## 4 Results and Discussion
### Selection of the Most Appropriate Rule  
The selection of the most appropriate rule is essential, as it helps agents reach an agreement. The definition of equity in the distribution of resources is not clear. To select the most appropriate allocation rule, we apply the method proposed by [19]. This method  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Province**} & **Power Allocation Using Homogeneous Weights** & **Power Allocation Using Heterogeneous Weights** & **Power Allocation Using Heterogeneous Weights** \\  & **(MW)** & **(MW)** & **(Based on the Population of the Province)** \\ \hline Punjab & 8771 & 9913 & 9621 \\ Sindh & 3438 & 2717 & 2983 \\ KPK & 2054 & 1633 & 1658 \\ Baluchistan & 962 & 962 & 962 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Power allocation using Nash bargaining theory.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Riparian**} & **PRO** & **CEA** & **CEL** & **Talmud** & **Piniles** \\  & **(\%)** & **(\%)** & **(\%)** & **(\%)** & **(\%)** & **Bargaining** \\  & & & & **(Homogenous** \\  & & & & **(Homogenous** \\  & & & & **Weights)** \\  & & & & **(\%)** & **(\%)** \\ \hline Punjab & 77 & 64 & 74 & 89 & 64 & 77 & 87 & 85 \\ Sindh & 77 & 100 & 50 & 68 & 74 & 87 & 69 & 75 \\ KPK & 77 & 100 & 50 & 50 & 100 & 100 & 79 & 80 \\ Baluchistan & 77 & 100 & 0 & 50 & 100 & 100 & 100 & 100 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Power allocation under five “bankruptcy rules”.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Province**} & **PRO (MW)** & **CEA (MW)** & **CEL (MW)** & **Talmud (MW)** & **Piniles (MW)** \\ \hline Punjab & 8840 & 7304 & 10,321 & 10,079 & 7304 \\ Sindh & 3027 & 3943 & 2917 & 2675.5 & 3943 \\ KPK & 1600 & 2054 & 1025 & 1027 & 2054 \\ Baluchistan & 751 & 962 & 0 & 481 & 962 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Power allocation under five “bankruptcy rules”.  
chooses a rule in which all stakeholders have the lowest dispersion of their total preferences on that rule. Therefore, the allocations are ranked in ascending order for each stakeholder separately. The priority vectors \(\Omega\) are set for this reason, with elements \(w_{i}\). Here, \(w_{i}\) is a vector with elements of \(\vartheta_{ji}\), where \(w_{i}\) is the preference vector, \(i\) is the number of rules, and \(j\) is the number of stakeholders. In the current study, \(1\leq i\leq 8\) and \(1\leq j\leq 4\).  
The priority vector set for our study was \(\Omega=\{w_{1}\), \(w_{2}\), \(\dots\) \(\dots\) \(w_{n}\}\) in which \(w_{1}\) = (4, 3, 4, 2), \(w_{2}\) = (7, 1, 1, 1), \(w_{3}\) = (6, 8, 6, 4), \(w_{4}\) = (1, 7, 5, 3), \(w_{5}\) = (7, 5, 1, 1), \(w_{6}\) = (5, 2, 1, 1), \(w_{7}\) = (2, 6, 3, 1) and \(w_{8}\) = (3, 4, 2, 1). The priority vector \(w_{i}\) was used for each bankruptcy rule. The priority vector with the lowest distance from the intermediate value is considered the best, denoted by \(\overline{w}\). The dispersion around the mean of vector \(i\), \(\delta_{i}\), is calculated using Equation (15) as follows:  
\[\delta_{i}=\frac{\sum_{j=1}^{n}(\vartheta_{ji}-\overline{w})^{2}}{n}=\frac{ \sum_{j=1}^{n}\left(\vartheta_{ji}-\frac{\sum_{j=1}^{n}\vartheta_{ji}}{n} \right)^{2}}{n} \tag{15}\]  
As an example, for the CEA rule, we have,  
\[\overline{w}_{2}=\frac{7+1+1+1}{4}=2.5\]  
\[\delta_{2}=\frac{(2.5-7)^{2}+(2.5-1)^{2}+(2.5-1)^{2}+(2.5-1)^{2}}{4}\]  
\[\delta_{4}=6.75\]  
Values of \(\delta_{i}\) for all rules is presented in Table 6. The rule with the lowest \(\delta_{i}\) is considered the best allocation rule. The allocated power among the provinces as a percentage of their power demand is presented in Table 5. Table 6 shows that CEA and Piniles rules are ranked last, whereas the Proportionate rule is ranked first. It should be noted that the Proportionate rule is considered best here only according to the current scenario. Thus, with a change in the supply-demand gap, the best rule will also change.