## 4 Spatial Cooperative Game Models and Their Applications in Terms of Locales  
Initial versions of spatial cooperative games in urban development/regeneration were presented in [PERSON] and [PERSON], [PERSON], and [PERSON] et al. [32; 41; 43]. The numbers of players and associated strategies in these studies were within the limit and capabilities of hand processing; however, once the number of players exceeds six, the hand-based capabilities of spatial analyses become quite limited and even technically impossible. As part of this research, once the necessary scripts were completed,the number of players was raised to 27, after which the number of strategies needing to be listed for the cooperative game models was increased to the hundreds of thousands. In this way, two different levels of analyses (i.e., local and global) have become technically practical. These analyses take into account both the size and scale of the players as well as their spatial neighborhood relations. The first analysis occurs over the urban development/regeneration building blocks [41] and the second one over the multinational coalitions/unions.  
In the context of the two selected characteristic case studies, the cooperative game models measure: (i) the significance of geographic locations and their associated neighborhood relations, (ii) the possible impacts of individual sizes and scales and their collective contributions to coalitions, and (iii) the unilateral and/or combined effects of players on the allocations from the grand coalition benefits. The critical players in a geographic setting are identified and sorted in this way. The model applications have been pursued over the following two different case areas:  
1. A three-player (\(N=3\)) hypothetic cooperative game case on a three-lot urban building block. The selected minimal number of players allows the proposed models to be presented succinctly. The model presentation contains every detail concerning the individual, sub-coalition, and grand coalition rationalities in the forms of equations, tables, and graphs.
2. A 24-player multinational cooperative game designed to evaluate the European Union countries possessing spatial contiguity and unity. The developed model determines varying critical positions of players in the union for the selected case area.  
The second characteristic case for the EU utilizes two different approaches to measure the significance of players' geographic locations and neighborhood relations alongside their relative sizes/scales. Given the geographic setting presenting all locations and neighborhood relations, the analyses are based on (i) utilizing the players' actual sizes/scales for the incremental multipliers, and (ii) assuming and utilizing equal sizes/scales for all players in order to eliminate the size/scale factor and determine the isolated impacts based on the players' geographic characteristics. In other words, the first approach observes the combined effects of players' geographic and size/scale aspects, while the second approach only observes the isolated effects of their geographic characteristics. In order to have comparable allocation results from these two different approaches, the second approach uses equal size/scale values for all players as obtained from the base grand coalition size factor (\(\sum_{i\in N}A_{i}\)) divided by the number of players (\(N\)). This way the grand coalition benefits/returns can be ascertained to be the same for the two different approaches. The allocation values provide results that can be compared among the players and between the approaches.  
Furthermore, for better understanding and in order to obtain comparable results, all players' returns (\(x_{i}\)) have been normalized with their original sizes/scales (\(A_{i}\)) by dividing the game theoretical allocation values to their original actual or assumed sizes, (\(x_{i}/A_{i}\)). In this way, the returns from spatial cooperative game are computed for each per unit of size and presented for every player.