Recent work demonstrated the existence of Boolean functions for which Shapley values provide misleading information about the relative importance of features in rule-based explanations. Such misleading information was broadly categorized into a number of possible issues. Each of those issues relates with features being relevant or irrelevant for a prediction, and all are significant regarding the inadequacy of Shapley values for rule-based explainability. This earlier work devised a brute-force approach to identify Boolean functions, defined on small numbers of features, and also associated instances, which displayed such inadequacy-revealing issues, and so served as evidence to the inadequacy of Shapley values for rule-based explainability. However, an outstanding question is how frequently such inadequacy-revealing issues can occur for Boolean functions with arbitrary large numbers of features. It is plain that a brute-force approach would be unlikely to provide insights on how to tackle this question. This paper answers the above question by proving that, for any number of features, there exist Boolean functions that exhibit one or more inadequacy-revealing issues, thereby contributing decisive arguments against the use of Shapley values as the theoretical underpinning of feature-attribution methods in explainability.

翻译：近期研究证明了存在布尔函数，使得沙普利值在基于规则的 explanations 中提供关于特征相对重要性的误导信息。这种误导信息被大致归为若干可能的问题。每个问题都与特征在预测中相关或不相关有关，并且对于沙普利值在基于规则的可解释性中的不适用性具有重要意义。该先前研究设计了一种暴力搜索方法，用于识别定义在少量特征上的布尔函数及其相关实例，这些实例表现出揭示不适用性的问题，从而作为沙普利值不适用于基于规则的可解释性的证据。然而，一个悬而未决的问题是，对于具有任意大量特征的布尔函数，这种揭示不适用性的问题出现的频率有多高。显然，暴力搜索方法不太可能为解决这一问题提供见解。本文通过证明对于任意数量的特征，都存在布尔函数表现出一个或多个揭示不适用性的问题，从而回答了上述问题，并为反对将沙普利值作为可解释性中特征归因方法的理论基础提供了决定性论据。