An Asymmetric Bargaining Model for Natural-Gas Distribution

Abstract

For the sustainable socio-economic growth, the energy supply is one of the foundations for any country. The gas shortage is one of the most significant impediments to any emerging country’s economic progress, making it a contested and disputed resource. In the middle of a supply–demand mismatch, distributing limited available gas across administrative units/provinces with competing requirements is a key challenge. In this work, an asymmetric gas allocation bargaining model is proposed under gas shortage to resolve natural gas-related disputes among Pakistan’s administrative units/provinces. Each administrative unit/province is characterized by its gas demand. Results show that the Nash bargaining theory, when applied under equal and bargaining weights, can address the supply–demand mismatches of the gas sector in Pakistan. Such an approach could help policymakers to make a fair gas-supply management system during gas shortage periods and would help in resolving the disputes between the provinces.

1. Introduction

In today’s modern society, natural gas is one of the main energy sources. The majority of the natural gas globally is supplied to customers/end consumers through a pipeline-based distribution system [1]. Natural gas is a major energy source in Pakistan, followed by hydroelectricity, liquid fuels, coal, and nuclear energy. According to the annual reports of the Government of Pakistan (GoP) and the national electric power regulatory authority (NEPRA), the share of natural gas crossed 50% of the primary energy supply in the country in 2004 [2]. The demand for natural gas in Pakistan due to the surge in population has increased, resulting in the serious shortage of gas supply. As a result, the supply has failed to match the demand [3]. Due to shortage and rapidly diminishing indigenous gas supply in Pakistan, liquified natural gas (LNG) is being imported to cover this supply–demand mismatch. As per latest report of Pakistan Economic Survey in 2019, the share of LNG in Pakistan has now reached 20% in the gas-supply mix. Pakistan is now the 13th largest LNG importing county of the world [4].

Pakistan’s socio-economic development is heavily dependent on the gas supply. Therefore, the allocation of gas among the provinces must be fair and efficient. The shortage of gas and its allocation among different stakeholders can be addressed from different perspectives. One such technique is the Game-theoretic approach which can be helpful for equitable resource allocation among the stakeholders. This approach can help the decision-makers, stakeholders, and planners make the right decision by analyzing and addressing the supply–demand gap for different sectors. Natural resources shared among the countries, states, or provinces can be a major source of conflict and cooperation [5,6,7], and natural gas is one of these resources.

Nash bargaining solution and bankruptcy concept are the two approaches that allocate the scarce resource among the various stakeholders. Bankruptcy is described in economics as a scenario in which all resources are insufficient to fulfill the stakeholders’ demands. The condition during the gas shortage is similar to the concept of bankruptcy. Therefore, the bankruptcy problem concept can be utilized for gas distribution among various Pakistan provinces. In a typical bankruptcy problem, the scarce resources are distributed among the various gas demanding clients/stakeholders using the classical bankruptcy rules and Nash bargaining solution. The constrained equal award, proportional, and constrained equal loss are the most utilized bankruptcy rules [8,9,10]. These rules satisfy several desirable properties, and each rule can be used in different bankruptcy scenarios. Bankruptcy theory and Nash bargaining has been applied to various resource allocation scenarios by various researchers [5,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26].

The available approaches of resources optimal allocation mainly focus on mathematical programming based on group rationality, nucleolus solution concept, and noncooperative game based on individual rationality and of cooperative game based on both individual rationality and group rationality [27]. Cooperative game is a suitable method for the natural resource allocation which achieves both individual profit optimization and group profit optimization [28]. Cooperative game is a transferable utility game which requires strict conditions and requires all the agents to cooperate. The Nash bargaining approach is able to achieve the cooperation among the agents through a noncooperative way. In Nash bargaining solution, two or more players or agents infinitely offer an allocation scheme of payoff or cost from a joint collaboration until both parties accept or disagree with the scheme. If an agreement-of-allocation scheme is achieved, it means that this game has a cooperative deal, and the problem has a cooperative solution, i.e., a Nash bargaining solution (NBS); however, if it comes out with a breakdown, it won’t lead to cooperation [29]. The bargaining problems can be solved by various methods, but much desired properties, such as flexibility, invariance under change of scale, unanimity, and pareto optimality, can be satisfied by the Nash bargaining solution [30,31]. Various other researchers [28,32,33,34,35] used the Nash bargaining solution for the management and allocation of scarce resources. Also, game theory, in recent years, has been used in increasingly more diverse scenarios and applications. For example, it has been applied in the implementation and construction of biogas plants [36]. Apart from this, game theory has also been applied to solve matrix games with hesitant fuzzy payoffs and to determine the market share of competitors in the telecommunication industry [37,38].

