Stability of Efficient International Agreements on Solar Geoengineering

Abstract

Solar geoengineering (SG) may have the potential to reduce extreme climate damages worldwide. Yet, international coordination will make the difference between success and failure in leveraging it. Using a simple game-theoretic framework, we investigate whether the stability of an efficient, self-enforcing international agreement on SG is attainable. We demonstrate that side payments from countries less vulnerable to climate change to those more vulnerable can guarantee the stability of an efficient agreement. The size of the side payments will vary within a zone of possible agreement, which will change depending on certain key assumptions. For example, assuming stronger mitigation reduces the necessary payments. Alternatively, asymmetry in national damages from SG over-provision vs. under-provision justifies larger payments; here, the welfare-optimal strategy may be deployment that makes no one worse off. We also show that an agreement may be stable without side payments if deployment costs are substantial and counter-SG is available, while a moratorium may be socially optimal if SG brings substantial global non-excludable fixed costs.

Appendices

A Default Specification

In the default model specification, damage function for country $i \in N$, $i=1,2, \dots n$, reads:

$$\begin{aligned} D_i = \frac{\alpha }{2} (G_N - k_i)^2, \end{aligned}$$

where $G_N$ is realized SG level and $k_i$ - country $i's$ optimal SG level. The damage function can be rewritten as:

$$\begin{aligned} D_i = \frac{\alpha }{2} G_N ^2 -\alpha G_N k_i+d_i \end{aligned}$$

with $d_i \equiv \frac{1}{2} \alpha k_i^2$ - country $i's$ climate damages in the absence of SG.

In the following analytical derivations we offer a solution in a more general form, where parameter $d_i$ is explicit. It must, however, satisfy the following condition:

$$\begin{aligned} d_i \ge \frac{1}{2} \alpha k_i^2. \end{aligned}$$

This condition ensures that damages cannot be negative, i.e., SG cannot be used to bring benefits beyond reduced climate damages. The value $d_i$ such that $d_i > \frac{1}{2} \alpha k_i^2$ describes the case when the damage compensation potential of SG differs across countries. For example, $d_i = \frac{1}{2 \cdot 0.9} \alpha k_i^2$ means that SG can be used to compensate for $90\%$ of climate damages in each country (in each - different level of SG). Here, residual damages would be largest (in terms of its levels) for the country most vulnerable as illustrated in Fig. 8.

Fig. 8

Larger $d_i$. Default damage functions (dashed line) versus damage functions that assume that SG can be used to reduce $90\%$ of region $i's$ climate damages when SG is used at a level that is optimal for the region i (solid line)

1.1 A.1 Third-Stage Solution

Signatories

Consider an arbitrary coalition $S \subseteq N$. Total SG deployment of coalition members amounts to $G_S = \sum _{i \in S} g_i$; the average level of SG preferences of coalition members is ${\bar{k}}_S = \sum _{i \in S} k_i/s$, where s - number of coalition members.

In the third stage of the game, coalition members minimize their total damages $D_S$:

$$\begin{aligned} \min _{G_S} D_S = \frac{1}{2} s \alpha (G_S + G_{N \setminus S})^2 - \alpha (G_S + G_{N \setminus S}) \sum _{i \in S} k_i + \sum _{i \in S} d_i \end{aligned}$$

Coalition S reaction function:

$$\begin{aligned} G_S = {\left\{ \begin{array}{ll} {\bar{k}}_S- G_{N \setminus S} &{} \textit{if } G_{N \setminus S} <{\bar{k}}_S \\ 0 &{} \textit{if } G_{N \setminus S} \ge {\bar{k}}_S\\ \end{array}\right. } \end{aligned}$$

Non-signatories

The non-signatory $i \in N\setminus S$ minimizes its individual damage function:

$$\begin{aligned} \min _{g_i} D_i = \frac{1}{2} \alpha (g_i + G_{N\setminus i})^2 - \alpha k_i (g_i + G_{N\setminus i}) + d_i \end{aligned}$$

The non-signatory i’s reaction function reads:

$$\begin{aligned} g_i = {\left\{ \begin{array}{ll} k_i - G_{N \setminus i} &{} \textit{if } G_{N \setminus i} < k_i \\ 0 &{} \textit{if } G_{N \setminus i} \ge k_i \\ \end{array}\right. } \end{aligned}$$

Total SG Deployment Level and Associated Damage Costs

Consider country j to be the country with the largest preferred SG level among non-signatories. Thereby, an arbitrary coalition S may be“active”, i.e. define the global thermostat, if it’s optimal deployment level exceeds $k_j$ and “passive” otherwise.

