## 4 Spatial Cooperative Game Models and Their Applications in Terms of Locales
### Evaluations Between Scenarios  
The previous section had two separate spatial game scenarios: the first involving the comprehensive effects of both the economic and geographic aspects of countries regarding allocating the grand coalition benefits, and the second involving the isolated effects of countries' geographic aspects in terms of their locations and neighborhood relations regarding allocating the grand coalition benefits. In the first one, countries' base GDPs are proportional to their actual sizes, and in the second one countries' base GDPs are all the same size. This current section makes analysis-based comparisons of their inter-scenarios.  
* _Comparing the minimum returns between two different base GDP size scenarios_: After retaining \(Z_{\text{Max}}\), the normalized minimum returns (\(X_{\text{Min}}\)/GDP_) presented in Tables 9 and 10 and graphed in Figures 7 and 9 show EU countries' relative powers/magnitudes. The countries' economic size and scale factors as well as geographic aspects bring Germany to the primary position in the EU (Table 9 and Figure 7). This is quite consistent with current EU politics and practices. Germany is followed by France, Italy, and Spain, with Poland and Netherlands being the other significant countries in the EU. Upon excluding economic size factors from the analyses by assuming an equal base GDP for all countries, Figure 9 clearly reflects the geographic factors. The results show France, Germany, Hungary, and Poland to be the four most distinctive players in the EU geography. In the first scenario, the economic scale factor dominates the relative weight of countries' geographic aspects. To illustrate, Hungary's normalized benefit falls from 3.4 to 1.0 (Tables 9 and 10). Technically, this result can also be attributed to the characteristics of the reward function (Equation (11)), which provides size and scale benefits when base size \(A_{S}\) reaches $215 billion.
* _Comparing the maximum returns between two different base GDP size scenarios_: After retaining \(Z_{\text{Max}}\), the normalized maximum returns (\(X_{\text{Max}}\)/GDP_) presented in Tables 9 and 10 and graphed in Figures 8 and 10 show the potential for extra benefit from being in the cooperative game framework. The estimated country minimum values come from their own characteristics; however, the maximum values come from the constraint structures based on other countries' individual and sub-group strategies and associated benefit expectations. If the constraints are permissive, some members of the grand coalition may attain higher benefits as long as other players have excessions. In this respect, some of the countries that have connections with large size/scale countries or connections with large numbers of countries may attain extra benefits, such as Hungary and Denmark. Germany, Poland, France, Romania, Lithuania, Sweden, Spain, and Bulgaria, despite not being the countries receiving the highest returns, do receive sufficiently large benefits (Table 9 and Figure 8). In the second scenario assuming equal base GDP sizes, the highest returns are provided to those countries that enable a connection to Central Europe. Denmark enables a connection with the Scandinavian countries of Sweden and Finland; Hungary enables the link with the Balkan countries of Greece, Bulgaria, and Romania; and Poland enables a connection with the Baltic countries of Estonia, Latvia, and Lithuania (Table 10, Figures 9 and 10). After these, France, Denmark, Romania, and Lithuania are other countries that provide peripheral access to Central Europe.