## 4 A Numerical Example  
This numerical example assumes the scenario of a city planner's decision-making process in choosing three transportation options: road, subway, and bike lanes. The planner considers five criteria: cost, accessibility, environmental impact (EI), safety, and economic development (ED). The case highlights the interdependent nature of these alternatives and their evaluation as a cooperative game. First, let an expert quantify the PCM of the criteria, leading to the calculation of the maximum eigenvalue, priority vector, CI, and CR. These calculations ensure the consistency of the criteria evaluation. First, assume the PCM of the criteria can be quantified by an expert as follows: \begin{tabular}{c c c c c c} \hline
**PCM** & **Cost** & **Accessibility** & **EI** & **Safety** & **ED** \\ \hline Cost & 1 & 1/2 & 1 & 1/3 & 1 \\ Accessibility & 2 & 1 & 2 & 1 & 3 \\ EI & 1 & 1/2 & 1 & 1/3 & 1 \\ Safety & 3 & 1 & 3 & 1 & 4 \\ ED & 1 & 1/3 & 1 & 1/4 & 1 \\ \hline \end{tabular}   
Using the eigenvector method, we can derive the maximum eigenvalue \(\lambda_{max}\) = 5.0364 and the corresponding eigenvector, i.e., priority vector, as [0.1223, 0.2895, 0.1223, 0.3592, 0.1067]. Then, we can derive CI = 0.0091 and CR = 0.0081, where RI = 1.12, which is less than 0.1, justifying the consistency of the criteria. Next, we can use the AHP to derive the scores of alternatives concerning each criterion. We assume the PCMs of the alternatives for each criterion are as follows: \begin{tabular}{c c c c c c} Cost & Road & Subway & Bike lanes & Accessibility & Road & Subway & Bike lanes \\ Road & 1 & 1/3 & 1/5 & Road & 1 & 1/3 & 1/2 \\ Subway & 3 & 1 & 1/2 & Subway & 3 & 1 & 2 \\ Bike lanes & 5 & 2 & 1 & Bike Lanes & 2 & 1/2 & 1 \\ EI & Road & Subway & Bike Lanes & Safety & Road & Subway & Bike Lanes \\ Road & 1 & 1/2 & 1/3 & Road & 1 & 1/3 & 1/4 \\ Subway & 2 & 1 & 1/2 & Subway & 3 & 1 & 1/2 \\ Bike Lanes & 3 & 2 & 1 & Bike Lanes & 4 & 2 & 1 \\  & & & ED & Road & Subway & Bike Lanes & \\  & & & Road & 1 & 1/2 & 1/3 & \\  & & & Subway & 2 & 1 & 3/4 & \\  & & & Bike Lanes & 3 & 4/3 & 1 & \\ \end{tabular}   
We can now calculate the eigenvectors for each PCM, which represent the scores of the alternatives for each criterion, and obtain the scores of the alternatives for each criterion as shown in Table 3.  
Next, to account for the interdependence between the alternatives and evaluate them in the context of cooperative game theory, we calculated the Shapley values using the worth of each coalition, as shown in Table 4 \begin{table}
\begin{tabular}{c c c c c c} \hline
**Alternatives** & **Cost** & **Accessibility** & **EI** & **Safety** & **ED** \\ \hline Road & 0.1095 & 0.1634 & 0.1634 & 0.1220 & 0.1677 \\ Subway & 0.3090 & 0.5396 & 0.2970 & 0.3196 & 0.3487 \\ Bike Lanes & 0.5816 & 0.2970 & 0.5396 & 0.5584 & 0.4836 \\ \hline Weight & 0.1223 & 0.2895 & 0.1223 & 0.3592 & 0.1067 \\ \end{tabular}
\end{table} 
Table 3: Scores of alternatives for each criterion. Furthermore, we calculated the marginal distribution of each alternative, taking into account all possible permutations of the alternatives, as shown in Table 5.  
Based on these calculations, the Shapley values for the three transportation alternatives are shown in Table 6.  
The city government can use these Shapley values with the AHP method to evaluate the alternatives based on multiple criteria while accounting for their interdependent nature. This will provide a more comprehensive evaluation of the transportation options, helping the city make better-informed decisions for urban planning. Since the AHP and Shapley's scores play a critical role in the final budget allocation, we used normalized geometric averages (NGAs) to combine the two scores, as shown in Table 7.  
In conclusion, by combining the AHP and Shapley values, the city government can evaluate the transportation alternatives based on multiple criteria while accounting for their interdependent nature.