Values of Games for Information Decomposition

TL;DR

This paper explores the mathematical foundations of information decomposition using coalitional game theory, extending axiomatic characterizations to more general settings and solution concepts.

Contribution

It generalizes the concept of values in coalitional games to boolean algebras, providing new axiomatic characterizations relevant to information decomposition.

Findings

Extended axiomatic framework for coalitional game values

Analyzed classes of values like random-order and sharing values

Provided theoretical insights into information attribution methods

Abstract

The information decomposition problem requires an additive decomposition of the mutual information between the input and target variables into nonnegative terms. The recently introduced solution to this problem, Information Attribution, involves the Shapley-style value measuring the influence of predictors in the coalitional game associated with the joint probability distribution of the input random vector and the target variable. Motivated by the original problem, we consider a general setting of coalitional games where the players form a boolean algebra, and the coalitions are the corresponding down-sets. This enables us to study in detail various single-valued solution concepts, called values. Namely, we focus on the classes of values that can represent very general alternatives to the solution of the information decomposition problem, such as random-order values or sharing values. We extend the axiomatic characterization of some classes of values that were known only for the standard coalitional games.