The motivation behind this study is to address the potential shortage of indigenous natural-gas supply in Pakistan. As per reports, natural-gas demand will rise about three times during 2012–2030 [39]. The urgent need of gas policy revision has also been asserted by various authors [2,40]. In the backdrop of depletion of natural-gas resources, its sensible allocation becomes very crucial. Historical variations in the gas-allocation policy within the country need to be understood in a statistical way to for the proper gas allocation. In this paper, the bankruptcy concept and Nash bargaining solution have been applied to address the supply–demand gap in the gas sector.

The rest of the article is organized as follows. Section 2 discusses the current conditions of gas supply in Pakistan. Section 3 discusses the current gas supply–demand gap in Pakistan. Section 4 discusses the Nash bargaining theory. Finally, Section 5 and Section 6 discuss the result and conclude the paper.

2. The Natural-Gas Scenario in Pakistan

Pakistan has witnessed a significant rise in energy demand in the last two decades. The supply–production of energy has failed to cope with the demands, resulting in an increase in the supply–demand gap. As per the latest population census of 2020, Pakistan’s population has reached 200 million [41]. This energy crisis has also affected the country’s economic growth [42,43]. Pakistan largely depends on gas and oil resources to meet its escalating energy demands. The natural-gas industry of Pakistan, with an offshore area of 827,268 square kilometers, is equipped with numerous exploration and production companies. Since its operations in 1952, the gas industry of Pakistan has discovered a significant gas field at Sui in the province of Baluchistan with a reserve of 0.31 trillion cubic meters (Tcm) [44]. In those days, a 16-inch transmission line used to bring gas from the province of Baluchistan to Karachi, Sindh, over a total length of 559 km. By 1960, the gas industry in Pakistan matured very quickly, and various other gas transmission and distribution lines were designed and installed by indigenous sources. The gas industry of Pakistan is the most developed industry in Pakistan. It has the earliest transmission system in the country, with more than 50 years of operational experience [44].

The natural-gas reserves of Pakistan are less than those of Russia and Iran. Pakistan, having a gas potential of 0.5 Tcm, is currently ranked at 29th position for its gas reserves in the world. The gas reserves in Pakistan account for almost 0.2% of the world’s gas reserves [45]. The GoP is further enhancing the gas reserves by investments in the development and drilling of existing gas–supply systems and the discovery of new gas reservoirs. In the coming years, the demand for natural gas will surge. Therefore, the supply–demand gap will also increase. The main gas fields of the country include Sui, Uch, Zamzama, Badin, Manzalai, Mari, Kandhkot, Bhit, Sawan, and Qadirpur. Out of the four provinces of Pakistan, Sindh is the highest gas supplier with 64% of the total gas, followed by Baluchistan with 17%. In contrast, Khyber Pakhtunkhwa (KPK) and Punjab supply 9 and 3%, respectively [46].

