# The Allocation of Greenhouse Gas Emission in European Union through Applying the Claims Problems Approach

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

**endowment**that has to be allocated among them. Each agent has a

**claim**,

**${c}_{i}\in {\mathbb{R}}_{+}$**on it. Let

**c**$\equiv {\left({c}_{i}\right)}_{i\in N}$ be the claims vector.

**$\mathcal{B}$**the set of all claims problems.

_{2}emission right by [16,18], these rules include Proportional, Constrained Equal Award, Constrained Equal Losses, Talmud, Adjusted Proportional, and $\alpha -minimal$.

**Proportional (P)**[22] divides the emission budget proportionally among countries according to their projections. In this rule for each $(E,c)\in \mathcal{B}$ and each $i\in N$, ${P}_{i}(E,c)\equiv \lambda {c}_{i},$ where $\lambda =E/\sum _{i\in N}{c}_{i}.$

**Constrained Equal Award (CEA)**[22] divides the emission budget equally to all countries’ provided that none of them receive more than their projections. The process is as follows: If the average emission budget exceeds the projection of one country, the rule fully satisfies the country’s projection, excludes this country from the allocation process, and continues to allocate the remaining emission budget equally to the rest of the countries.

**Constrained Equal Losses (CEL)**[22] proposes to divide the loss (difference between aggregate projections and emission budget) equally to all countries given that no country receives a negative amount. The allocation each country receives is the difference between her projection and the loss which is divided by $CEL$. If this difference is negative for a country, the country’s allocation will be zero and she leaves the allocation process. $CEL$ divides the loss equal to the remaining countries.

**Talmud (T)**[23] proposes a combination of CEA and CEL. This rule focuses on the half-sum of aggregate projections. If the emission budget is less than or equal to the half-sum of projections, CEA is applied. Countries receive the average emission budget or half of their projection (if the average emission budget is greater than half of the projection). If the emission budget is greater than the half-sum of projections, the following process is conducted: countries receive half of their projections. The projections and emission budget are revised down by these initial allocations and CEL is applied to these revised amounts.

**Adjusted Proportional (AP)**[24] has been introduced in two steps. In the first step, AP assigns to each country a minimal right (${m}_{i}$). This minimal right is the remaining emission budget when the projections of the rest countries have been satisfied, with respect to the condition of ${m}_{i}(E,c)=max\left\{0,E-{\sum}_{j\ne i}{c}_{j}\right\}$. In the second step, the projections are revised down by the minimal rights. Then the remaining emission budget is assigned proportionally among countries based on their revised projections.

**$\mathbf{\alpha}$-minimal ($\mathbf{\alpha}$-min)**[25] proposes to give each country a minimal amount equal to the lowest projection, in the case that the emission budget is enough. After revising down the projections by minimal amounts, the rule distributes the remaining emission budget proportionally among countries according to their revised projections. However, if the emission budget is not sufficient to give all countries the minimal amount, this rule recommends dividing the emission budget equally among countries.

**non-negativity**), ${\phi}_{i}(E,c)\ge 0$, for all $i\in N$. Second, countries cannot receive more than their projections (

**claim-boundedness**), ${\phi}_{i}(E,c)\le {c}_{i}$, for all $i\in N$. Third, the whole emission budget should be divided among countries. (

**efficiency**) ${\sum}_{i\in N}{\phi}_{i}(E,c)=E$.

**Equal treatment of equals**states that countries with the same projections should receive an equal amount of emission budget. For each $(E,c)\in \beta $ and $i,j\subseteq N$, if ${c}_{i}={c}_{j}$ then ${\phi}_{i}(E,c)={\phi}_{j}(E,c)$.

**Anonymity**says the allocation of emission budget exclusively depends on countries’ projections. Other factors such as the identity of the countries cannot affect the emission allocation. For each $(E,c)\in \beta $ each $\pi \in {\prod}^{N}$ and each $i\in N$, ${\phi}_{\pi \left(i\right)}(E,{\left({c}_{\pi \left(i\right)}\right)}_{i\in N})=\phi (E,c)$, where ${\prod}^{N}$ is the permutations of N.

