# Games with Adaptation and Mitigation

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Models and Methods: Games with Mitigation and Adaptation

#### 2.1. Modeling Framework

_{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 q

_{i}in terms of the pollution x

_{i}as

_{i}= A

_{i}x

_{i}y

_{i}

^{k}, i = 1,…, n,

_{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 k and completely useless at k = 0.

_{i}= B

_{i}X

^{2}to the country i [2,6,40], which can be reduced by the country’s adaptation spending z

_{i}as

_{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 z

_{i}in Equation (2) is in line with the majority of related studies [3,4,5,6,19,20,25,41].

_{i}= q

_{i}− y

_{i}− z

_{i}between the output and mitigation and adaptation investments. The objective of a country i is to maximize the individual welfare, measured by the difference between the consumption utility ${C}_{i}{}^{1-\eta}$ and the monetarized disutility (2) of environmental damages:

#### 2.2. Competitive and Cooperative Games

_{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].

_{i}≥ 0, y

_{i}≥ 0, and z

_{i}≥ 0, i = 1, …, n.

_{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.**

_{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:

_{C}= x

_{N}(1, B

_{w}), y

_{C}= y

_{N}(1, B

_{w}), z

_{C}= z

_{N}(1, B

_{w}),

_{w}= Bn

^{2}.

**Proof.**

_{i}= x

_{C}, y

_{i}= y

_{C}, z

_{i}= z

_{C}, i = 1, …, n to Equation (5), we obtain

_{C}, y

_{C}, z

_{C}) coincides with the solution of the one-country model

_{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 (x

_{1}, y

_{1}, z

_{1}) to Equation (8). It justifies the formulas (7).

## 3. Results: Comparative Analysis

**k = 0, a = 0**(no adaptation and no mitigation). Then, the game (4) becomes

_{i}and z

_{i}. 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

_{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.

#### 3.1. Model with Mitigation

_{i}and x

_{i}and setting derivatives to zero, we obtain the explicit formulas for Nash equilibrium solution:

**Theorem**

**2.**

**Proof.**

_{i}y

_{i}

^{k}in Equation (1) grows faster than the mitigation cost y

_{i}, which leads to the infinite output and pollution.

#### 3.2. Model with Adaptation

**Theorem**

**3.**

_{cr}, then the optimal adaptation Z

_{N}= 0 and${x}_{N}$ is determined by Equation (10).

**Proof.**

_{i}and z

_{i}to zero, we obtain the following system of two nonlinear equations in x

_{i}and z

_{i}:

_{i}= x, z

_{i}= z, I = 1, …, n, and the system of Equations (29) and (30) becomes

_{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.

_{cr}, let $z=\frac{v-1}{a}$ = 0, then v = 1 and ${x}_{N}=\frac{2\left(1+D\right)}{Aan}$ in Equation (25). Therefore, a

_{cr}is determined from Equations (26) and (27) as Equation (24).

_{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).

_{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.

_{c}is the unique solution of the nonlinear equation over [1,∞):

_{c}is given by Equation (12) at $0<a\le {a}_{C}$. The cooperative payoff is:

_{C}is larger in the cooperative scenario: a

_{C}> a

_{cr}.

**Corollary**

**1.**

**Proof.**

**Theorem**

**4.**

_{N}increases and pollution x

_{N}decreases when n increases. For n >> 1,

_{N}and adaptation z

_{N}increase when A increases, in both competitive and cooperative cases. For A >> 1,

**Proof.**

_{N}, x

_{N}, and E follow directly from Equations (38) and (39).

#### 3.3. Model with Mitigation and Adaptation

**Theorem**

**5.**

_{cr},

_{N}→ ∞ and y

_{N}→ ∞. The game (4) has no finite solution at k ≥ (η + 1)/2.

