The Sets of Power

TL;DR

This paper unifies various importance measures across domains like voting, argumentation, and databases, showing they are instances of a general framework based on monotonic predicates, and explores new possibilities for their application.

Contribution

It introduces a general framework for importance measures applicable across multiple domains and demonstrates how existing measures fit into this framework, enabling new applications.

Findings

Importance measures can be computed using monotonic predicates.

Existing measures are special cases of the general framework.

New importance measures can be devised for unexplored domains.

Abstract

Measures of voting power have been the subject of extensive research since the mid 1940s. More recently, similar measures of relative importance have been studied in other domains that include inconsistent knowledge bases, intensity of attacks in argumentation, different problems in the analysis of database management, and explainability. This paper demonstrates that all these examples are instantiations of computing measures of importance for a rather more general problem domain. The paper then shows that the best-known measures of importance can be computed for any reference set whenever one is given a monotonically increasing predicate that partitions the subsets of that reference set. As a consequence, the paper also proves that measures of importance can be devised in several domains, for some of which such measures have not yet been studied nor proposed. Furthermore, the paper highlights several research directions related with computing measures of importance.