Classification aggregation: a quantitative impossibility theorem

2026-05-16

AI Summary

This study investigates whether dictatorship remains unavoidable in collective multi-class classification when surjectivity—requiring every class to be nonempty—is satisfied only with high probability and the aggregation function is far from constant. By extending impossibility theorems from social choice theory to a general setting featuring probabilistic surjectivity and non-degenerate aggregation functions, and leveraging the combinatorial framework of Alekseev and Filmus together with probabilistic methods under symmetric i.i.d. assumptions, the authors demonstrate that even if surjectivity holds merely with probability $1-\varepsilon$, dictatorship is still inevitable provided the aggregation function is not approximately constant. This result is further generalized to the setting of equivalence relation aggregation.

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 $1-ε$ (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.

Problem

• classification aggregation
• surjectivity
• impossibility theorem
• dictatorship
• aggregation functions

Innovation

• impossibility theorem
• classification aggregation
• surjectivity
• dictatorship
• symmetric i.i.d. distribution