$$\begin{aligned} \begin{array}{ c l l } &{} \hbox {Active coalitions:} {\bar{k}}_S > k_j &{} \hbox {Passive coalitions:} {\bar{k}}_S < k_j \\ G_N &{} {\bar{k}}_S &{} k_j \\ D_i &{} \alpha {\bar{k}}_S (0.5 {\bar{k}}_S -k_i) +d_i &{} \alpha k_j( 0.5 k_j - k_i ) +d_i \\ D_N &{} \alpha n {\bar{k}}_S ( 0.5 {\bar{k}}_S - {\bar{k}}_N ) +d_N &{} \alpha n k_j ( 0.5 k_j - {\bar{k}}_N ) +d_N \\ \end{array} \end{aligned}$$

1.2 A.2 The Proposition Proof

Consider an arbitrary active coalition $S \subseteq N$. Optimal level of SG for this coalition is $\bar{k_S}$. Let us denote by $DR^S$ the set of drivers: $j \in DR^S$ if $j \in S$ and $k_j > \bar{k_S}$, and by $ND^S$ the set of non-drivers: $i \in ND^S$ if $i \in S$ and $k_i \le \bar{k_S}$.

The proposition states that the total gain of non-drivers from cooperation (relative to the no-cooperation case) exceeds the sum of the gains of all drivers from their unilateral deviation from the cooperation:

$$\begin{aligned} \sum _{i \in ND^S} (D_i(\emptyset ) - D_i(S)) \ge \sum _{j \in DR^S} (D_j(S) - D_j(S\setminus j)) \end{aligned}$$

Substitute the considered functional form of damages, here $k_h$ denotes the highest SG preference of all considered countries from set N:

$$\begin{aligned} \frac{\alpha }{2} \sum _{i \in ND^S} \left( k_h - \bar{k_S} \right) \left( \bar{k_S} + k_h - 2 k_i \right) \ge \frac{\alpha }{2} \sum _{j \in DR^S} \left( k_j - \bar{k_S} \right) ^2 \end{aligned}$$

Proof

By definition, $k_h$ is the largest value of all nationally-optimal SG levels. It is larger than the average of optimal levels for all non-drivers: $k_h \ge {\bar{k}}_{ND^S}$. Multiplying both sides by the number of non-drivers in a coalition S, we arrive at:

$$\begin{aligned} |ND^S| k_h \ge k_{ND^S} \end{aligned}$$

where $|ND^S|$ is a number of non-drivers in a coalition S, $k_{ND^S} = \sum _{i \in ND^S} k_i$ is the sum all SG amounts preferred by non-drivers. Then we add them up and subtract from the right hand side (RHS) of the sum of all preferences of coalition members ($k_S$):

$$\begin{aligned} |ND^S| k_h \ge k_{ND^S} + \underbrace{( k_{ND^S} + k_{DR^S})}_{k_S} - \underbrace{(|DR^S| +|ND^S| ) {\bar{k}}_S}_{k_S} \end{aligned}$$

After some reshuffling we arrive at:

$$\begin{aligned} |ND^S| k_h + |ND^S| {\bar{k}}_S - 2k_{ND^S} \ge k_{DR^S} - |DR^S| {\bar{k}}_S \end{aligned}$$

We then rewrite this inequality using summation operator as:

$$\begin{aligned} \sum _{i \in ND^S} \left( k_h + {\bar{k}}_S - 2 k_i \right) \ge \sum _{j \in DR^S} \left( k_j - {\bar{k}}_S \right) \end{aligned}$$

.

Multiplying both sides by $\left( k_h - {\bar{k}}_S \right) \ge 0$, the difference between the largest preferred SG level among all countries in N and the average of coalition member’s preferred SG level.

$$\begin{aligned} \left( k_h - {\bar{k}}_S \right) \sum _{i \in ND^S} \left( {\bar{k}}_S + k_h - 2 k_i \right) \ge \left( k_h - {\bar{k}}_S \right) \sum _{j \in DR^S} \left( k_j -{\bar{k}}_S \right) \end{aligned}$$

By definition $k_h \ge k_i ~ \forall i \in N$, the RHS is greater than or equal to $\sum _{j \in DR^S} \left( k_j - {\bar{k}}_S\right) ^2$. Therefore, the following holds:

$$\begin{aligned} \left( k_h - {\bar{k}}_S \right) \sum _{i \in ND^S} \left( {\bar{k}}_S + k_h - 2 k_i \right) \ge \sum _{j \in DR^S} \left( k_j - {\bar{k}}_S\right) ^2 \end{aligned}$$