**Order preservation**[23] means the emission allocation assigned to countries with larger projections cannot be smaller than the emission allocation of countries with lower projections. For each $(E,c)\in \beta $ and each $i,j\in N$ such that ${c}_{i}\ge {c}_{j}$, then ${\phi}_{i}(E,c)\ge {\phi}_{j}(E,c)$. Likewise, countries with larger projections bear an equal or larger amount of loss than countries with lower projections. ${c}_{i}-{\phi}_{i}(E,c)\ge {c}_{j}-{\phi}_{j}(E,c)$.

**Claims monotonicity**states if a country increases its projection, the allocation assigned to this country cannot be less than the initial amount. For each $(E,c)\in \beta $, $i\in N$ and each ${c}_{i}^{\prime}>{c}_{i}$ we have ${\phi}_{i}(E,{c}_{i}^{\prime},{c}_{-i})\ge {\phi}_{i}(E,c)$.

**Composition down**[26] is a property for situations in which the emission budget is reduced after allocation due to re-evaluation of the emission budget or setting more strict emission reduction rules. For instance, the EU decides to increase the percentage of emission reduction by 2030 to more than 55%, while the emission budget of 55% reduction has been allocated before this announcement. In this case, we have two choices: First, we cancel the initial emission budget allocation and reallocate the new amount of the emission budget. Second, we consider the initial emission allocation assigned to each country as their claims (rather than considering the projections) and divide the new emission budget by these new claims. for each $(E,c)\in \mathcal{B},$ each $i\in N$, and each $0\le {E}^{\prime}\le E,$ ${\phi}_{i}({E}^{\prime},c)={\phi}_{i}({E}^{\prime},\phi (E,c))$.

**Invariance under claims truncation**[27] imposes an upper bound to countries’ projections. If a country’s projection is greater than the emission budget, the exceeding part will be ignored. This property says that countries cannot request more than the available emission budget. For each $(E,c)\in \beta $ and each $i\in N$, ${\phi}_{i}(E,c)={\phi}_{i}(E,{(min{c}_{i},E)}_{i\in N})$.

## 3. Results and Discussion

#### 3.1. The Allocation of total GHG Emission

**Gini index (Gi):**[28] Gini index is a statistic dispersion indicator that evaluates the degree of inequality in a resource allocation. Its value is in the interval of 0 and 1, 0 indicates perfect equality and 1 represents extreme inequality. Given an n-dimensional endowment, the Gini index is defined as follow:

**Coefficient of Variation (CV):**For analyzing the stability of the allocation results, the countries’ historical emissions should be considered. Countries decide about the fairness of a rule by comparing their historical emission with the emission allocation a particular rule assigns them. Countries would accept a division rule when they are ensured about the fairness of that rule [29]. If the amount they receive by a rule is far from their historical emissions, countries would refuse that rule. For this purpose, we apply the Coefficient of Variation Index to evaluate the weights of countries [30]. CV calculates the dispersion of allocation around the mean. Formally, we can define it as:

#### 3.2. The Annual Allocation of GHG Emission

## 4. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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Emission Budget | Projection (WEM) | Projection (WAM) |
---|---|---|

25,662.12 | 33,027.50 | 30,927.98 |

**Table 2.**Properties and division rules. (rows represent division rules and columns show the properties).

Properties/Rules | P | CEA | CEL | T | AP | $\mathit{\alpha}$-min |
---|---|---|---|---|---|---|

Equal treatment of equals | Yes | Yes | Yes | Yes | Yes | Yes |

Anonymity | Yes | Yes | Yes | Yes | Yes | Yes |

Order preservation | Yes | Yes | Yes | Yes | Yes | Yes |

Claims monotonicity | Yes | Yes | Yes | Yes | Yes | Yes |

Composition down | Yes | Yes | Yes | No | No | Yes |

Invariance under claims truncation | No | Yes | No | Yes | No | No |

Country | Projection | P | CEA | CEL | T | AP | $\mathit{\alpha}$-min |
---|---|---|---|---|---|---|---|