_{cr}, then the optimal adaptation z

_{N}= 0, while pollution x

_{N}and mitigation y

_{N}are given by Equations (16) and (17). The payoff is

**Proof.**

_{i}, y

_{i}, and z

_{i}equal to zero and taking the symmetry condition (6) into consideration, we obtain the following system of three nonlinear equations

_{i}= x, y

_{i}= y, z

_{i}= z, I = 1, …, n. In the new variable v = 1 + az, this system is reduced to one nonlinear Equation (47) in v. Let us rewrite Equation (47) as

_{cr}denote the smallest critical value of the parameter of a when the solution v ≥ 1 exists. To find a

_{cr}, let Z

_{N}= 0, then v = 1, Z

_{N}= 0 by Equation (46), and ${y}_{N}=\frac{2k\left(1+D\right)}{an}$. Substituting those values to (47), we obtain the formula (45) for a

_{cr}.

^{1−2k+η}, so $\underset{v\to \infty}{\mathrm{lim}}f(v)=\infty $ at k < (η + 1)/2, and $\underset{v\to \infty}{\mathrm{lim}}f(v)=0$ at k > (η + 1)/2. Therefore, the Equation (52) is guaranteed to have a solution 1 ≤ v < ∞ only at k < (η + 1)/2. Similarly to Equation (33), the first derivative f′(v) is positive at v ≥ 1 at natural conditions. Hence, the solution to the Equation (52) is unique if it exists. The payoff (48) is obtained from Equations (3) and (46).

_{cr}, then the optimal z

^{N}= 0 is a corner solution in [0, ∞), while the optimal x

^{N}and y

^{N}coincide with (19) and (20). The payoff (48) is the same as (18).

_{cr}, where

_{C}, 1 ≤ v

_{C}< ∞, is a unique solution of the nonlinear equation

_{cr}, then z

_{C}= 0, while x

_{C}and y

_{C}are found in Equations (19) and (20).

**Corollary**

**2.**

_{N}is smaller and mitigation y

_{N}is larger for a larger n. For n >> 1,

_{N}increases in n at$k<\eta /2$and decreases otherwise. The optimal ratio between adaptation and mitigation

**Proof.**

_{N}, y

_{N}, z

_{N}and, subsequently, (58).

_{N}/z

_{N}between mitigation and adaptation in the competitive strategy is smaller when more countries are involved in the game, and is larger when the country’s productivity becomes larger.

_{N}in competitive strategy is larger in absolute units for a larger number n if the mitigation effectiveness k is weak: k < η/2. At more effective mitigation: $\frac{\eta}{2}<k<\frac{1+\eta}{2}$, the adaptation becomes less relevant and decreases with n. If mitigation becomes even more effective: k ≥ (η + 1)/2, then the optimal output grows faster than the mitigation cost and leads to infinite output (as it was in the model (13) with mitigation only).

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**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.

Notation | Description |
---|---|

n, n ≥ 1 | the number of countries |

x_{i}, I = 1, 2, …, n | pollution intensity of the country i |

c_{i} | the consumption in the country i |

q_{i} | the production output of the country i |

A_{i} | a productivity factor |

y_{i}, I = 1, 2, …, n | mitigation investment in the country i |

k, 0 < k < 1 | the efficiency of mitigation investment |

z_{i}, I = 1, 2, …, n | adaptation investment in the country i |

a_{i}, a_{i} > 0 | the efficiency of adaptation investment |

η, 0 < η < 1 | the risk aversion parameter of utility function |

F | the payoff function |

B_{i}, B_{i} > 0 | vulnerability to environmental damage |

D_{i}, D_{i} > 0 | the non-avoidable damage in the country |

(x_{N}, y_{N}, z_{N}) | the Nash equilibrium solution of the game (4) |

(x_{C}, y_{C}, z_{C}) | the solution of the cooperative problem (5) |

σ | an auxiliary parameter defined by Equation (27) |

v | an auxiliary variable (in Theorems 3 and 5) |

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Hritonenko, N.; Hritonenko, V.; Yatsenko, Y. Games with Adaptation and Mitigation. *Games* **2020**, *11*, 60.
https://doi.org/10.3390/g11040060

**AMA Style**

Hritonenko N, Hritonenko V, Yatsenko Y. Games with Adaptation and Mitigation. *Games*. 2020; 11(4):60.
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**Chicago/Turabian Style**

Hritonenko, Natali, Victoria Hritonenko, and Yuri Yatsenko. 2020. "Games with Adaptation and Mitigation" *Games* 11, no. 4: 60.
https://doi.org/10.3390/g11040060