## 3 Methodology and Cooperative Game Theory Models
### Listing Cooperative Game Strategies and Computing Their Returns
#### 3.2.1 Listing Cooperative Game Strategies  
Given the sum of all players with \(N\), the total number of cooperative game theory strategies is computed using the formula \(2^{N}-1\). Table 1 has been prepared to show the total number of strategies and the extent of the problem size. And its accompanying graph shows the total number of strategies to grow exponentially as the number of players increases. After a certain \(N\) value (\(\approx\)7), the strategies first can no longer practically be listed by hand; later on, listing the number of strategies using personal computer programs becomes technically impossible for a higher \(N\) value (\(\approx\)19). [PERSON] utilized a hand-based approach for just six participants [43].  
The current research has prepared an algorithm for handling large numbers of players (\(N\)) and the associated strategies (\(2^{N}-1\)). Some strategies are observed to be equivalent to earlier narrower coalition strategies due to spatial disconnections. Spatial disconnections and disunities neither allow fruitful coalitions to take advantage of spatially uninterrupted flows nor, as a result, coalitions. In order to remove these redundant strategies based on the algorithm presented in Figure 1, a script has been developed for extensive multiparty geographic analyses as the technical base and contribution of this research. The flowchart demonstrates the sequential coalition listing and elimination procedures for redundant strategies: Step 1 involves generating and listing coalition strategies (\(2^{N}-1\)) by combining the procedures starting from the individual (\(i\)) to the larger subgroup (\(S\)) levels and ultimately to the grand coalition (\(N\)). Step 2 involves evaluating a strategy in line whether it is redundant or not. Step 3 involves the decision phase where, if the strategy in question is redundant (Yes), the strategy gets discarded before moving on to the next strategy in the list; if the strategy is not redundant (No), the associated coalition benefit (\(b_{S}\)) is computed and the associated LP constraint for the models is prepared. Step 4 also involves a decision phase that checks the availability of further strategies in the list where if it is not the last coalition strategy (grand coalition), then the script goes back to the next strategy in the list for further evaluations. All evaluation processes are noteworthy pursued, from the leastto the most comprehensive coalition strategies. This algorithm is designed to list the LP constraints in a way where the generated constraints text can be easily read and solved by an optimization solver within the General Algebraic Modeling System (GAMS).  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Number of Players** & **Total Number of Strategies** \\ **(N)** & **(2\({}^{\circ}\) – 1)** \\ \hline
1 & 1 \\ \hline
2 & 3 \\ \hline
3 & 7 \\ \hline
4 & 15 \\ \hline
5 & 31 \\ \hline
6 & 63 \\ \hline
7 & 127 \\ \hline
19 & 524,287 \\ \hline
24 & 16,777,215 \\ \hline
25 & 33,554,431 \\ \hline
26 & 67,108,863 \\ \hline
27 & 134,217,727 \\ \hline
28 & 268,435,455 \\ \hline
29 & 536,870,911 \\ \hline
30 & 1,073,741,823 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The association between the number of players (\(N\)) and the total number of strategies.  
Figure 1: An algorithm to eliminate redundant coalitions as well as the associated allocation constraints.  
For the practical application of the optimization models, having more players provides a better scope of analysis. The total number of achievable players becomes \(N=27\), which is much better for extensive and multiparty geographical analyses. This research selected 24 European Union countries for one case study, which means the number of players is slightly less than \(N=27\). Nonetheless, any continental or complete global analyses require many more players.