3. Pakistan’s Natural-Gas Production, Consumption, and Supply–Demand Gap

According to the BP Statistical Review of World Energy Report published in June 2017, the total natural-gas consumption in the world reached 3.551 Tcm in 2016 with an annual growth of 0.3%. According to a report [4], the share of natural gas in energy resources is going to increase globally. Some of the world’s largest gas producers are Iran, the United States, and Russia. Pakistan’s gas production in 2016, in comparison with some of the major gas-producing nations, is shown in Table 1. In 2016, Pakistan shared 1.2% of the global gas production. In 2016, 1464 BCF natural gas was produced by Pakistan, which increased to 1658 in 2020. Currently, Pakistan faces a huge shortfall of gas due to non-efficient distribution. The country’s economy has also been affected due to this inefficient allocation of gas. Figure 1 shows that gas reserves in Pakistan have been in sharp decline since 2005. It is evident from the figure that gas production and consumption were almost equal in 2015. The consumption and production trends, however, started to separate in 2016. To cope with this supply–demand gap, the government and policymakers must ensure that the gas allocation is done among the provinces of Pakistan in the most efficient way.

Table 1. Natural-gas production of the major gas producing countries compared with Pakistan [4].

Country	Total Gas Production (Bcm)	Per cent of World Production
Pakistan	41.5	1.2
China	138.4	3.9
Russia	579.4	16.3
Canada	152.0	4.3
United States	749.2	21.1
Qatar	181.2	5.1
Iran	202.4	5.7
Norway	116.6	3.3

Table 1. Natural-gas production of the major gas producing countries compared with Pakistan [4].

Country	Total Gas Production (Bcm)	Per cent of World Production
Pakistan	41.5	1.2
China	138.4	3.9
Russia	579.4	16.3
Canada	152.0	4.3
United States	749.2	21.1
Qatar	181.2	5.1
Iran	202.4	5.7
Norway	116.6	3.3

Figure 1. Pakistan’s natural-gas production, reserves, and consumption trend [45].

Figure 1. Pakistan’s natural-gas production, reserves, and consumption trend [45].

As per the report of the “Development Plan of Pakistan Oil and Gas Industry” in 2020, the supply–demand balance for years 2020–2028 [46], are shown in

Table 2. Gas demand and supply scenario from 2020–2028 [46].

	2020	2021	2022	2023	2024	2025	2026	2027	2028
Committed and anticipated Supply/domestic supply (BCF)	1220	1144	1044	946	859	783	692	649	612
LNG supply (BCF)	438	657	657	657	657	657	657	657	657
Iran–Pakistan (BCF)	0	0	0	0	0	96	274	274	274
TAPI (BCF)	0	0	0	0	354	451	484	484	484
Total supply (BCF)	1658	1801	1701	1957	1967	2026	2113	2069	2033
Punjab’s demand (BCF)	922	968	1013	1066	1119	1176	1236	1299	1363
Sindh’s demand (BCF)	667	699	734	770	809	851	893	939	985
Baluchistan’s demand (BCF)	184	194	205	215	226	237	247	261	272
KPK’s demand (BCF)	95	99	102	109	117	120	127	134	141
Other’s demand (BCF)	81	88	92	95	99	106	109	117	124
Total demand (BCF)	1949	2048	2146	2255	2370	2490	2612	2750	2885
Gap/shortfall (BCF)	291	247	445	298	403	464	499	681	852

. It can be seen from the table that domestic natural-gas production in Pakistan continues to decrease whereas the overall gas supply is increasing due to LNG supply, Iran–Pakistan gas pipeline, and The Turkmenistan–Afghanistan–Pakistan–India Pipeline (TAPI), also known as the Trans-Afghanistan Pipeline. TAPI is a natural-gas pipeline developed by the Galkynysh—TAPI Pipeline Company Limited, with the participation of the Asian Development Bank. However, despite increasing gas imports, the shortfall increases [46]. Because provincial demands are increasing swiftly, the supply is not coping with the demand.

4. An Asymmetric Hybrid Bankruptcy and Nash Bargaining Model for Gas Distribution

In this work, a hybrid model based upon bankruptcy theory with the Nash bargaining solution is proposed to tackle the gas-allocation problem [28,32,33,34,35].

