Bi-capacities -- Part I: definition, M\"obius transform and interaction

TL;DR

This paper introduces bi-capacities as a bipolar generalization of capacities, detailing their mathematical structure, M"obius transform, derivatives, and connections to cooperative game theory, laying the groundwork for decision-making models like CPT.

Contribution

It defines bi-capacities, develops their M"obius transform and derivatives, and extends concepts like the Shapley value and interaction index to this new framework.

Findings

Bi-capacities generalize capacities to bipolar scales.

M"obius transform and derivatives are adapted for bi-capacities.

Framework connects to cooperative game theory and decision models.

Abstract

Bi-capacities arise as a natural generalization of capacities (or fuzzy measures) in a context of decision making where underlying scales are bipolar. They are able to capture a wide variety of decision behaviours, encompassing models such as Cumulative Prospect Theory (CPT). The aim of this paper in two parts is to present the machinery behind bi-capacities, and thus remains on a rather theoretical level, although some parts are firmly rooted in decision theory, notably cooperative game theory. The present first part is devoted to the introduction of bi-capacities and the structure on which they are defined. We define the M\"obius transform of bi-capacities, by just applying the well known theory of M\" obius functions as established by Rota to the particular case of bi-capacities. Then, we introduce derivatives of bi-capacities, by analogy with what was done for pseudo-Boolean functions (another view of capacities and set functions), and this is the key point to introduce the Shapley value and the interaction index for bi-capacities. This is done in a cooperative game theoretic perspective. In summary, all familiar notions used for fuzzy measures are available in this more general framework.