Interaction transform of set functions over a finite set

Abstract

The paper introduces a new transform of set functions over a finite set, which is linear and invertible as the well-known Moebius transform in combinatorics. This transform leads to the interaction index, a central concept in multicriteria decision making. The interaction index of a singleton happens to be the Shapley value of the set function or, in terms of cooperative game theory, of the value function of the game. Properties of this new transform are studied in detail, and some illustrative examples are given.