A bankruptcy situation, or a claims problem, is defined by a tuple (N, E, c), where N = {1, …, n} is the set of stakeholders, E is the total gas available to satisfy the demands of all stakeholders and c = {c₁, …, c_n} are the claims [47,48]. To allocate the estate E among the agents in N, we define an allocations vector x = {x₁, …, x_n} as a real valued vector respecting the following properties: (1) rationality, (2) claim boundedness, and (3) efficiency. The asymmetric Nash bargaining theory is linked with the bankruptcy theory for gas distribution. While applying this methodology, the agent’s bargaining weights $(w_i,i=1,\dots .,n)$ and disagreement allocation points $(m_i,i=1,\dots .,n$) are also considered to ensure self-enforceability and equity in a bounded space. In such optimization problem, apart from satisfying a set of desirable properties, offers a unique solution and maximizes the area between the Pareto-optimal frontier (x) and the disagreement point (m_i). Therefore, this optimization technique helps us to prevent conflicts among the agents.

In this study, x⁻ is the agent’s allocated gas amount. The following three bankruptcy conditions must be satisfied provided in Equations (1)–(3):

$${\displaystyle \sum }_{i=1}^n{x}_i=E$$

(1)

Equation (1) requires that the sum of available resources must be precisely allocated between all the stakeholders/agents and is called the Pareto efficiency.

$$x_i\le c_i$$

(2)

Equation (2) helps to prevent the overuse of resources which might cause the tragedy of the commons and is called the Claim boundedness. Equation (3) ensures the non-negativity of the resources.

$$x_i\ge 0$$

(3)

In this study, disagreement points vector (d₁, d₂, …, di) are defined as the benefits of minimum gas allocation (I₁, I₂, …, I_n) to the riparians. It represents the minimum threshold value of the agent’s benefits. Therefore, the individual rationality requirements must be reflected before the cooperation of the followers so that the maximal and minimal solutions are satisfied.

The bankruptcy theory can solve the situation of minimum gas distribution to every province, which is applied in a gas-shortage period. The formula of minimum gas allocation to each province is given by Equation (4) below.

$$m_i=\max \left(0,E-{\displaystyle \sum }_{k\ne i}\left(c_i\right)\right)$$

(4)

Subject to:

$$E<{\displaystyle \sum }_{i=1}^n{c}_i$$

(5)

While using Equation (4), the minimum amount of gas allocated to any province may become zero, especially for the provinces with more minor claims. However, in reality, each province will have a minimum threshold value of gas λ_i in the process of gas distribution. For example, suppose we use bankruptcy theory. In that case, the minimum province allocated gas may be less than its threshold gas required value. Therefore, to avoid these circumstances, a formula is proposed to consider for the province minimum requirement:

$$m_i=\text{max}({\lambda }_i, E-{\displaystyle \sum }_{k \ne i}{c}_i)$$

(6)

In Equation (6), λ_i is the minimum gas requirement of each province. In this work, it is assumed that the minimum requirement of each province is half of the total gas demand.

In this optimization problem, the upper bound core of the problem is the respective gas claim of each province. According to [49], the gas-distribution optimization problem under the bankruptcy scenario can be defined as follows:

$$\mathit{\text{Maximize }}{N}^{wi}={\left(x_1-\left(E-{\displaystyle \sum }_{i\in N/\left\{1\right\}}{c}_i\right)\right)}^{w_1}{\left(x_2-\left(E-{\displaystyle \sum }_{i\in N/\left\{2\right\}}{c}_i\right)\right)}^{w_2}{\left(x_3-\left(E-{\displaystyle \sum }_{i\in N/\left\{3\right\}}{c}_i\right)\right)}^{w_3}\dots {\left(x_n-\left(E-{\displaystyle \sum }_{i\in N/\left\{n\right\}}{c}_i\right)\right)}^{w_n}$$

(7)

