Importance in systems with interval decisions

TL;DR

This paper extends the concept of importance measures from binary voting to systems with interval decisions, analyzing how individual agents influence the aggregated system state.

Contribution

It introduces two importance measures for systems with interval decisions based on classical voting power indices and analyzes their properties.

Findings

Two importance measures are proposed and analyzed.

Results are provided for specific classes of aggregation functions.

The measures generalize power indices to interval decision systems.

Abstract

Given a system where the real-valued states of the agents are aggregated by a function to a real-valued state of the entire system, we are interested in the influence or importance of the different agents for that function. This generalizes the notion of power indices for binary voting systems to decisions over interval policy spaces and has applications in economics, engineering, security analysis, and other disciplines. Here, we study the question of importance in systems with interval decisions. Based on the classical Shapley-Shubik and Penrose-Banzhaf index, from binary voting, we motivate and analyze two importance measures. Additionally, we present some results for parametric classes of aggregation functions.