Finally, by multiplying both sides by $0.5 \alpha$ we arrive at the following expression, which was to be demonstrated:

$$\begin{aligned} \frac{\alpha }{2} \sum _{i \in ND^S} \left( k_h - {\bar{k}}_S \right) \left( {\bar{k}}_S + k_h - 2 k_i \right) \ge \frac{\alpha }{2} \sum _{j \in DR^S} \left( k_j - {\bar{k}}_S\right) ^2 \end{aligned}$$

. $\square$

1.3 A.3 Illustration

Figure 9 illustrates our Proposition, which itself is in the spirit of Coase theorem. What differs from the Coase theorem case, is that we are dealing with public “gob" and thus country may alter between being a loser or winner dependent on the amount of "gob" supplied.

We consider two groups of countries: (i) non-drivers and (ii) potential free drivers. Note that marginal damages to non-drivers are negative at low SG values, indicating that non-drivers benefit from small amounts of SG due to the associated reduced climate damages. Similarly, marginal benefits to potential free driversis negative when SG levels are large. Thereby, there is a difference between the sum of damages to free drivers in Nash equilibrium (where some free drivers actually lose) and the sum of damages that potential free drivers bear from the unilateral deviations.

In Fig. 9, the green area shows the benefits to non-drivers from global cooperation relative to the Nash equilibrium case. Red area shows the benefits to potential free drivers in Nash equilibrium relative to the global cooperation case. Here, red area (triangle) in the negative zone indicates losses to potential drivers from SG over-provision in Nash equilibrium.

Fig. 9

Total marginal damages to nondrivers (green line) and the sum of marginal benefits to potential free drivers (red line) under a uniform distribution of optimal values $k^U = \{1,2,3,4,5,6,7 \}$ (left) and under right-modal distribution $k^R = \{1.0, 1.2, 4.5, 4.8, 5.3, 5.5, 5.7 \}$ (right)

B Asymmetric Damage Function

We define the degree of asymmetry B as a deviation (in degrees) of $D_i^{asym}$ from the default function $D_i$ at a point $G_N=k_i+1$. The resulting degree of asymmetry is:

$$\begin{aligned} B(\beta )=(arctan(\alpha +\beta )-arctan(\alpha ) ) \cdot \frac{180}{\pi } [degrees] \end{aligned}$$

where $\alpha =1.5$ and associated $arctan(\alpha )=56.3$ [degrees].

We consider two types of asymmetry: with relatively larger damages from SG over-provision and smaller damages from SG under-provision. This helps us to demonstrate the potential for the qualitative change in the results. The illustration of the considered asymmetric damage functions are presented in the Fig. 10.

Changes in the total amount of SG, benefits to ND of cooperation and benefits to DR from unilateral deviation for the left-side asymmetry are depicted in the Fig. 11. This figure demonstrates that ZOPA is positive and increasing with the degree of asymmetry.

We also offer results for the alternative measure of asymmetry, which is as follows: $(\beta -\alpha )/\alpha$. The corresponding graphs are presented in the Fig. 12.

Fig. 10

a Asymmetric damage functions with stronger damage for SG overprovision (dashed line) versus default damage functions (solid line). b Asymmetric damage functions with lower damage for SG underprovision (dashed line) versus default damage functions (solidline)

Fig. 11

Changes in socially optimal SG level (solid red line) and non-cooperative SG level (dashed red line) as a function of damage functions’ ‘left-side’ asymmetry, under alternative distributions of countries’ preferred SG level $k_i$, including: a left-modal, b uniform, c right-modal distributions, and d empirical distribution. Associated benefits to non-drivers from cooperation (blue line) and total benefits to drivers from unilateral deviation (green line), estimated as a reduction in damages relative to the no-SG case. Shaded area between the curves represents the zone of possible agreement

Fig. 12

Changes in socially optimal SG level (solid red line) and non-cooperative SG level (dashed red line) as a function of damage functions’ ‘right-side’ asymmetry measured as $(\beta -\alpha )/\alpha$, under alternative distributions of countries’ preferred SG level $k_i$, including: a left-modal, b uniform, c right-modal distributions, and d empirical distribution. Associated benefits to non-drivers from cooperation (blue line) and total benefits to drivers from unilateral deviation (green line), estimated as a reduction in damages relative to the no-SG case. Shaded area between the curves represents the zone of possible agreement

C Non-negligible Direct SG Deployment Costs

When assuming that the deployment costs are non-negligible, the damage function reads:

$$\begin{aligned} D_i = \frac{1}{2} \alpha G^2_N - \alpha k_i G_N + d_i + \frac{1}{2} z g_i^2 \end{aligned}$$

where z is the slope of marginal SG deployment costs and $g_i \ge 0$.