The model presented in Equation (7) is constraint by individual feasibility and rationality. As discussed above, the disagreement points and province claims serve as the lower and upper bounds, respectively. Therefore, Equation (7) can be reformulated as:

$$\mathit{\text{Maximize}}{N}^{wi}={\left(x_p-\left(E-{\displaystyle \sum }_{i\in N/\left\{P\right\}}{c}_i\right)\right)}^{w_p}.{\left(x_s-\left(E-{\displaystyle \sum }_{i\in N/\left\{S\right\}}{c}_i\right)\right)}^{w_S}.{\left(x_B-\left(E-{\displaystyle \sum }_{i\in N/B}{c}_i\right)\right)}^{w_B}.{\left(x_K-\left(E-{\displaystyle \sum }_{i\in N/K}{c}_i\right)\right)}^{w_K}.{\left(x_{ot}-\left(E-{\displaystyle \sum }_{i\in N/\left\{Ot\right\}}{c}_i\right)\right)}^{w_{Ot}}$$

(8)

Here, ${\displaystyle \sum }_{i=1}^n{w}_i=1$

In Equation (8):

$x_p$ = Punjab’s gas optimized distribution value.

I_P = Punjab’s minimum gas requirement.

$x_s$ = Sindh’s gas optimized distribution value.

I_S = Sindh’s minimum gas requirement.

$x_B$ = Baluchistan’s gas optimized distribution value.

I_B = Baluchistan’s minimum gas requirement.

$x_K$ = Khyber Pakhtunkhwa’s (KPK) gas optimized distribution value.

I_K = KPK’s minimum gas requirement.

$x_{ot}$ = Other area’s gas optimized distribution value.

I_ot = Other area’s minimum gas requirement.

For the above gas-allocation model, the following constraints are set:

• The gas allocation of each province must be equal to or more than its lower threshold value.

$$x_i\ge m_{i}i=1,2,\dots \dots ..,n$$

(9)

2.

The gas allocation of each province must be equal to or less than its upper threshold value (claim) but more than or equal to its lower threshold value.

$$m_i\le x_i\le c_i$$

(10)

3.

The total gas allocation must be equal to or less than the total available gas.

$${\displaystyle \sum }_{i=1}^n{x}_i\le E$$

(11)

After defining the objective function (Equation (8)) and setting the constraints (Equations (9)–(11)), the allocation of gas among the agents (provinces and administrative units) is done using the Nash bargaining solution. Figure 2 illustrates the working of the proposed approach under a gas shortage period. When the gas demand and its availability change with time, the disagreement points are defined again, using Nash bargaining theory, and resource allocation is done again.

Nine different scenarios (from 2020 to 2028) are considered for gas allocation, as mentioned in Table 3. First, the gas allocation for each year is computed using the Nash bargaining solution with equal weights. After that, the reallocation of gas is done using higher ‘gas production’. The province which generates/produces more gas would be given higher priority in allocation.

The core solutions are defined by rationality, efficiency, and marginality in a cooperative game, which are already defined in Equations (1)–(3). The rationality principle defines the lower bound of the user, whereas the marginality principle defines the upper bound. The lower core values are defined using Equation (6). Results are summarized in Table 3, in which x(i) represents the gas allocation for the ith user expressed in Mm³/y. The upper and lower core bounds, as stated in Table 3, can be considered the limits of “feasible values” that each stakeholder could accept.

Table 3. Gas-Allocation Core Solutions Scenarios (BCF/y with respect to available natural-gas resources).

