## 4 Spatial Cooperative Game Models and Their Applications in Terms of Locales
### Case I: The Three Player (\(N=3\)) Hypothetical Spatial Cooperative Game
Figure 2: Hypothetical three-player neighborhood design and the associated matrix [41]. Next, Table 4 has been prepared in order to list the possible block-based regeneration coalition scenarios in columns containing all the possible coalition strategies listed in the rows in Table 3. To illustrate, Scenario I is a three-player unilateral scenario, Scenario IIa involves one player unilateral (PC) and one bilateral (2-player; PA and PB) coalition. The row totals show the maximum attainable FAR benefits for each scenario (Scenarios I-III) in this development block. These totals also imply the critical party's involvements in coalition strategies. As can be seen from Table 4, Scenarios I and IIb have the lowest block return totals, while Scenario III has the highest block return total. The difference between Scenarios I and III can be attributed to the involvement of player PB because player PB has a strategic location that enables the coalition of players PA and PC. The same impact can also be seen in Scenarios IIa, IIb, and IIc, where the coalition with player PB provides higher FAR values. In summary, player PB's critical geographic role in such a minimal development block is clearly identified in Table 4, and this provides valuable clues about the extensive use of the model in larger development blocks, namely in multinational geographical settings with large numbers of countries/players. The same results are later observed in the cooperative game optimization model applications. Before performing the optimization operations, Table 5 was designed to present the simplest form of cooperative game linear programming model constraints in equation form. Table 5 provides graphic representations of all possible coalition strategies and their associated model constraints alongside the equation numbers consistent with the optimization equations for Models I-IV. In Table 5, due to Equations (2.1) and (2.3) having the same role as Equation (3.2) in the equation system, Equation (3.2) has been specifically highlighted in gray. Because of its neutral effect in determining objective value, this equation can be considered unnecessary and has thus been removed from the constraint list. Though seeming to be a minor intervention, this removal implies a mathematical solution for technical limitations when the number of players reaches the twenties, and software programs and computers fall short of being able to handle the strategy listing and associated benefit calculation assignments. The number of redundant equations also exponentially increases alongside increases in the number of players. Removing these redundant equations allows for larger numbers of players to be included in the models and better spatial analyses to be made, especially in multiparty/multinational settings. This procedure has been algorithmically presented in Figure 1. Finally, because this representative case only has three players, all constraints and possible allocation solutions for any maximization problems can be graphically presented in a triangular graph (Figure 3). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Strategies**} & \multirow{2}{*}{**Parcels**} & **Return** & **Individual** & **Individual and Bilateral Coalition** & **Grand Coalition** \\  & & **Strategies** & & **Strategies** & & **Strategy** \\ \hline \multicolumn{2}{c}{Scenarios:} & \multicolumn{2}{c}{I} & IIa & IIb & IIc & III \\ \hline \multirow{3}{*}{Individual} & _PA_ & \(b_{A}\) & 1000 & & 1000 & \\  & _PB_ & \(b_{B}\) & **1000** & & **1000** & \\  & _PC_ & \(b_{C}\) & 1000 & 1000 & & \\ \hline \multirow{3}{*}{Sub-Group} & _PA and PB_ & \(b_{AB}\) & & 3000 & & \\  & _PA and PC_ & \(b_{AC}\) & & & 2000 & \\  & _PB and PC_ & \(b_{BC}\) & & & 3000 & \\ \hline Grand & _PA, PB, and PC_ & \(b_{ABC}\) & & & & 6000 \\ \hline \multicolumn{2}{c}{Return totals for the building block:} & **3000** & 4000 & **3000** & 4000 & 6000 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Coalitional strategies and possible scenario return totals (FAR m\({}^{2}\)) [41].