## 4 Spatial Cooperative Game Models and Their Applications in Terms of Locales
### Case I: The Three Player (\(N=3\)) Hypothetical Spatial Cooperative Game
For such a small problem, this single core vector solution is not an unexpected outcome; however, for the coming case studies with larger numbers of players (\(N\) = 24 and \(N\) = 25), the \(P\_Max}\) and \(P\_Min}\) values may be different and may provide valuable insights. The single core vector solutions show player PB to receive twice as high a return compared to players PA and PC. This difference reflects the importance of players having a strategic location in this simple development block, and the models designed in a simple spatial cooperative game framework technically uncover these characteristics. Table 7 uses the optimization results provided in Table 6 and normalizes the maximum and minimum return values by dividing them by the base area sizes. The normalized values show that, for each unit of base parcel area, player PB receives two units of return. On the other hand, players PA and PC receive only a single unit of return. Player PB technically can say, \"If I do not cooperate, you or the government cannot receive any bonus reward at all; therefore, I rationally expect to receive a 2000 m\({}^{2}\) return from this cooperative game theory formation\". The same results can be observed in Figures 4 and 5. Instead of withdrawing \(Z_{Max}\) = 2000 from the core, if it is allocated to the players, their maximum return values would be 2000 m\({}^{2}\) units more than the values presented in Table 7 (\(PA_{Max}\) = 3000; \(PB_{Max}\) = 4000; \(PC_{Max}\) = 3000), and the per unit returns can be tripled for the edge parcels and quadrupled for the interim parcel. All allocation tables and figures show that the developed spatial cooperative game models and associated analyses have unearthed the critical roles players' strategic locations have in the formation of spatially extensive coalitions. The simplest form of the linear programming models technically works quite well in identifying and sorting the critical players. The models' extensive uses in the coming section will provide more insights for \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{1}{c}{**Initial Base Values**} & \multicolumn{3}{c}{**Model III (Z = 2000)**} & \multicolumn{3}{c}{**Model IV (Z = 2000)**} \\ \hline
**Parcel** & **Area (m\({}^{2}\))** & **PA\({}_{Max}\)** & **PB\({}_{Max}\)** & **PC\({}_{Max}\)** & **PA\({}_{Min}\)** & **PB\({}_{Min}\)** & **PC\({}_{Min}\)** \\ \hline PA & 1000 & **1000** & 1000 & 1000 & **1000** & 1000 & 1000 \\ PB & 1000 & 2000 & **2000** & 2000 & 2000 & **2000** & 2000 \\ PC & 1000 & 1000 & 1000 & **1000** & 1000 & 1000 & **1000** \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Results from the cooperative game optimization models [41]. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{1}{c}{**Initial Base Values**} & \multicolumn{3}{c}{**Model III-IV (Z = 2000)**} & \multicolumn{3}{c}{**Normalized Values**} \\ \hline
**Parcel** & **Area (m\({}^{2}\))** & **Minimum** & **Maximum** & **P\({}_{Min}\)/P\({}_{Area}\)** & **P\({}_{Max}\)/P\({}_{Area}\)** \\ \hline PA & 1000 & 1000 & 1000 & 1.0 & 1.0 \\ PB & 1000 & 2000 & 2000 & 2.0 & 2.0 \\ PC & 1000 & 1000 & 1000 & 1.0 & 1.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Normalized cooperative game optimization results [41]. prospective research. The same presentation procedures and techniques are used in the coming sections for the extensive cases with larger numbers of players. Measuring the Spatial Importance of Countries' Geographic Locations and Economic Sizes in Multinational Coalitions  
The urban regeneration problem has clear incremental bonus rules and regulations regarding building blocks and the associated parcels, and municipal authorities use these practically [32]. However, at the international level, no such clear-cut return/bonus rules exist for use among cooperating countries. Conversely, absolute benefits and welfare gains certainly exist among trading and cooperating multinational parties regarding the efficient use of resources.