## 3 Methodology and Cooperative Game Theory Models
### Mathematical Forms of the Cooperative Game Models
#### 3.1.1 Model I  
Model I is a linear programming optimization model that has been designed to check whether a core solution exists in a cooperative game setting and contains all the constraints, some of which are selectively used in the subsequent models (Models II-IV). Let \(i\) be the indices for involved parties/players (I\(\rightarrow\)\(N\)), \(S\) be all possible sub-coalition formations (\(\forall\ S\in N\)). Next, \(b_{i}\) is the individual benefit derived from \(i\)'s unilateral action; \(b_{S}\) is the sub-coalition benefit derived from the cooperative action of \(S\) subset of \(N\) players; and \(b_{N}\) is the grand coalition benefit derived from the cooperation of all players. All \(b_{i}\), \(b_{S}\), and \(b_{N}\) are the parameters externally computed and loaded into the models. Next, \(x_{i}\) is the benefit apportioned to the \(i\)th member of the grand coalition from the grand coalition benefit \(b_{N}\). Moreover, \(Z\) is an additional party benefit or cost from the formation of the coalition in the form of a tax or subsidy. In addition to \(N\) players, this party \(Z\) can be a moderating and/or facilitating supreme authority (e.g., government).  
Given these sets, parameters, and variables, \(F\) is the objective function and is a function of either \(x_{i}\) or \(Z\) (see Equation (1)). Equations (2)-(5) are the constraints defining the individual (Equation (2)), subgroup (Equation (3)), and grand coalition (Equations (4) and (5)) rationalities. Individual rationalities are the constraints for each player \(i\) expecting \(x_{i}\) amount of return from the grand coalition benefit \(b_{N}\) whose least expected benefit is \(b_{i}\) (Equation (2)). Next, the sub-coalition rationalities are the constraints for multilaterality acting subgroup \(S\) expecting an \(\sum_{i\in S}x_{i}\) amount of return from the grand coalition benefit \(b_{N}\) whose least expected benefit is \(b_{S}\) (Equation (3)). These are the necessary conditionsfor being a part of a cooperative game, but these are not sufficient conditions. Finally, the total of allocations, \(\sum_{i\in N}x_{i}\), are equal to the grand coalition benefit, \(b_{N}\) (Equation (4)), with Equations (2)-(4) being used to calculate the core constraints. When all these constraints are satisfied in an optimization model, one or multiple core solutions are present. When adding the supreme authority to the model through the variable \(Z\), the total of allocations, \(\sum_{i\in N}x_{i}+Z\), equals the grand coalition benefit, \(b_{N}\) (Equation (5)). Using \(Z\) as a maximized objective function (i.e., using Equation (5) instead of Equation (4) as the constraint in the model) relaxes (or contracts) the core constraints minimally by guaranteeing the existence of a core solution [42]. The succeeding Models II-IV are forms of game theoretical linear programming (LP) models that differ from Model I in terms of their objective functions.  
Model I Maximize \(F(x_{i},Z)\) (1) Subject to  
\[x_{i}\geq b_{i}\ \forall i\in N\ (\text{Individual Rationality}) \tag{2}\]  
\[\sum_{i\in S}x_{i}\geq b_{S}\ \forall S\subset N\ (\text{Sub-Group Rationality}) \tag{3}\]  
\[\sum_{i\in N}x_{i}\ =\ b_{N}\ (\text{Grand Coalition}) \tag{4}\]  
\[\sum_{i\in N}x_{i}+Z\ =\ b_{N}\ (\text{Grand Coalition with tax/subsidy}) \tag{5}\]  
Model II Maximize \(F=Z\) (6)  
Subject to Equations (2), (3) and (5)  
Model III (repeats for every \(i\))  
\[\text{Maximize }F=x_{i} \tag{8}\]  
Subject to Equation (7)  
Model IV (repeats for every \(i\))  
\[\text{Minimize }F=x_{i} \tag{9}\]  
Subject to Equation (7)