1.1 C.1 Third-Stage Solution

Signatories

Minimization problem for members of a coalition $S \subseteq N$:

$$\begin{aligned} \min _{g_i} D_S = 0.5 s \alpha (G_S+G_{N\setminus S})^2 - \alpha k_S (G_S+G_{N\setminus S}) +d_S + 0.5 z \sum _{i=1}^s g_i^2 \end{aligned}$$

FOC:

$$\begin{aligned}\frac{\partial D_S}{\partial g_i} = s \alpha (G_S+G_{N\setminus S}) - \alpha k_S + z g_i = 0 \end{aligned}$$

Adding it up for the coalition members $i \in S$, we arrive at:

$$\begin{aligned} s^2 \alpha (G_S+G_{N\setminus S}) - \alpha s k_S + z G_S= 0 \end{aligned}$$

Reaction function for a coalition S then reads:

$$\begin{aligned} G_S = {\left\{ \begin{array}{ll} \frac{ \alpha s^2 }{\alpha s^2 + z} ({\bar{k}}_S- G_{N \setminus S}) &{} \textit{if } G_{N \setminus S} <{\bar{k}}_S \\ 0 &{} \textit{if } G_{N \setminus S} \ge {\bar{k}}_S\\ \end{array}\right. } \end{aligned}$$

Non-signatories

Non-signatory $i \in N\setminus S$ minimizes it’s individual damage function as follows:

$$\begin{aligned} \min _{g_i} D_i = 0.5 \alpha (g_i + G_{N\setminus i})^2 - \alpha k_i (g_i + G_{N\setminus i}) + d_i + 0.5 z g_i^2 \end{aligned}$$

FOC:

$$\begin{aligned}\frac{\partial D_i}{\partial g_i} = \alpha (g_i + G_{N\setminus i}) - \alpha k_i + z g_i = 0 \end{aligned}$$

The associated reaction function reads:

$$\begin{aligned} g_i = {\left\{ \begin{array}{ll} \frac{\alpha }{\alpha +z} ( k_i -G_{N\setminus i}) &{} G_{N\setminus i} \le k_i \\ 0 &{} G_{N\setminus i} >k_i \\ \end{array}\right. } \end{aligned}$$

1.2 C.2 Selected Illustrations

Fig. 13

Effect of the marginal deployment costs on individual and total levels of SG deployment in (i) global cooperation (orange lines); (ii) Nash equilibrium (blue lines); (iii) unilateral deviation (green lines)

Fig. 14

Effect of the marginal deployment costs on individual losses in (i) global cooperation (orange line); (ii) Nash equilibrium (blue line); (iii) unilateral deviation (green line). Individual incentives to deviate are indicated by the area in red, incentives to cooperate - the area in green

Figure 13 shows individual and total levels of SG deployment in global cooperation, Nash equilibrium and under the unilateral deviation case. The Figure is presented for a uniform distribution of optimal values $k^U = \{1,2,3,4,5,6,7 \}$ for countries 2–7.

For countries with very low preferred SG levels (1 and 2), the deviation always results in zero individual SG deployment and an increase in total deployment, since otherl coalition members have larger average over preferred SG levels. The difference between the total SG deployment in cooperation and deviation decreases with marginal deployment costs, z.

Preferences of countries 3 and 4 are the closest to the socially optimal SG deployment level. Therefore, the difference between total SG level in the cooperative case and their unilateral deviation is tiny. This creates incentives to free ride: individual SG deployment level (deviation) is either zero or is very low.

Countries 5–7 have large preferred SG level and free drive when marginal deployment costs are relatively low: their individual SG deployment level in the case of deviation determines the total SG level. However, increasing costs of deployment curb their free driving ability and the SG deployment level decreases below it’s cooperative level. This is the case because the optimal SG level of other coalition members is lower. As a result, these countries do not have an incentive to free drive when SG deployment costs are substantial. However, as deployment costs are increasing, countries may become free riders. In the considered range of marginal deployment costs this happens only for the country 5 shown in Fig. 14.

