as follows: max xi,yi FN(xi,yi)=Axiyik−yi1−η−B(1+D)       n X i=1 xi       2 ,. . . for all i =1, . . . ,n. (15) Differentiating Equation (15) in y i and x i and setting derivatives to zero, we obtain the explicit formulas for Nash equilibrium solution: yN= A2(1−η)k1+η 2B(1+D)n(1−k)η!1/(1+η−2k) , (16) xN=yN1−k kA (17) and the related payoff FN=yN1−kkη−1 2(1−k)η(2−2k−n(1−η)). (18) By Theorem 1, the solution of the related cooperative problem (5) is yC= A2(1–η)k1+η 2B(1+D)n2(1+k)η!1/(1+η–2k) , (19) xC=yC1–k kA , (20) FC=yC1−ηkη−1 2(1−k)η(1+η−2k). (21) Here and thereafter, the notation z(v) ~ f(v) describes asymptotic behavior of the function z(v) when vis large and means that lim v→∞ z(v) f(v)=const ,0. Theorem 2. Let k <( η +1)/2. Then, in both competitive game (15) and its cooperative case, the optimal emission x, mitigation y, and payoffF increase when A and/or η increase, but decrease when n and/or B increase. The mitigation /pollution ratio increases and is convex when A increases: y x∼A 1+η 1+η−2k. (22) The global emission in the competitive case X =nx ~ n(η−k)/(1+η) decreases in n at k < η and increases at η<k<(η+1)/2. At k →(η+1)/2, the optimal x→∞ and y→∞. Proof. Follows from formulas (16)–(21).  Games 2020,11, 60 7 of 16 The game (13) and related cooperative problem have no finite solution at k≥(η+1)/2. By Equations (16) and (19), the optimal mitigation yis positive in both competitive and cooperative scenarios, but it is small and much smaller than emission, x<< y, for weak economies with A<< 1 . The optimal emission (17) of an individual country is always larger in the presence of mitigation than with no mitigation in both competitive and social optimum scenarios. However, it is not so for the competitive optimal payoff(18). By Theorem 2, the relation between the mitigation effectiveness parameter kand risk aversion η essentially affects both optimal competitive and cooperative strategies. The sign of k −η determines whether the global pollution increases or decreases when the number nof countries becomes larger. The optimal emission, mitigation, and payoffs are finite at 0<k<(η+1)/2 , but they increase indefinitely when k → ( η +1)/2. At k ≥ ( η +1)/2, mitigation is so effective that the optimal output Ax i y ik in Equation (1) grows faster than the mitigation cost y i , which leads to the infinite output and pollution. 3.2. Model with Adaptation With adaptation, the competitive and cooperative strategies become richer, but the analytic complexity increases. For clarity, let us first consider the problems (4) and (5), with adaptation but without mitigation. Then, the competitive game (4) becomes max xi,zi FN(xi,zi)=(Axi−zi)1−η−B 1 1+azi +D!       n X i=1 xi       2 ,. . . for all i =1, . . . ,n. (23) Theorem 3. Let a>acr = 22+η(1+D)2+η (1−η)A2Nη 1/(1+η) , (24) then the Nash equilibrium solution of the competitive game (23) is: xN=2v+Dv2 Aan >0, zN=v−1 a>0, (25) where v, 1 ≤v<∞, is the unique solution of the nonlinear equation 2D nv2−v1−2 n+1η (1+Dv)2=σ, (26) with σ=(1−η)A2a1+η 4B. (27) If a <acr, then the optimal adaptation ZN=0 and xNis determined by Equation (10). The payoffis FN=(AxN−zN)−η"AxN 1−n(1−η) 2!