Malta | 30.11 | 23.40 | 30.11 | 0.00 | 15.05 | 23.28 | 30.11 |

Cyprus | 79.16 | 61.51 | 79.16 | 0.00 | 39.58 | 61.21 | 67.95 |

Sweden | 108.77 | 84.51 | 108.77 | 0.00 | 54.38 | 84.11 | 90.79 |

Luxembourg | 115.98 | 90.12 | 115.98 | 0.00 | 57.99 | 89.69 | 96.35 |

Slovenia | 120.76 | 93.83 | 120.76 | 0.00 | 60.38 | 93.38 | 100.03 |

Estonia | 121.97 | 94.77 | 121.97 | 0.00 | 60.98 | 94.32 | 100.97 |

Lithuania | 138.88 | 107.91 | 138.88 | 0.00 | 69.44 | 107.39 | 114.01 |

Latvia | 140.36 | 109.06 | 140.36 | 0.00 | 70.18 | 108.54 | 115.15 |

Croatia | 207.86 | 161.51 | 207.86 | 0.00 | 103.93 | 160.74 | 167.22 |

Finland | 278.29 | 216.23 | 278.29 | 0.00 | 139.15 | 215.20 | 221.55 |

Slovakia | 405.30 | 314.92 | 405.30 | 50.99 | 202.65 | 313.41 | 319.52 |

Denmark | 428.22 | 332.72 | 428.22 | 73.91 | 214.11 | 331.14 | 337.20 |

Bulgaria | 499.75 | 388.30 | 499.75 | 145.44 | 249.88 | 386.45 | 392.37 |

Portugal | 499.77 | 388.32 | 499.77 | 145.46 | 249.88 | 386.47 | 392.39 |

Hungary | 602.23 | 467.93 | 602.23 | 247.92 | 301.12 | 465.70 | 471.42 |

Ireland | 685.67 | 532.76 | 685.67 | 331.36 | 342.83 | 530.22 | 535.79 |

Austria | 734.51 | 570.71 | 734.51 | 380.20 | 367.25 | 567.99 | 573.46 |

Romania | 798.82 | 620.68 | 798.82 | 444.51 | 399.41 | 617.72 | 623.07 |

Greece | 801.66 | 622.88 | 801.66 | 447.35 | 400.83 | 619.91 | 625.26 |

Czech | 1173.08 | 911.47 | 1173.08 | 818.77 | 677.29 | 907.13 | 911.76 |

Belgium | 1260.61 | 979.48 | 1260.61 | 906.30 | 764.82 | 974.81 | 979.27 |

Netherlands | 1764.67 | 1371.14 | 1764.67 | 1410.36 | 1268.88 | 1364.60 | 1368.09 |

Spain | 2964.72 | 2303.57 | 2933.14 | 2610.41 | 2468.93 | 2292.58 | 2293.76 |

Italy | 3641.23 | 2829.21 | 2933.14 | 3286.92 | 3145.44 | 2815.72 | 2815.60 |

France | 3732.83 | 2900.38 | 2933.14 | 3378.52 | 3237.04 | 2886.55 | 2886.26 |

Poland | 3787.25 | 2942.67 | 2933.14 | 3432.94 | 3291.46 | 2928.64 | 2928.24 |

Germany | 7905.04 | 6142.16 | 2933.14 | 7550.73 | 7409.25 | 6235.22 | 6104.55 |

Country | Projection | P | CEA | CEL | T | AP | $\mathit{\alpha}$-min |
---|---|---|---|---|---|---|---|

