## 3 AHP/ANP with Shapley Values
### Method for the AHP/ANP with Shapley Values  
We detail the mathematical formulation of Shapley values, a concept from cooperative game theory, to distribute resources equitably among alternatives in MCDM. This integration offers a novel approach to account for alternatives' collective and interconnected nature, enhancing the decision-making process. The proposed method comprises the following steps:  
1. Problem Definition and Initial Assessment with the AHP/ANP: Initially, the problem is defined, including identifying the criteria and alternatives. Depending on the nature of these criteria and alternatives (independent or interdependent), the AHP or ANP is employed to assess the weights of the criteria and scores of the alternatives. This process involves constructing PCMs and calculating criteria weights and alternative scores through eigenvector extraction. Consistency ratios are evaluated to ensure the reliability of judgments. 2. Quantification of Coalition Values: Parallel to the AHP/ANP assessment, we determine the coalition values among the alternatives. This involves consulting domain experts to understand possible coalitions among the alternatives and quantifying the value each coalition brings. These coalition values are essential for computing Shapley values, as they represent the worth of each alternative's contribution to potential cooperative groupings. 3. Calculating Shapley Values: With the coalition values at hand, we calculate the Shapley values for each alternative. The Shapley value, \(\phi_{i}(v)\), for alternative \(i\) is determined using the formula that incorporates the sum of marginal contributions of alternative \(i\) across all possible coalitions, weighted by the factorial of the sizes of the coalitions and the set of all alternatives. This step is crucial for acknowledging the contributions of individual alternatives to the collective utility of coalitions they are a part of. The Shapley value is a mathematical concept employed in cooperative game theory to distribute resources among players based on their contributions. In multi-criteria decision-making, the Shapley value aids in allocating resources among alternatives by considering their interdependence. The mathematical formulation of the Shapley value for an alternative \(i\) is given by \[\phi_{i}(v)=\sum_{\mathcal{S}\subseteq N\{i\}}\frac{|S|!(N-|S|-1)!}{N!}(v(S \cup\{i\})-v(S))\] (4) where \(\phi_{i}(v)\) is the Shapley value of alternative \(i\); \(N\) is the set of all alternatives; \(S\) is a subset of \(N\) that does not include alternative \(i\); \(v(S)\) is the coalition value function, which assigns a value to each coalition \(S\); and \(\,1\,S\,\) and \(\,1\,N\,|\,\) are the cardinalities of the sets \(S\) and \(N\), respectively. The Shapley value is calculated by summing the marginal contributions of alternative \(i\) across all possible coalitions \(S\) that do not include \(i\). The marginal contribution of \(i\) to a coalition \(S\) is the difference between the coalition value with \(i\), i.e., \(v(S\cup\{i\})\), and the coalition value without \(i\), i.e., \(v(S)\). The binomial coefficient then weights the sum to account for the different ways that the coalition can be formed. 4. Synthesizing AHP/ANP and Shapley Values: After obtaining the scores from the AHP/ANP and the Shapley values, these are synthesized to achieve a final ranking of the alternatives. This synthesis is performed using normalized geometric averages, allowing for a balanced integration of the criteria-based evaluation and the cooperativecontributions captured by the Shapley values. Such a holistic synthesis provides a comprehensive viewpoint that accounts for both the intrinsic merits of alternatives and their cooperative dynamics. 5. Final Decision-Making: By leveraging the synthesized scores, decision-makers can make a more informed and equitable resource allocation decision.