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：近期研究证明了布尔函数的存在性，其中Shapley值在基于规则的推理解释中会误导特征相对重要性的判断。此类误导性信息被广泛归纳为若干潜在问题，每个问题均涉及特征对预测的相关性或无关性，且均对Shapley值在基于规则的推理可解释性中的不适用性具有重大意义。该前期研究设计了一种暴力枚举方法，用于识别定义在少量特征上的布尔函数及其关联实例，这些实例呈现出此类暴露不适用性的问题，从而佐证了Shapley值对基于规则的可解释性的不适用性。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，此类暴露不适用性的问题出现的频率如何。显然，暴力枚举法难以揭示解决此问题的洞见。本文通过证明以下结论回答了上述问题：对于任意数量的特征，始终存在表现出一个或多个暴露不适用性问题的布尔函数，从而为反对将Shapley值作为可解释性中特征归因方法理论基石的主张提供了决定性论据。