## 4 Spatial Cooperative Game Models and Their Applications in Terms of Locales
### Case I: The Three Player (\(N=3\)) Hypothetical Spatial Cooperative Game  
A hypothetical urban regeneration area exists that has one building block consisting of three neighboring parcels (_PA_, _PB_, and _PC_; see Figure 2). Each parcel/lot is owned separately by one player in this urban spatial cooperative game. If each player prefers unilateral action, they may only use the limited base floor area ratio (FAR), which is equal to 1. The municipal government endorses comprehensive projects rather than three unilateral parcel-based projects. If possible, even one single block-based project would be preferred in order to solve various development and inefficiency problems such as common usage of green areas, car parking, and security management. One endorsement method is to provide larger FAR values, which is a strategy extensively utilized in Istanbul for solving the _Gecekondu_ [squatter housing] problem. Table 2 provides the decision rules regarding FAR bonuses. For this hypothetical case, the density bonus decision rules provide two cooperating neighboring parties a FAR of 1.5, which is a 50% increase in development rights compared to unilateral strategies. In the same direction, the FAR value reaches 2.0 for the grand coalition containing all three parties, and this is a 100% increase in development rights compared to unilateral strategies. This model setting has a sufficiently large FAR subsidy for encouraging the grand coalition. Overall, an increase in the FAR schedule is seen to occur that encourages larger and more extensive coalitions. The fact that cooperating parties are required to be spatially contiguous neighborhoods needs to be mentioned. To illustrate, the sub-coalition strategy of PA and PC is an unproductive coalition that cannot have a unified project able to benefit from scale economies; therefore, they cannot receive a FAR greater than the unilateral FAR values. The benefit calculations (\(b_{i}\), \(b_{S}\), \(b_{N}\)) with FAR bonuses are clearly illustrated in Table 3. In this table, the row with a gray background represents an unproductive coalition strategy. To overcome the technical limitations and thus increase the number of players, these types of neutral strategies are found to be unnecessary with regard to the optimization constraints and were later eliminated from the strategy list. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Lower Threshold** & \multicolumn{2}{c}{**Coalition Project Area**} & \multicolumn{2}{c}{**Upper Threshold**} & \multicolumn{1}{c}{**Floor Area Ratio**} \\
**(m\({}^{2}\))** & & **(m\({}^{2}\))** & & **(m\({}^{2}\))** & **(FAR)** \\ \hline
0 & \(<\) & A & \(\leq\) & 1000 & 1.0 \\
1000 & \(<\) & A & \(\leq\) & 2000 & 1.5 \\
2000 & \(<\) & A & & 2.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Floor Area Ratio (FAR) thresholds for a building block in an urban regeneration area [41]. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Strategies** & \multicolumn{2}{c}{**Returns from Strategies**} \\ \hline \multirow{3}{*}{Individual} & \(b_{A}\) & A (1000 m\({}^{2}\)) without bonus \\  & \(b_{B}\) & B (1000 m\({}^{2}\)) without bonus \\  & \(b_{C}\) & C (1000 m\({}^{2}\)) without bonus \\ \hline \multirow{3}{*}{Sub-Group} & \(b_{AB}\) & AB (1000 \(\text{+}\) 1000 \(\text{=}\) 2000 m\({}^{2}\)) 50\% bonus (2000 \(\times\) 1.5 \(\text{=}\) 3000 m\({}^{2}\)) \\ \cline{2-3}  & \(b_{AC}\) & AC (1000 \(\text{+}\) 1000 \(\text{=}\) 2000 m\({}^{2}\)) unproductive coalition 0\% bonus (2000 \(\times\) 1\(\text{=}\) 2000 m\({}^{2}\)) \\ \cline{2-3}  & \(b_{BC}\) & BC (1000 \(\text{+}\) 1000 \(\text{=}\) 2000 m\({}^{2}\)) 50\% bonus (2000 \(\times\) 1.5 \(\text{=}\) 3000 m\({}^{2}\)) \\ \hline Grand & \(b_{ABC}\) & ABC (1000 \(\text{+}\) 1000 \(\text{+}\) 1000 \(\text{=}\) 2000 m\({}^{2}\)) 100\% bonus (3000 \(\times\) 2.0 \(\text{=}\) 6000 m\({}^{2}\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Computation of returns for the listed coalition strategies using FAR bonuses [41].