Year 2020	Year 2021	Year 2022
x(Pun) + x (Sin) + x(Bal) + x(Kpk) + x(Ot) = 1658 631 ≤ x(P) ≤ 922 376 ≤ x(S) ≤ 667 92≤ x(B) ≤ 184 48 ≤ x(K) ≤ 95 41 ≤ x(Ot) ≤ 81	x(Pun) + x (Sin) + x(Bal) + x(Kpk) + x(Ot) = 1801 721 ≤ x(P) ≤ 968 452 ≤ x(S) ≤ 699 97 ≤ x(B) ≤ 194 50 ≤ x(K) ≤ 99 44 ≤ x(Ot) ≤ 88	x(Pun) + x (Sin) + x(Bal) + x(Kpk) + x(Ot) = 1701 568 ≤ x(P) ≤ 1013 367 ≤ x(S) ≤ 734 103 ≤ x(B) ≤ 205 51 ≤ x(K) ≤ 102 46 ≤ x(Ot) ≤ 92
Year 2023	Year 2024	Year 2025
x(Pun) + x (Sin)+ x(Bal) + x(Kpk) + x(Ot) = 1603 533 ≤ x(P) ≤ 1066 385 ≤ x(S) ≤ 770 108 ≤ x(B) ≤ 215 55 ≤ x(K) ≤ 109 48 ≤ x(Ot) ≤ 95	x(Pun) + x (Sin) + x(Bal)+ x(Kpk) + x(Ot)= 1870 619 ≤ x(P) ≤ 1119 405 ≤ x(S) ≤ 809 113 ≤ x(B) ≤ 226 59 ≤ x(K) ≤ 117 50 ≤ x(Ot) ≤ 99	x(Pun) + x (Sin) + x(Bal) + x(Kpk) + x(Ot) = 1987 673 ≤ x(P) ≤ 1176 426 ≤ x(S) ≤ 851 119 ≤ x(B) ≤ 237 60 ≤ x(K) ≤ 120 53 ≤ x(Ot) ≤ 106
Year 2026	Year 2027	Year 2028
x(Pun) + x (Sin) + x(Bal) + x(Kpk) + x(Ot) = 2107 731 ≤ x(P) ≤ 1236 447 ≤ x(S) ≤ 893 124 ≤ x(B) ≤ 247 64 ≤ x(K) ≤ 127 55 ≤ x(Ot) ≤ 109	x(Pun) + x (Sin) + x(Bal) + x(Kpk) + x(Ot) = 2063 650 ≤ x(P) ≤ 1299 470 ≤ x(S) ≤ 939 131 ≤ x(B) ≤ 261 67 ≤ x(K) ≤ 134 59 ≤ x(Ot) ≤ 117	x(Pun) + x (Sin) + x(Bal) + x(Kpk) + x(Ot) = 2026 682 ≤ x(P) ≤ 1363 492 ≤ x(S) ≤ 985 136 ≤ x(B) ≤ 272 71 ≤ x(K) ≤ 141 62 ≤ x(Ot) ≤ 124

Table 3. Gas-Allocation Core Solutions Scenarios (BCF/y with respect to available natural-gas resources).

