The Sets of Power
Joao Marques-Silva, Carlos Mencía, Raúl Mencía
TL;DR
The paper addresses unifying measures of relative importance across domains by modeling them with monotone predicates over a reference set $N$ and reducing many problems to Minimal Sets over a Monotone Predicate (MSMP). It shows that standard power indices such as Shapley-Shubik, Banzhaf, and Deegan-Packel can be instantiated as Uninstantiated Measures of Importance (UMIs) within this framework, using a common characteristic function $v(S)=ITE(WinC(S),1,0)$ and the criticality notion $Crit_s(i,S)$. The paper provides running examples (Dominating sets and Sardaukar training) and discusses case studies, along with exact and approximate computation methods for MSPs/MBPs and their use as explanations in XAI. It closes with directions for future work, suggesting new application domains (e.g., prime implicants, inconsistent linear programs) and the extension of UMIs to additional monotone predicates.
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.