−xN#(28) Proof. Setting the partial derivatives of FN(xi,zi) in Equation (23) with respect to x i and z i to zero, we obtain the following system of two nonlinear equations in xiand zi: ∂FN(xi,zi) ∂ei =A(1−η)(Axi−zi)−η−2B 1 1+azi +D!       n X i=1 ei       =0, (29) Games 2020,11, 60 8 of 16 ∂FN(xi,zi) ∂ei =−(1−η)(Axi−zi)−η+aB (1+azi)2       n X i=1 xi       2 =0 (30) Since the countries are identical, their competitive strategy is the same: x i =x,z i =z,I=1, . . . ,n, and the system of Equations (29) and (30) becomes A(1−η)(Ax −z)−η=2Bnx1 1+az +D, (31) (1−η)(Ax −z)−η=aB (1+azi)2(nx)2. (32) Formulas (25) follow from Equations (31) and (32) after expressing them via the new auxiliary variable v=1 +az . Excluding xfrom the system of Equations (31) and (32), we obtain one nonlinear Equation (26) in v. To analyze the existence and uniqueness of its solution, let us rewrite Equation (26) as f(v)=σ, (33) where f(v)=2Dv2/n−v1−2 n+1η(1+Dv)2and σis defined by (27). A typical shape of the function f(v) is shown with the solid line in Figure 1. Because of the constraint z i≥ 0 in Equation (23), we are interested in the solution vof the Equation (33) only in the interval [1, ∞ ). It is easy to see that there is no solution v ≥ 1 if the right-hand side σ of Equation (33) is small. Let acr denote the smallest critical value of the parameter awhen a solution v≥1 exists. Games 2020, 11, x FOR PEER REVIEW 8 of 15 󰇛1−𝜂󰇜󰇛 𝐴 𝑥−𝑧󰇜=𝑎𝐵 󰇛1+𝑎𝑧󰇜󰇛𝑛𝑥󰇜. (32) Formulas (25) follow from Equations (31) and (32) after expressing them via the new auxiliary variable v = 1+𝑎𝑧. Excluding x from the system of Equations (31) and (32), we obtain one nonlinear Equation (26) in v. To analyze the existence and uniqueness of its solution, let us rewrite Equation (26) as f(v) = σ , (33) where 𝑓󰇛𝑣󰇜=󰇡2𝐷𝑣/𝑛−𝑣󰇡1−󰇢+1󰇢󰇛1+𝐷𝑣󰇜 and σ is defined by (27). A typical shape of the function f(v) is shown with the solid line in Figure 1. Because of the constraint z i ≥ 0 in Equation (23), we are interested in the solution v of the Equation (33) only in the interval [1, ∞). It is easy to see that there is no solution v ≥ 1 if the right-hand side σ of Equation (33) is small. Let a cr denote the smallest critical value of the parameter a when a solution v ≥1 exists. To find a cr , let 𝑧=  = 0, then v = 1 and 𝑥=󰇛󰇜  in Equation (25). Therefore, a cr is determined from Equations (26) and (27) as Equation (24). The payoff (28) is obtained combining Equations (23) and (29). Next, we find the derivative f’(v) and obtain that f’(v) > 0 at v ≥ 1 at natural conditions. Therefore, the solution v to the nonlinear Equation (33) is unique in the interval [1, ∞). If 0 < a < a crN , then the optimal z N = 0 is a corner solution in [0, ∞), while x N coincides with Equation (10). As expected, the resulting payoff (28) in this case is the same as (11). The Theorem is proven. □ By Theorem 3, the economy must be productive enough to engage into adaptation activities. The critical value a cr positively depends on the climate vulnerability B and residual damage D. Thus, the larger B and D are, the more economically powerful a country should be to profitably engage in adaptation. Figure 1. The function f(v) in the nonlinear Equations (26) (solid curve) and (47) (dashed curve). The acceptable interval (1, ∞) of the solution v* is indicated in bold gray. To find a cr , let z=v−1 a =0, then v=1 and xN=2(1+D) Aan in Equation (25). Therefore, a cr is determined from Equations (26) and (27) as Equation (24). Games 2020,11, 60 15 of 16 14. Hritonenko, N.; Yatsenko, Y. Modeling of environmental adaptation: Amenity vs productivity and modernization. Clim. 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