## 1 Introduction
### Literature Review  
A review of the existing literature about Shapley values shows that the researches from [PERSON] et al. [55; 59] were the first to propose them in the explanation of how featurescontribute to the prediction of classification and regression machine learning algorithms, posing the basis for XAI. They showed that Shapley values can reveal the influence of features, no matter the concept that the AI algorithm learns. An issue that affected all the algorithms based on the Shapley values was the high computational cost, but [PERSON] et al. [52] proposed a solution to make the computation feasible, even for a large number of input data and features, by sampling the permutations of the inputs randomly and estimating the cost functions related to the evaluated algorithms via the Monte Carlo simulation. The work by [PERSON] et al. [47] introduced the kernel SHAP framework, which involved the standard Shapley value algorithm and the local interpretable model-agnostic explanation (LIME) algorithm, to assign each feature an importance value for a particular prediction. Further developments of the SHAP method were carried out by [PERSON] et al. [53], who introduced the extension to the kernel SHAP to reduce the computation time and lead to more reliable predictions when features are correlated. [PERSON] et al. [56], who developed an algorithm able to explain the contribution of mixed (continuous, discrete, and categorical) and dependent features using conditional inference trees, and reported an improvement in the computation time when the features are dependent, with respect to [PERSON] et al. The kernel SHAP and the extension to the kernel SHAP were successfully used in studies regarding many different subjects to explain the outcome of both classifiers and the regression ML algorithms. [PERSON] et al. [60] used the Shapley values to find causal connections between socioeconomic metrics and the spread of COVID-19 in the different phases of the pandemic. [PERSON] et al. [34] applied the SHAP method to explain the forecasts of the air quality in terms of the NO\({}_{2}\) concentration made by the long short-term memory (LSTM) deep learning model, while [PERSON] et al. [61] explained the predictions of a machine-learning-based model for sea-level projections. The SHAP method can also be applied to finance, as it was performed in the study by [PERSON] et al. [57], who wanted to assess the feature importance in the evaluation of the portfolio performance to genomics. This is similar to the work by [PERSON] et al. [62], who explained the importance of interpretable machine learning (iML) for making the prediction of the ML algorithm understandable to human, when the observed phenomena are highly complex and dependent on a large number of variables, and to aviation, as cited in the research by [PERSON] et al. [63]. They employed the SHAP method to highlight the most important features affecting the outcome of the XGBoost algorithm for the real-time evaluation of the airport runway conditions. The SHAP method can also be used for the multi-label classification problem, as it was reported in [PERSON] et al. [64] and [PERSON] et al. [65]: the former proposed and tested a multi-label learning fused with the Shapley value (SHAPFS-ML), while the latter used the SHAP method to explain the predictions about the students' learning style. The SHAP explanation was also used in medicine ([PERSON] et al. [66], [PERSON] et al. [67]), psychology ([PERSON] et al. [68], [PERSON] et al. [69]), and engineering ([PERSON] et al. [70], [PERSON] et al. [71])  
In the field of feature selection, the research from [PERSON] et al. [72] was the first to present contribution selection algorithms (CSAs) that were based on Shapley values to select the most relevant features. The CSA was tested on seven different datasets and the authors reported its capability to improve the performance of their classifier, competing with the most used feature selection methods, especially in those cases where features interacted with each other. [PERSON] et al.