Continuous Classification Aggregation

TL;DR

This paper proves that in a fuzzy group identification setting with a continuum of individuals $I=[0,1]$ and $m\ge 3$ objects classified into $2\le p\le m$ types, any aggregation rule that is optimal, independent, and zero-unanimous must take the form of a weighted arithmetic mean, extending finite-agent results to a continuum via a Riesz representation approach. The authors establish a rigorous characterization by leveraging Cauchy-functional arguments and measure representations, showing that social classifications arise from integrating individual classifications against a probability measure on $I$. They also provide a complete characterization for the special case $m=p=2$, and derive corollaries linking the axioms to anonymity and non-dictatorship. The work bridges fuzzy set theory, social choice, and functional-analytic methods, offering a robust foundation for continuum-based group identification models with practical implications for fair and consistent aggregation of vague classifications.

Abstract

We prove that any optimal, independent, and zero unanimous fuzzy classification aggregation function of a continuum of individual classifications of $m\ge 3$ objects into $2\le p\le m$ types must be a weighted arithmetic mean. We also provide a characterization for the case when $m=p=2$.

Figures (3)

• Figure 1: $c(x_1)$.
• Figure 2: $c(x_2)$.
• Figure 3: Logic Flow

Theorems & Definitions (6)

• proof
• proof
• proof
• proof
• proof
• proof