Figure 14 depicts individual losses in cooperation, Nash equilibrium and unilateral deviation cases for countries 2–7. Unilateral deviation occurs if the associated losses are smaller than in the cooperative case. This difference is depicted in Fig. 14 by the area. The area in green indicates the case when a country is better off in the cooperative state. Figure 14 demonstrates that countries with large preferred SG levels (countries 5–7) have an incentive to deviate when SG deployment costs are low. They also may have an incentive to act as free riders if deployment costs are substantial. In the considered range of marginal deployment costs, it happens only for the country 5.

Countries whose preferred SG level is close to the socially optimal value (i.e., countries 3 and especially 4) have the strongest free-riding incentive. For these countries deviation represents costs with no benefits and thus is not rational. Countries with small preferred SG level, such as country 2, have small incentive to free ride at a relatively large deployment costs.

D Counter SG with Non-negligible Deployment Costs

Damage function:

$$\begin{aligned} D_i = \frac{1}{2} \alpha G^2_N - \alpha k_i G_N + d_i + \frac{1}{2} z g_i^2 \end{aligned}$$

Where z is the slope of marginal SG or counter-SG deployment costs. The main difference with the previous case is that $g_i$ may be both positive and negative: $g_i \in R$.

1.1 D.1 Third-Stage Solution

Minimization problems are similar to the previous case with non-negligible costs. The difference is in the reaction functions for the case where $g_i$ is not restricted to be non-negative:

Reaction function for a coalition S reads:

$$\begin{aligned} G_S = \frac{ \alpha s^2 }{\alpha s^2 + z} ({\bar{k}}_S- G_{N \setminus S}) \end{aligned}$$

Non-signatory $i \in N\setminus S$ reaction function reads:

$$\begin{aligned} g_i = \frac{\alpha }{\alpha + z} ( k_i - G_{N\setminus i}) \end{aligned}$$

1.2 D.2 Illustrations

Fig. 15

Impact of the upper limit of counter- SG on individual levels of SG deployment in (i) global cooperation (grey dashed lines); (ii) Nash equilibrium (three upper graphs); (iii) unilateral deviation (three lower graphs). Columns refer to three considered distributions of k: a left-modal, b uniform, and c right-modal distributions

Fig. 16

Impact of the upper limit of counter- SG on individual losses in (i) global cooperation (dashed lines); (ii) Nash equilibrium (three upper graphs); (iii) unilateral deviation (three lower graphs). Columns refer to three considered distributions of k: a left-modal, b uniform, and c right-modal distributions

Figure 15 shows individual SG deployment levels for the scenarios of global cooperation, Nash equilibrium and unilateral deviation, where marginal costs of both SG and counter-SG deployment aret set to $z = 0.1$. The behavior of countries is intuitive: non-drivers counteract SG deployment activities of drivers (subject to a specified limit). In Nash equilibrium (upper graphs) some countries may switch from counter-SG to SG deployment as the CSG limit increases. In unilateral deviation (lower graphs) countries are clearly divided into two groups: (i) drivers that deploy SG beyond cooperative optimum, and (ii) non-drivers that counteract SG deployment. Note that a country whose optimal SG level almost coincides with the global optimal level (country 4 in the uniform distribution, middle column) tends to free ride in the case of a unilateral deviation, avoiding either activity.

Figure 16 depicts countries’ individual losses in Nash equilibrium and in the case of a unilateral deviation. For comparison, individual losses in cooperation are indicated by dashed lines. Color stays the same for each country.

E Exogenous Mitigation

Consider the global share of emissions abatement $\sum _{i \in N}{\bar{a}} = {\bar{A}}_N \in [0, 1]$.

Damage function then reads:

$$\begin{aligned} D_i = \frac{\alpha }{2} G^2_N - (1-{\bar{A}}_N) \alpha k_i G_N + (1-{\bar{A}}_N)^2 d_i + \frac{c_i}{2.6}{{\bar{a}}}_i^{2.6} \end{aligned}$$

Where $c_i$ in the parameter of abatement costs of country i.

