# An Asymmetric Bargaining Model for Natural-Gas Distribution
## 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 \(N=\{1,\ldots,n\}\) is the set of stakeholders, \(E\) is the total gas available to satisfy the demands of all stakeholders and \(c=\{c_{1},\ldots,c_{n}\}\) are the claims [47; 48]. To allocate the estate E among the agents in \(N\), we define an allocations vector \(x=\{x_{1},\ldots,x_{n}\}\) 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 \((w_{i},i=1,\ldots,n)\) and disagreement allocation points \((m_{i},i=1,\ldots,n)\) 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):  
\[\sum_{i=1}^{n}x_{i}=E \tag{1}\]  
Equation (1) requires that the sum of available resources must be precisely allocated between all the stakeholders/agents and is called the Pareto efficiency. \[x_{i}\leq c_{i} \tag{2}\]  
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. \[x_{i}\geq 0 \tag{3}\]  
In this study, disagreement points vector \((d_{1},d_{2},\ldots,di)\) are defined as the benefits of minimum gas allocation \((I_{1},\)\(I_{2},\)\(\ldots,\)\(I_{n})\) 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. \[m_{i}=\max\Biggl{(}0,\ E-\sum_{k\
eq i}(c_{i})\Biggr{)} \tag{4}\]  
Subject to:  
\[E<\sum_{i=1}^{n}c_{i} \tag{5}\]  
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 \(\lambda_{i}\) in the process of gas distribution. For example, suppose we use bankruptcy theory. In that case, the minimum provinceallocated 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:  
\[m_{i}=\max(\lambda_{i},\ E-\sum\limits_{k\
eq i}c_{i}) \tag{6}\]  
In Equation (6), \(\lambda_{i}\) 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:  
\[