Classification aggregation: a quantitative impossibility theorem

TL;DR

This paper proves a quantitative impossibility theorem for classification aggregation, showing that under certain probabilistic surjectivity conditions, aggregation mechanisms must be nearly dictatorial unless they are constant.

Contribution

It extends existing impossibility results to probabilistic surjectivity and characterizes mechanisms that always produce surjective outcomes without assumptions.

Findings

Aggregation mechanisms are nearly dictatorial under probabilistic surjectivity.

Characterization of all mechanisms with always surjective outcomes.

Impossibility results for aggregating equivalence relations.

Abstract

A group of individuals wishes to classify $m$ objects into $n$ categories in such a way that no class is left empty, a condition known as surjectivity. The opinions of the individuals are aggregated separately for each object using an aggregation function that can depend on the object. Maniquet and Mongin showed that if the aggregation functions are unanimous and the outcome must always be surjective, then the aggregation mechanism is dictatorial. Cailloux et al. showed that the same holds even if unanimity is relaxed to citizen sovereignty (each object can be classified into any category). We show that similar results hold even if we only require the outcome to be surjective with probability $1-\epsilon$ (with respect to an arbitrary symmetric i.i.d. distribution), provided that the aggregation functions are far from being constant. On the way, we characterize all aggregation mechanisms whose outcome is always surjective without any assumptions on the aggregation functions. Our approach uses a general result of Alekseev and Filmus which has wider applicability. We illustrate this by proving a similar impossibility result for aggregating equivalence relations.