Signatories

Minimization problem for members of a coalition $S \subseteq N$ reads:

$$\begin{aligned} \min D_S = \frac{1}{2} s \alpha (G_S + G_{N\setminus S})^2 - (1-{\bar{A}}_N) \alpha k_S (G_S + G_{N\setminus S}) + (1-{\bar{A}}_N)^2 d_S + \sum _{i \in S}\frac{c_i}{2.6}{{\bar{a}}}_i^{2.6} \end{aligned}$$

Non-signatories

Non-signatory $i \in N\setminus S$ minimize their individual damage function as following:

$$\begin{aligned} \min D_i = \frac{\alpha }{2} G^2_N - (1-{\bar{A}}_N) \alpha k_i G_N + (1-{\bar{A}}_N)^2 d_i + \frac{c_i}{2.6}{{\bar{a}}}_i^{2.6} \end{aligned}$$

1.1 E.1 Amount of Transfers

Minimum amount of transfers to sustain the cooperation is the sum of benefits from deviation of all free drivers:

$$\begin{aligned} \sum _{j \in DR} (D_j(N) - D_j(N\setminus j)) = 0.5 \alpha (1-{\bar{A}}_N)^2 \sum _{j \in DR} \left( k_j - \frac{k_N}{n} \right) ^2 \end{aligned}$$

Note that the actual transfers size may exceed this level.

F Fixed Costs Associated with SG Deployment

Damage function:

$$\begin{aligned} D_i = \frac{1}{2} \alpha G^2_N - \alpha k_i G_N + d_i + FC(G_N) \end{aligned}$$

Where

$$\begin{aligned}FC(G_N) = {\left\{ \begin{array}{ll} 0 &{} \textit{if } G_N =0 \\ FC&{} \textit{if } G_N > 0 \end{array}\right. } \end{aligned}$$

1.1 F.1 The Proposition Proof

The difference between the default specification and the modification with fixed costs of deployment is that a corner solution may come up. Here, corner solution means moratorium on SG. We consider an active coalition S with the optimal deployment level $\bar{k_S}$. We distinguish three cases depending on the magnitude of fixed costs, FC:

• $FC < 0.5 \alpha (\bar{k_S})^2$ - Deployment equilibrium (interior solution) in both coalition and no cooperation. The case is similar to the default specification, without FC.

• $0.5 \alpha (\bar{k_S})^2 \le FC < 0.5 \alpha k_h^2$ - Moratorium on SG in coalition and deployment in non-cooperative scenario.

• $FC \ge 0.5 \alpha k_h^2$ - Moratorium on SG in both coalition and non-cooperative scenario: no collective action problem.

The interesting case (and thus our focus in the following) is when $0.5 \alpha (\bar{k_S})^2 \le FC < 0.5 \alpha k_j^2$.

Individual damages
	$ND: k_i \le (2FC/\alpha )^{0.5}$
$D_i(S)$	$d_i$
$D_i(S \setminus j)$	$0.5 \alpha k_j^2 - \alpha k_i k_j + FC + d_i$
$D_i(S \setminus j) - D_i(S)$	$0.5 \alpha k_j (k_j - 2 k_i) +FC$
	$DR: k_j > (2FC/\alpha )^{0.5}$
$D_j(S)$	$d_j$
$D_j(S \setminus j)$	$- 0.5 \alpha k^2_j + FC + d_j$
$D_j(S) - D_j(S \setminus j)$	$0.5 \alpha k_j^2 - FC$

Proposition reads: $\sum _{j \in DR^S} (D_j(S) - D_j(S\setminus j)) \le \sum _{i \in ND^S} (D_i(\emptyset ) - D_i(S))$

After we substitute the considered functional form of damages, where $k_h$ denotes the largest preferred SG level of all considered countries from set N, it reads:

$$\begin{aligned} \sum _{j \in DR^S} (\frac{\alpha }{2} k_j^2 - FC) \le \sum _{i \in ND^S} (\frac{\alpha }{2} k_h (k_h - 2 k_i) +FC ) \end{aligned}$$

Proof

$|DR^S|$ is the number of drivers in a coalition S and $|ND^S|$ is the number of non-drivers in a coalition S.

$$\begin{aligned} \frac{\alpha }{2} \sum _{j \in DR^S} k_j^2 - |DR^S|\cdot FC \le \frac{\alpha }{2} |ND^S| k^2_h - \alpha k_h k_{ND^S} + |ND^S|\cdot FC \end{aligned}$$

We can rewrite as follows:

$$\begin{aligned} \frac{\alpha }{2} \sum _{j \in DR^S} k_j^2 \le \frac{\alpha }{2} |ND^S| k^2_h - \alpha k_h k_{ND^S} + s FC \end{aligned}$$