Malta | 30.11 | 24.98 | 30.11 | 0.00 | 15.05 | 24.51 | 30.11 |

Cyprus | 75.29 | 62.47 | 75.29 | 0.00 | 37.65 | 61.28 | 67.39 |

Luxembourg | 90.77 | 75.32 | 90.77 | 0.00 | 45.38 | 73.87 | 80.16 |

Sweden | 108.77 | 90.25 | 108.77 | 0.00 | 54.38 | 88.52 | 95.02 |

Lithuania | 119.70 | 99.32 | 119.70 | 0.00 | 59.85 | 97.42 | 104.03 |

Estonia | 120.66 | 100.12 | 120.66 | 0.00 | 60.33 | 98.20 | 104.83 |

Slovenia | 132.02 | 109.54 | 132.02 | 0.00 | 66.01 | 107.44 | 114.20 |

Latvia | 132.76 | 110.16 | 132.76 | 0.00 | 66.38 | 108.05 | 114.81 |

Croatia | 198.58 | 164.77 | 198.58 | 0.00 | 99.29 | 161.61 | 169.12 |

Finland | 264.21 | 219.22 | 264.21 | 27.70 | 132.10 | 215.03 | 223.28 |

Slovakia | 360.87 | 299.43 | 360.87 | 124.36 | 180.44 | 293.70 | 303.03 |

Denmark | 428.22 | 355.31 | 428.22 | 191.71 | 214.11 | 348.51 | 358.61 |

Portugal | 448.84 | 372.42 | 448.84 | 212.33 | 224.42 | 365.29 | 375.62 |

Bulgaria | 483.58 | 401.24 | 483.58 | 247.07 | 241.79 | 393.56 | 404.29 |

Hungary | 558.34 | 463.28 | 558.34 | 321.83 | 279.17 | 454.41 | 465.97 |

Ireland | 627.44 | 520.61 | 627.44 | 390.93 | 336.65 | 510.64 | 522.99 |

Greece | 672.19 | 557.74 | 672.19 | 435.68 | 381.40 | 547.06 | 559.92 |

Austria | 699.54 | 580.43 | 699.54 | 463.03 | 408.75 | 569.32 | 582.48 |

Romania | 757.57 | 628.58 | 757.57 | 521.06 | 466.78 | 616.55 | 630.37 |

Czech | 1057.91 | 877.79 | 1057.91 | 821.40 | 767.12 | 860.98 | 878.19 |

Belgium | 1151.02 | 955.04 | 1151.02 | 914.51 | 860.23 | 936.76 | 955.02 |

Netherlands | 1769.12 | 1467.91 | 1769.12 | 1532.61 | 1478.33 | 1439.80 | 1465.04 |

Spain | 2486.89 | 2063.47 | 2486.89 | 2250.38 | 2196.10 | 2023.96 | 2057.30 |

Italy | 3159.62 | 2621.66 | 3159.62 | 2923.11 | 2868.83 | 2571.47 | 2612.40 |

Poland | 3356.11 | 2784.69 | 3242.70 | 3119.60 | 3065.32 | 2731.38 | 2774.53 |

France | 3732.83 | 3097.27 | 3242.70 | 3496.32 | 3442.04 | 3037.98 | 3085.38 |

Germany | 7905.04 | 6559.11 | 3242.70 | 7668.53 | 7614.25 | 6924.81 | 6528.04 |

**Table 5.**Gini Index (Gi) and Coefficient of Variation (CV) for the total GHG emission allocation (WEM scenario).

Criterion | P | CEA | CEL | T | AP | ${\mathit{\alpha}}_{min}$ |
---|---|---|---|---|---|---|

Gi | 0.63 | 0.55 | 0.75 | 0.70 | 0.63 | 0.63 |

CV | 1.45 | 1.10 | 1.82 | 1.74 | 1.47 | 1.44 |

**Table 6.**Gini Index (Gi) and Coefficient of Variation (CV) for the total GHG emission allocation (WAM scenario).

Criterion | P | CEA | CEL | T | AP | ${\mathit{\alpha}}_{min}$ |
---|---|---|---|---|---|---|

Gi | 0.64 | 0.57 | 0.74 | 0.71 | 0.64 | 0.63 |

CV | 1.50 | 1.17 | 1.79 | 1.76 | 1.55 | 1.50 |

Year | Emission Budget | Projection (WEM) | Projection (WAM) |
---|---|---|---|

2021 | 3,023,064.45 | 3,393,431.76 | 3,326,137.63 |

2022 | 2,921,541.73 | 3,408,314.62 | 3,305,192.39 |

2023 | 2,820,019.00 | 3,408,667.03 | 3,269,079.93 |

2024 | 2,718,496.27 | 3,377,677.61 | 3,200,327.53 |

2025 | 2616,973.55 | 3,356,436.31 | 3,143,477.10 |

2026 | 2,515,450.82 | 3,310,391.92 | 3,077,607.66 |

2027 | 2,413,928.10 | 3,268,123.97 | 3,012,592.09 |

2028 | 2,312,405.37 | 3,217,965.41 | 2,938,835.50 |

2029 | 2,210,882.64 | 3,173,982.99 | 2,872,139.24 |

2030 | 2,109,359.92 | 3,112,504.27 | 2,782,595.63 |

**Table 8.**The aggregate annual GHG emission allocation 2021–2030 (WEM), numbers are in megatonnes (Mt).

Country | Projection | P | CEA | CEL | T | AP | $\mathit{\alpha}$-min |
---|---|---|---|---|---|---|---|

