## 4 Spatial Cooperative Game Models and Their Applications in Terms of Locales
### In-Scenario Evaluations
* _Models with assumed equal country economic sizes:_ In the second scenario, given the same geographic setting in Figure 6, the country base GDPs (Table 10) are assumed to be $266.6 billion for each country, with their total equaling the sum of the base GDPs (Table 9) in the first scenario. By eliminating the relative country sizes from the analyses in this way, only the impacts of strategic locations and neighborhood relations are kept in the focus of the analyses. In other words, the equal sizes allow for a controlled experiment to be performed focusing on spatial geography.  
In the first step, Model II was applied to test the availability of a core solution and to measure the size of the tax return (\(Z_{Mix}\)). Given the same core constraints, the maximized tax value (\(Z_{Mix}\)) is found to be $408.16 billion, which is much higher than the earlier scenario. That implies that among the lopsided players, the larger a country's GDP, the larger the scale economies and in-country free trade within the largest country allow reward benefits to be retained alongside its largest \(A_{i}\). On the other hand, when the countries have uniform equal economic base GDP sizes, they earn much higher grand coalition benefits, which can then be allocated to players or be retained as tax. Keeping the estimated maximum tax (\(Z_{Mix}\) = $408.16 billion) as a constant (Z) in Equation (5), some of the countries' benefits from optimizations may slightly vary. In order to estimate the range of these changes, Models III and IV are separately applied to each country (\(X_{AT}\), \(\ldots\)\(X_{EE}\)), Model III for the maximum and Model IV for the minimum (\(X_{-Mix}\) and \(X_{-Min}\)). The results are presented in Table 10. Later, these extreme values are normalized by dividing them by the base GDP sizes in order to allow them to be compared in terms of per unit economic size base. These normalized values are also added in the last two columns of Table 10. In order to present the significance of geographic locations and neighborhood relations, these values are then used in Figures 9  
Among these minimization (\(X_{Min}\)) scenarios, Table 10 shows the country returns per unit economic base size (\(X_{Min}\)/\(GDP\_\)) to vary between 1.1 and 3.5. The highest minimum returns are observed for France, Germany, Hungary, and Poland, and these are presented as a graph in Figure 9. Their returns are 3.5-3.4 times larger than their base unit economic sizes. Figure 9 clearly presents the geographic key aspects of these countries, showing them to have direct neighborhood connections to Central Europe and also to provide a bridge between the peripheral and central European countries. Consequently, by excluding the size and scale factors, the EU countries' geographically distinctive aspects and relative powers are clearly reflected in the spatial cooperative game analyses. Among the maximization (\(X_{Max}\)) scenarios, Table 10 shows the country returns per unit economic base size (\(X_{Max}\)/\(GDP\_\)) to vary between 2.0 and 8.1. The highest returns are seen for Germany, Hungary, and Poland, at 8.1 times their base economic size, followed by France, Denmark, Romania, and Lithuania at 6.4 times their base economic size (see Table 10 and Figures 9 and 10). The minimum values represent the least expected grand coalition benefits reflecting the significance of countries' geographic locations. Because the base economic sizes are assumed to be equal, higher return for a country implies a more critical location in the EU geography. In this respect, France, Germany, Hungary, and Poland are distinctively strong countries. The maximum values are the maximum economic benefits reachable from the grand coalition given the constraints of all individual and sub-group rationalities. If the minimum and maximum values are the same, then individual and sub-coalition constraints do not permit any margin of change in the maximized and minimized allocations.