Assumming $0.5 \alpha (\bar{k_S})^2 \le FC$, it follows:

$$\begin{aligned} \frac{\alpha }{2} \sum _{j \in DR^S} k^2_j \le \frac{\alpha }{2} |ND^S| k^2_h - \alpha k_h k_{ND^S} + \frac{\alpha }{2} s \left( \bar{k_S} \right) ^2 \end{aligned}$$

Multiply by $2/\alpha$ and rewrite LHS in an identical form:

$$\begin{aligned} \sum _{j \in DR^S} \left( k_h k_j - (k_h - k_j)k_j \right) \le |ND^S| k^2_h - 2 k_h k_{ND^S} + s \left( \bar{k_S} \right) ^2 \end{aligned}$$

Open the summation operator:

$$\begin{aligned} k_h k_{DR^S} - \sum _{j \in DR^S} \left( (k_h - k_j)k_j \right) \le |ND^S| k^2_h - 2 k_h k_{ND^S} + s \left( \bar{k_S} \right) ^2 \end{aligned}$$

Can be rewritten as follows:

$$\begin{aligned} - \sum _{j \in DR^S} \left( (k_h - k_j)k_j \right) + k_h k_{ND^S} \le s k^2_h - |DR^S| k^2_h - k_h k_S + \frac{k_S^2}{s} \end{aligned}$$

Combine RHS in the square of a difference:

$$\begin{aligned} |DR| k^2_h - \sum _{j \in DR^S} \left( (k_h - k_j)k_j \right) + k_h k_{ND^S} - k_h k_S \le s \left( k^2_h - 2 k_h \frac{k_S}{s} + \left( \frac{k_S}{s} \right) ^2 \right) \end{aligned}$$

Rewrite LHS in an identical form:

$$\begin{aligned} \sum _{j \in DR^S} (k_h (k_h - k_j)) - \sum _{j \in DR^S} \left( (k_h - k_j)k_j \right) + k_h k_{DR^S} + k_h k_{ND^S} - k_h k_S \le s \left( k_h - \frac{k_S}{s} \right) ^2 \end{aligned}$$

Cancel out three terms from LHS:

$$\begin{aligned} \sum _{j \in DR^S} (k_h (k_h - k_j)) - \sum _{j \in DR^S} \left( (k_h - k_j)k_j \right) \le s \left( k_h - \frac{k_S}{s} \right) ^2 \end{aligned}$$

The following is true by the definition:

$$\begin{aligned} \sum _{j \in DR^S} (k_h - k_j)^2 \le s \left( k_h - \bar{k_S} \right) ^2 \end{aligned}$$

$\square$

G Countries with Distinct Decision-Making Weights

Consider an arbitrary active coalition $S \subseteq N$. We introduce country-specific weights, $w_i$, which are normalized to one in a considered coalition S: $\sum _{i \in S} w_i =1$. As we are interested in decision making within one considered coalition, we can normalize weights to one in any coalition we analyze. Then, cooperative solution is the result of the minimization of weighted sum of damage functions:

$$\begin{aligned} \min _{G_S} \sum _{i \in S} w_i D_i \end{aligned}$$

The solution reads:

$$\begin{aligned} k_{wgt} \equiv \sum _{i \in S} w_i k_i \end{aligned}$$

where $DR^S$ is the set of drivers: $j \in DR^S$ if $j \in S$ and $k_j > k_{wgt}$, and $ND^S$ is the set of non-drivers: $i \in ND^S$ if $i \in S$ and $k_i \le k_{wgt}$.

1.1 G.1 The Proposition Proof

The Proposition reads:

$$\begin{aligned} \sum _{j \in DR^S} w_j (D_j(S) - D_j(S\setminus j)) \le \sum _{i \in ND^S} w_i(D_i(\emptyset ) - D_i(S)) \end{aligned}$$

In the considered specification it takes the following form:

$$\begin{aligned} \frac{\alpha }{2} \sum _{j \in DR^S} w_j (k_j - k_{wgt})^2 \le \frac{\alpha }{2} \sum _{i \in ND^S} w_i (k_h - k_{wgt})( k_h + k_{wgt}- 2 k_i) \end{aligned}$$

Proof

Since $k_h$ denotes the largest preferred SG level, the following inequality holds:

$$\begin{aligned} \sum _{i \in ND^S} w_i k_i \le \sum _{i \in ND^S} w_i k_h \end{aligned}$$