Malta | 30.11 | 22.85 | 30.11 | 0.00 | 26.95 | 22.86 | 30.11 |

Cyprus | 79.16 | 60.99 | 79.16 | 0.00 | 71.31 | 61.03 | 67.95 |

Sweden | 108.77 | 87.48 | 108.77 | 0.00 | 99.73 | 87.51 | 94.32 |

Luxembourg | 115.98 | 88.97 | 115.98 | 0.00 | 104.26 | 89.02 | 95.70 |

Slovenia | 120.76 | 92.10 | 120.76 | 0.00 | 108.46 | 92.15 | 98.76 |

Estonia | 121.97 | 93.68 | 121.97 | 0.00 | 109.77 | 93.73 | 100.37 |

Lithuania | 138.88 | 108.63 | 138.88 | 0.23 | 125.91 | 108.69 | 115.25 |

Latvia | 140.36 | 104.62 | 140.36 | 0.00 | 124.72 | 104.69 | 111.15 |

Croatia | 207.86 | 159.70 | 207.86 | 5.32 | 187.03 | 159.79 | 165.87 |

Finland | 278.29 | 217.62 | 278.29 | 26.23 | 252.29 | 217.73 | 223.42 |

Slovakia | 405.30 | 311.47 | 405.30 | 74.48 | 364.66 | 311.63 | 316.40 |

Denmark | 428.22 | 336.49 | 428.22 | 101.32 | 389.14 | 336.65 | 341.40 |

Bulgaria | 499.75 | 390.00 | 499.75 | 149.90 | 452.86 | 390.20 | 394.43 |

Portugal | 499.77 | 403.29 | 499.77 | 168.57 | 459.74 | 403.47 | 407.92 |

Hungary | 602.23 | 466.18 | 602.23 | 239.45 | 543.40 | 466.42 | 469.96 |

Ireland | 685.67 | 529.00 | 685.67 | 322.90 | 617.76 | 529.28 | 532.23 |

Austria | 734.51 | 567.38 | 734.51 | 371.73 | 662.40 | 567.68 | 570.31 |

Romania | 798.82 | 620.65 | 798.82 | 436.03 | 722.28 | 620.98 | 623.25 |

Greece | 801.66 | 621.47 | 801.66 | 438.88 | 724.09 | 621.79 | 624.03 |

Czech | 1173.08 | 921.42 | 1173.08 | 810.30 | 1065.12 | 921.87 | 921.99 |

Belgium | 1260.61 | 963.89 | 1260.61 | 897.83 | 1131.79 | 964.42 | 963.44 |

Netherlands | 1764.67 | 1374.69 | 1764.67 | 1401.89 | 1598.57 | 1375.38 | 1371.37 |

Spain | 2964.72 | 2289.44 | 2739.55 | 2601.94 | 2473.20 | 2290.63 | 2278.63 |

Italy | 3641.23 | 2822.37 | 2939.66 | 3278.45 | 2792.57 | 2823.82 | 2807.41 |

France | 3732.83 | 2886.75 | 2961.27 | 3370.04 | 2829.77 | 2888.24 | 2871.20 |

Poland | 3787.25 | 2902.18 | 2956.14 | 3424.48 | 2816.08 | 2903.73 | 2885.97 |

Germany | 7905.04 | 6218.81 | 3069.07 | 7542.26 | 4808.34 | 6208.75 | 6179.30 |

Criterion | P | CEA | CEL | T | AP | $\mathit{\alpha}$-min |
---|---|---|---|---|---|---|

Gi | 0.63 | 0.55 | 0.75 | 0.58 | 0.63 | 0.63 |

CV | 1.46 | 1.10 | 1.83 | 1.27 | 1.46 | 1.45 |

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**MDPI and ACS Style**

Salekpay, F.
The Allocation of Greenhouse Gas Emission in European Union through Applying the Claims Problems Approach. *Games* **2023**, *14*, 9.
https://doi.org/10.3390/g14010009

**AMA Style**

Salekpay F.
The Allocation of Greenhouse Gas Emission in European Union through Applying the Claims Problems Approach. *Games*. 2023; 14(1):9.
https://doi.org/10.3390/g14010009

**Chicago/Turabian Style**

Salekpay, Foroogh.
2023. "The Allocation of Greenhouse Gas Emission in European Union through Applying the Claims Problems Approach" *Games* 14, no. 1: 9.
https://doi.org/10.3390/g14010009