Shapley and Banzhaf Vectors of a Formal Concept

We propose the usage of two power indices from cooperative
game theory and public choice theory for ranking attributes of closed sets,
namely intents of formal concepts (or closed itemsets). The introduced
indices are related to extensional concept stability and based on counting generators, especially those that contain a selected attribute. The
introduction of such indices is motivated by the so-called interpretable
machine learning, which supposes that we do not only have the class
membership decision of a trained model for a particular object, but also
a set of attributes (in the form of JSM-hypotheses or other patterns)
along with individual importance of their single attributes (or more complex constituent elements). We characterise computation of Shapley and
Banzhaf values of a formal concept in terms of minimal generators and
their order filters, provide the reader with their properties important for
computation purposes, and show experimental results.