Then we add the term $\sum _{i \in S} w_i k_i$ to both RHS and LHS, arriving at:

$$\begin{aligned} \sum _{i \in S} w_i k_i + \sum _{i \in ND^S} w_i k_i \le \sum _{i \in ND^S} w_i k_h + \sum _{i \in S} w_i k_i \end{aligned}$$

We then rewrite the LHS and substitute $k_{wgt} \equiv \sum _{i \in S} w_i k_i$ in the RHS:

$$\begin{aligned} \sum _{j \in DR^S} w_j k_j + 2 \sum _{i \in ND^S} w_i k_i \le \sum _{i \in ND^S} w_i k_h + k_{wgt} \end{aligned}$$

As country-specific weights in a coalition are normalized to one, we may use the following equality $\sum _{i \in DR^S}w_i +\sum _{i \in ND^S}w_i =1$:

$$\begin{aligned} \sum _{j \in DR^S} w_j k_j - \sum _{j \in DR^S} w_j k_{wgt} \le \sum _{i \in ND^S} w_i k_h + \sum _{i \in ND^S} w_ik_{wgt}- 2 \sum _{i \in ND^S} w_i k_i \end{aligned}$$

Now we take the sum operator out of brackets:

$$\begin{aligned} \sum _{j \in DR^S} w_j (k_j - k_{wgt}) \le \sum _{i \in ND^S} w_i ( k_h + k_{wgt}- 2 k_i) \end{aligned}$$

We then multiply both RHS and LHS by $(k_h - k_{wgt} ) >0$:

$$\begin{aligned} (k_h - k_{wgt} ) \sum _{j \in DR^S} w_j (k_j - k_{wgt}) \le (k_h - k_{wgt}) \sum _{i \in ND^S} w_i ( k_h + k_{wgt}- 2 k_i)\end{aligned}$$

As $k_h \ge k_j ~ ~ \forall j\in N$, we have $(k_h - k_{wgt} ) \ge (k_j - k_{wgt})$ , and therefore:

$$\begin{aligned} (k_j - k_{wgt}) \sum _{j \in DR^S} w_j (k_j - k_{wgt}) \le (k_h - k_{wgt} ) \sum _{j \in DR^S} w_j (k_j - k_{wgt}) \le (k_h - k_{wgt} ) \sum _{j \in ND^S} w_j ( k_h + k_{wgt}- 2 k_i) \end{aligned}$$

Or just:

$$\begin{aligned} \sum _{j \in DR^S} w_j (k_j - k_{wgt})^2 \le (k_h - k_{wgt} ) \sum _{j \in ND^S} w_j ( k_h + k_{wgt}- 2 k_i) \end{aligned}$$

Finally, by multiplying both sides by $0.5\alpha$ we arrive at the expression, which was to be demonstrated:

$$\begin{aligned} \frac{\alpha }{2} \sum _{j \in DR^S} w_j (k_j - k_{wgt})^2 \le \frac{\alpha }{2} \sum _{i \in ND^S} w_i (k_h - k_{wgt})( k_h + k_{wgt}- 2 k_i) \end{aligned}$$

$\square$

H Empirical Calibration

To identify individual preferences of countries, we take the following steps:

• 1. From the paper by Rickels et. al (2020), we adopt the following estimates:

  $V_i(0)$ - absolute impact on country GVA in the absence of SG deployment

  $V_i(G^*)$ - absolute impact on country GVA in the optimal level of SG deployment

  We then use these data to find the normalized difference:

  $$\begin{aligned} \frac{V_i(G^*) - V_i(0)}{V_i(0)} = V_i \end{aligned}$$

• 2. We use the normalized difference in countries’ GVA to estimate the normalized difference in countries’ damages when $SG = 0$ and $SG = G^*$ as formulated in our model:

  $$\begin{aligned} 2/\alpha \cdot (D_i(0) - D_i(G^*)) = 2 G^* k_i - G^{*2}= V_i \end{aligned}$$

  . Following the approach in the paper by Rickels et al.,(2020), we define the optimal level of SG ($G^*$) as GDP-weighted average of countries’ preferences $k_i , i \in N$:

  $$\begin{aligned} G^* = \sum _{i \in N} \frac{GDP_i}{\sum _{j \in N} GDP_j} k_i \end{aligned}$$

• 3. We solve the system of equations for each country $i \in N$ :

  $$\begin{aligned} 2 G^* k_i - G^{*2} = V_i \end{aligned}$$

• 4. We put a lower bound of 0 on $k_i$.

• 5. Normalize preferred SG levels for all countries to meet the global average of 4 W/m².