## 4 Spatial Cooperative Game Models and Their Applications in Terms of Locales
### Case I: The Three Player (\(N=3\)) Hypothetical Spatial Cooperative Game
Core Existence Test and Core Size  
Figure 3 provides the triangular graph solution prior to the solver-based optimization model applications for the Model IV application maximizing player _PA_'s return (\(X_{A}\)) from the grand coalition without violating individual (red lines), sub-group, or grand coalition rationalities. Figure 3a shows the maximized \(X_{A}\) to be 3000 m\({}^{2}\) and the remaining allocations \(X_{B}\) and \(X_{C}\) to be 2000 and 1000 m\({}^{2}\), respectively. This solution is presented by a dot at the right corner of the highlighted core area. For Figure 3b, when a 2000 m\({}^{2}\) benefit is retained as tax in the model without violating the base core constraints from Table 5, one single core solution occurs. Next, the allocation values for \(X_{A}\), \(X_{B}\), and \(X_{C}\) are found to be 1000, 2000, 1000 m\({}^{2}\), respectively. When the number of players is greater than three, graphical solutions and presentations become technically impossible. Thus, the solver-based model applications become a technical necessity. \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline
**Strategies** & **Allocation Variables (m\({}^{2}\))** & \multicolumn{3}{c}{**Returns from Strategies (m\({}^{2}\)) \({}^{**}\)**} & **Equations** & **Strategies in Graphs** \\ \hline Individual (unilateral) & \(X_{A}\) & & \(\geq\) & \(b_{A}=1000\) & Equation (2.1) & \(\blacksquare\) \\  & \(X_{B}\) & & \(\geq\) & \(b_{B}=1000\) & Equation (2.2) & \(\blacksquare\) \\  & & \(X_{C}\) & \(\geq\) & \(b_{C}=1000\) & Equation (2.3) & \(\blacksquare\) \\ \hline Sub-Group Coalition (Bilateral) & \(X_{A}\) & + \(X_{B}\) & & \(\geq\) & \(b_{AB}=3000\) & Equation (3.1) & \(\blacksquare\) \\  & \(X_{A}\) & & + \(X_{C}\) & \(\geq\) & \(b_{AC}=2000\) & Equation (3.2) \({}^{**}\) & \(\blacksquare\) \\  & & \(X_{B}\) & + \(X_{C}\) & \(\geq\) & \(b_{BC}=3000\) & Equation (3.3) & \(\blacksquare\) \\ \hline Grand coalition & \(X_{A}\) & + \(X_{B}\) & + \(X_{C}\) & & = & \(b_{ABC}=6000\) & Equation (4) & \(\blacksquare\) \\ \hline Grand coalition with (\(Z\)) & \(X_{A}\) & + \(X_{B}\) & + \(X_{C}\) & + \(Z\) & = & \(b_{ABC}=6000\) & Equation (5) & \(\blacksquare\) \\ \hline \hline \end{tabular}  
* Values are obtained from Table 3. ** Equation (3.2) is a redundant strategy as its role in the equation system is strictly played by Equations (2.1) and (2.3); therefore, its removal does not change the optimal value of any designed objective function in the optimization problem.  
\end{table} 
Table 5: Core constraints in explicit equation forms and the returns from all strategies * ([PERSON] et al., 2018). Figure 3: Model III application with the core and maximized \(X_{A}\) (10\({}^{3}\)): (**a**) Tax is zero (\(Z=0\)): (**b**) Tax is two thousand (\(Z=2000\)) ([PERSON] et al., 2018). Model II is the first one used in the computer-based model applications and has been designed to test the availability of a core solution. The objective function maximizes \(Z\), given the associated constraints in Model II. The maximized objective value \(Z_{Max}\) is found to be 2000 in Table 6, which is consistent with the graphed solution presented in Figure 3. This value of \(Z_{Max}\) = 2000 shows that a core solution exists, with the grand collation providing a 2000 m\({}^{2}\) taxable return that can be retained or later allocated to the players. Later on, the \(Z_{Max}\) (2000) is used as a constant in Equation (5), with Models III and IV then being operated separately for each player's return maximization and minimization given the same constraints. Table 6 presents the obtained maximum and minimum values (\(P\_Max}\) and \(P\_Min\)) for each player. Because this is a problem limited to three players, both the maximum and minimum values are found be equal for each player (\(PA_{Max}=PA_{Min}=1000\); \(PB_{Max}=PB_{Min}=2000\); \(PC_{Max}=PC_{Min}=1000\)) after deducting \(Z_{Max}\) = 2000 from the grand coalition.