Year 2020	Year 2021	Year 2022
x(Pun) + x (Sin) + x(Bal) + x(Kpk) + x(Ot) = 1658 631 ≤ x(P) ≤ 922 376 ≤ x(S) ≤ 667 92≤ x(B) ≤ 184 48 ≤ x(K) ≤ 95 41 ≤ x(Ot) ≤ 81	x(Pun) + x (Sin) + x(Bal) + x(Kpk) + x(Ot) = 1801 721 ≤ x(P) ≤ 968 452 ≤ x(S) ≤ 699 97 ≤ x(B) ≤ 194 50 ≤ x(K) ≤ 99 44 ≤ x(Ot) ≤ 88	x(Pun) + x (Sin) + x(Bal) + x(Kpk) + x(Ot) = 1701 568 ≤ x(P) ≤ 1013 367 ≤ x(S) ≤ 734 103 ≤ x(B) ≤ 205 51 ≤ x(K) ≤ 102 46 ≤ x(Ot) ≤ 92
Year 2023	Year 2024	Year 2025
x(Pun) + x (Sin)+ x(Bal) + x(Kpk) + x(Ot) = 1603 533 ≤ x(P) ≤ 1066 385 ≤ x(S) ≤ 770 108 ≤ x(B) ≤ 215 55 ≤ x(K) ≤ 109 48 ≤ x(Ot) ≤ 95	x(Pun) + x (Sin) + x(Bal)+ x(Kpk) + x(Ot)= 1870 619 ≤ x(P) ≤ 1119 405 ≤ x(S) ≤ 809 113 ≤ x(B) ≤ 226 59 ≤ x(K) ≤ 117 50 ≤ x(Ot) ≤ 99	x(Pun) + x (Sin) + x(Bal) + x(Kpk) + x(Ot) = 1987 673 ≤ x(P) ≤ 1176 426 ≤ x(S) ≤ 851 119 ≤ x(B) ≤ 237 60 ≤ x(K) ≤ 120 53 ≤ x(Ot) ≤ 106
Year 2026	Year 2027	Year 2028
x(Pun) + x (Sin) + x(Bal) + x(Kpk) + x(Ot) = 2107 731 ≤ x(P) ≤ 1236 447 ≤ x(S) ≤ 893 124 ≤ x(B) ≤ 247 64 ≤ x(K) ≤ 127 55 ≤ x(Ot) ≤ 109	x(Pun) + x (Sin) + x(Bal) + x(Kpk) + x(Ot) = 2063 650 ≤ x(P) ≤ 1299 470 ≤ x(S) ≤ 939 131 ≤ x(B) ≤ 261 67 ≤ x(K) ≤ 134 59 ≤ x(Ot) ≤ 117	x(Pun) + x (Sin) + x(Bal) + x(Kpk) + x(Ot) = 2026 682 ≤ x(P) ≤ 1363 492 ≤ x(S) ≤ 985 136 ≤ x(B) ≤ 272 71 ≤ x(K) ≤ 141 62 ≤ x(Ot) ≤ 124

Bargaining Weights Determination

The optimization problem presented in Equation (8) is utilized for the gas-distribution problem in Pakistan, and three cases are analyzed for each year. First of all (first case), equal bargaining weights are considered to optimize the function. However, each province is different regarding its socio-economic and gas production status. Therefore, various bargaining weights are considered depending upon each province’s gas production to emphasize the significance of various bargaining weights in the second case. As per the latest available data, the gas production, along with the bargaining weights of Pakistan’s every province, is shown in

Table 4. Production of gas in provinces of Pakistan, number of consumers, and their bargaining weights.

	Provinces
	Punjab	Sindh	Baluchistan	KPK	Other Areas	Total
Percentage share in total production	4	62.6	21.4	11	1	100
Bargaining weight	0.04	0.626	0.214	0.11	0.01	1.00
Number of consumers (domestic, commercial and industrial), in millions	5.80	2.70	0.28	0.87	0.2	9.85
Bargaining weight	0.588	0.274	0.028	0.088	0.022	1.00

[2,3,44,46]. The bargaining weights for Punjab, Sindh, Baluchistan, KPK, and other areas come out to be 0.04, 0.626, 0.214, 0.11, and 0.01, respectively. All the mentioned bargaining weights are directly proportional to their gas production. Therefore, the greater the province’s gas production, the high bargaining weight. In the last case, the number of consumers (domestic, commercial, and industrial) are considered for computing the bargaining weights of the provinces, as shown in Table 4.

5. Results and Discussion

Figure 3 shows the results of the Nash bargaining solution. The Nash bargaining is applied for all three cases from 2020–2028. In the first case, equal bargaining weights are assigned to each province. Whereas in the second case, the parameter of gas production is used to assign the bargaining weights. In the last case, the number of consumers (domestic, commercial, and industrial) are used to assign the bargaining weights for each province. Figure 3 shows the gas distribution among each province as a percentage of gas demand under all three cases for the years 2020–2028 using the Nash bargaining solution.

Table 5. (a) Gas allocation in BCM using equal weights for years 2020–2028. (b) Gas allocation in BCM using bargaining weights (Gas Production) for years 2020–2028. (c) Gas allocation in BCM using bargaining weights (Number of gas consumers) for years 2020–2028.

