Aggregation of evaluations without unanimity

TL;DR

This paper extends foundational results on judgment aggregation to broader settings, characterizing all possible aggregators for symmetric predicates over binary inputs, unifying previous research directions.

Contribution

It combines and extends the work of Dokow-Holzman, Szegedy-Xu, and Mossel to characterize all aggregators for symmetric predicates over {0,1}.

Findings

All aggregators for symmetric predicates over {0,1} are characterized.

Unified framework for aggregation without unanimity assumption.

Extends previous theorems to broader predicate classes.

Abstract

Dokow and Holzman determined which predicates over $\{0, 1\}$ satisfy an analog of Arrow's theorem: all unanimous aggregators are dictatorial. Szegedy and Xu, extending earlier work of Dokow and Holzman, extended this to predicates over arbitrary finite alphabets. Mossel extended Arrow's theorem in an orthogonal direction, determining all aggregators without the assumption of unanimity. We bring together both threads of research by extending the results of Dokow-Holzman and Szegedy-Xu to the setting of Mossel. As an application, we determine, for each symmetric predicate over $\{0,1\}$, all of its aggregators.