## 4 A Numerical Example
The calculated normalized geometric averages show that subway (0.4245) is the preferred option, followed by bike lanes (0.3585) and road (0.2170), where the scores of the alternatives indicate the decision of the budget allocation. Compared to the AHP's result, bike lanes should be the most important alternative, and our method displays different perspectives on the problem. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Alternative** & **Shapley Value** & **Normalized Shapley Value** \\ \hline Road & 123.333 & 0.3083 \\ Subway & 175.833 & 0.4396 \\ Bike Lanes & 100.833 & 0.2521 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Shapley values of the alternatives. \begin{table}
\begin{tabular}{l c} \hline \hline
**Coalition** & **Worth** \\ \hline Road & 100 \\ Subway & 150 \\ Bike Lanes & 75 \\ (Road, Subway) & 260 \\ (Road, Bike Lanes) & 185 \\ (Subway, Bike Lanes) & 240 \\ (Road, Subway, Bike Lanes) & 400 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The worth of each coalition. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Permutation** & **Road** & **Subway** & **Bike Lanes** \\ \hline (1, 2, 3) & 100 & 160 & 140 \\ (1, 3, 2) & 100 & 85 & 215 \\ (2, 1, 3) & 150 & 110 & 140 \\ (2, 3, 1) & 150 & 90 & 160 \\ (3, 1, 2) & 75 & 110 & 215 \\ (3, 2, 1) & 75 & 165 & 160 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Marginal distribution of the alternatives. Compared to the conventional AHP, the proposed method integrating the AHP/ANP with Shapley values provides a more comprehensive and equitable evaluation of alternatives by considering their cooperative potential and contributions to coalitions, which can lead to a rank reversal and a more informed decision-making process in resource allocation problems. Next, using the above example, we considered an interdependence between criteria and cooperative alternatives. First, we needed to create our problem structure, which consisted of two clusters: Criteria and Alternatives. The Criteria cluster contains five criteria: cost, accessibility, EI, safety, and ED. The Alternatives cluster contains three transportation options: road, subway, and bike lanes. The interdependence between the two clusters and goal, given by an expert, can be depicted as shown in Figure 2.  
According to Figure 2, we can design our supermatrix as follows:  
\[\begin{array}{cccc}\text{Goal}&\text{C}\text{C}\text{C}\text{C}\text{C  
The comparison reveals that the ranking of the alternatives differed slightly between the combined AHP and ANP and the Shapley value methods. In the previous situation, the preferred option was subway (0.4245), followed by bike lanes (0.3585) and road (0.2170). On the other hand, the ANP and Shapley method also indicated subway as the preferred option (0.4147), followed by bike lanes (0.3657) and road (0.2197). Both methods suggested that subway is the most suitable transportation alternative. However, the ANP and Shapley method provided a slightly different perspective on the problem by accounting for the interdependence between the criteria and alternatives, which was not captured by the AHP method alone. In conclusion, the proposed method showed different rankings with respect to the AHP or ANP and incorporates the information of coalition between the alternatives to adjust the final decision of the budget allocation.