## 4 Spatial Cooperative Game Models and Their Applications in Terms of Locales
### In-Scenario Evaluations  
Two scenarios exist regarding the EU case. The first one is designed to measure both locations and neighborhood relations alongside their varying economic sizes, and the second one is designed to disregard countries' relative sizes and scales in order to focus only on locations and neighborhood relations using the assumed equal base GDPs. In both analyses, the critical players in the EU are identified using spatial cooperative game models. * _Models with varying country economic sizes_: In the cooperative game model applications, Model II was first applied to 24 European Union countries (see Table 8 and Figure 6) to test the availability of a core solution. The maximized tax value (\(Z_{Mxx}\)) was found to \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{3}{c}{**Start**} & \multicolumn{4}{c}{**Models III and IV \({}^{*}\)**} \\  & \multicolumn{4}{c}{**(Z = 408.16)**} \\ \hline
**Code** & **Country** & **\(\boldsymbol{GDP}\) (S/\(\boldsymbol{4}_{i}\)-\(\boldsymbol{10^{4}}\))** & \(X_{\text{M}ix}\) & \(X_{\text{M}ix}\) & \(X_{\text{M}ix}\)/\(GDP_{\text{\tiny{}^{*}}}\) & \(X_{\text{M}ix}\)/\(GDP_{\text{\tiny{}^{*}}}\) \\ \hline AT & Austria & 266.6 & 545 & 545 & 2.00 & 2.00 \\ CZ & Czechia & 266.6 & 545 & 545 & 2.00 & 2.00 \\ DE & Germany & 266.6 & 915 & 2148 & 3.40 & 8.10 \\ HU & Hungary & 266.6 & 915 & 2148 & 3.40 & 8.10 \\ IT & Italy & 266.6 & 545 & 545 & 2.00 & 2.00 \\ SK & Slovakia & 266.6 & 545 & 545 & 2.00 & 2.00 \\ SI & Slovenia & 266.6 & 545 & 545 & 2.00 & 2.00 \\ PL & Poland & 266.6 & 915 & 2148 & 3.40 & 8.10 \\ BE & Belgium & 266.6 & 545 & 545 & 2.00 & 2.00 \\ DK & Denmark & 266.6 & 284 & 1719 & 1.10 & 6.40 \\ FR & France & 266.6 & 932 & 1719 & 3.50 & 6.40 \\ LU & Luxembourg & 266.6 & 545 & 545 & 2.00 & 2.00 \\ NL & Netherlands & 266.6 & 545 & 545 & 2.00 & 2.00 \\ HR & Croatia & 266.6 & 545 & 545 & 2.00 & 2.00 \\ RO & Romania & 266.6 & 284 & 1719 & 1.10 & 6.40 \\ LT & Lithuania & 266.6 & 284 & 1719 & 1.10 & 6.40 \\ SE & Sweden & 266.6 & 284 & 1202 & 1.10 & 4.50 \\ ES & Spain & 266.6 & 284 & 1202 & 1.10 & 4.50 \\ BG & Bulgaria & 266.6 & 284 & 1202 & 1.10 & 4.50 \\ LV & Latvia & 266.6 & 284 & 1202 & 1.10 & 4.50 \\ FI & Finland & 266.6 & 284 & 545 & 1.10 & 2.00 \\ PT & Portugal & 266.6 & 284 & 545 & 1.10 & 2.00 \\ GR & Greece & 266.6 & 284 & 545 & 1.10 & 2.00 \\ EE & Estonia & 266.6 & 284 & 545 & 1.10 & 2.00 \\ \hline \hline \end{tabular} \({}^{*}\) Each minimum and maximum value is derived from a separate optimization operation.  
\end{table} 
Table 10: Optimization results—Equal GDP. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{3}{c}{**Start**} & \multicolumn{4}{c}{**Models III and IV \({}^{*}\)**} \\  & \multicolumn{4}{c}{**(Z = 39.04)**} \\ \hline
**Code** & **Country** & **\(\boldsymbol{GDP}\) (S/\(\boldsymbol{4}_{i}\)-\(\boldsymbol{10^{4}}\))** & \(X_{\text{M}ix}\) & \(X_{\text{M}ix}\) & \(X_{\text{M}ix}\)/\(GDP_{\text{\tiny{}^{*}}}\) & \(X_{\text{M}ix}\)/\(GDP_{\text{\tiny{}^{*}}}\) \\ \hline FR & France & 1030 & 1657 & 5003 & **1.60** & 4.90 \\ LU & Luxembourg & 22 & 22 & 39 & 1.00 & 1.80 \\ NL & Netherlands & 333 & 379 & 1153 & 1.10 & 3.50 \\ HR & Croatia & 37 & 37 & 93 & 1.00 & 2.50 \\ RO & Romania & 174 & 174 & 1001 & 1.00 & 5.80 \\ LT & Lithuania & 33 & 33 & 166 & 1.00 & 5.00 \\ SE & Sweden & 187 & 187 & 858 & 1.00 & 4.60 \\ ES & Spain & 641 & 891 & 2517 & 1.40 & 3.90 \\ BG & Bulgaria & 55 & 55 & 440 & 1.00 & 8.00 \\ LV & Latvia & 19 & 19 & 69 & 1.00 & 3.60 \\ FI & Finland & 88 & 88 & 279 & 1.00 & 3.20 \\ PT & Portugal & 113 & 113 & 369 & 1.00 & 3.30 \\ GR & Greece & 108 & 108 & 351 & 1.00 & 3.20 \\ EE & Estonia & 15 & 15 & 15 & 1.00 & 1.00 \\ \hline \hline \end{tabular} \({}^{*}\) Each minimum and maximum value is derived from a separate optimization operation.  
\end{table}