(a)
Punjab	Sindh	Baluchistan	KPK	Others
Year 2020
777	522	184	95	81
Year 2021
845	576	194	99	88
Year 2022
751	551	205	102	92
Year 2023
702	502	205	102	92
Year 2024
821	607	226	117	99
Year 2025
886	639	237	120	106
Year 2026
954	670	247	127	109
Year 2027
865	686	261	134	117
Year 2028
839	650	272	141	124
(b)
Punjab	Sindh	Baluchistan	KPK	Others
Year 2020
663	667	184	95	47
Year 2021
756	699	194	99	88
Year 2022
605	734	205	102	55
Year 2023
552	682	209	107	53
Year 2024
658	809	226	117	60
Year 2025
715	851	237	120	64
Year 2026
774	893	247	127	66
Year 2027
679	923	261	134	66
Year 2028
705	857	261	135	68
(c)
Punjab	Sindh	Baluchistan	KPK	Others
Year 2020
854	478	184	95	47
Year 2021
986	578	110	90	54
Year 2022
901	522	119	101	59
Year 2023
812	515	121	97	59
Year 2024
986	576	130	114	64
Year 2025
1059	606	137	118	67
Year 2026
1134	635	143	124	70
Year 2027
1053	658	150	127	74
Year 2028
1025	652	152	122	75

a–c show the results in tabular form.

It can be seen from Figure 3 and Table 5a–c that when the gas allocation is done using the homogenous weights, the allocation of Punjab and Sindh is reduced, whereas the smaller provinces/administrative units get 100% of their claims. It is due to the reason that the claims of these provinces are already less. In the second case, when the allocation of gas is done using bargaining weights 2 (based on the gas production), the allocation of Punjab is considerably reduced, whereas the allocations for Sindh and Baluchistan are considerably increased for all the years. It is because these two provinces have the country’s largest gas reserves, which is also evident in Table 4. Finally, in the third case, when the gas allocation is done using the heterogeneous weights 1 (based on the number of gas consumers), the allocations of Punjab and Sindh are increased. It is due to the reason that both these two provinces have the largest number of consumers. On the other hand, other provinces have considerably fewer consumers than Punjab and Sindh; hence, their allocation is reduced.

6. Conclusions and Recommendations

In this work, the hybrid Nash bargaining theory and the bankruptcy concept are applied for the efficient reallocation of gas among Pakistan’s provinces/administrative units. This approach considers the principles of sustainability, efficiency, and equity. These three critical properties are incorporated to ensure that resource allocation is reasonable and equitable. While using equal weights, factors such as the population of each province, its socio-economic condition, and gas production are not taken into consideration. Therefore, some agents (provinces) may not accept this allocation. It is therefore recommended that the resource allocation rules must incorporate these factors. Therefore, in this study, the allocation of gas among the provinces/administrative units is also done using heterogeneous weights that take into account the ‘number of gas consumers’ and the ‘gas production’. However, the gas allocation among the agents is a complex task that cannot be solely solved by applying mathematical methods. Therefore, negotiations between the agents are recommended, which would help them to reach an agreement.

The limitations of this study may be addressed in in future studies and some additional influential factors should be considered, such as the socio-political aspects of the provinces, the reliable relative weights of the provinces based on their political influence. It is also recommended to apply multi-criteria decision analysis (MCDA) to combine all the relative weights to be considered. Historical gas consumption structure in Pakistan consists of seven main consumer groups, that is, residential, commercial, cement industry, fertilizer, power, general industry, and transport. Industrial sector includes general industry and cement industry. Miscellaneous sector includes commercial consumers and transport consumers (compressed natural gas, CNG). Therefore, sectoral based gas allocations should also be considered for future studies.

The study also has some immediate policy implications. It can be seen that despite the increase in gas production and gas import, the supply–demand gap will continue to increase and, hence, the disputes between the provinces will increase. It is therefore, expected that the research findings will help a great deal in resolving the gas disputes between the administrative units/provinces of Pakistan and other countries having similar problem of gas shortage. Apart from this, in the negotiation process, the bargaining power of the agents (provinces), will play a role to ensure fair and equitable allocation of gas.