Applying Fourier Analysis to Judgment Aggregation

TL;DR

This paper extends classical social choice results to judgment aggregation using Fourier analysis, identifying logical functions like OR, AND, XOR as key non-dictatorial aggregation functions under certain constraints.

Contribution

It introduces the concept of 'normal pairs' of functions and applies Fourier analysis to characterize judgment aggregation rules in the logical setting.

Findings

Identifies OR, AND, XOR as the only non-dictatorial functions in certain cases

Provides a Fourier-analytic framework for judgment aggregation

Compares new results with existing theorems in the literature

Abstract

The classical Arrow's Theorem answers "how can $n$ voters obtain a collective preference on a set of outcomes, if they have to obey certain constraints?" We give an analogue in the judgment aggregation framework of List and Pettit, answering "how can $n$ judges obtain a collective judgment on a set of logical propositions, if they have to obey certain constraints?" We abstract this notion with the concept of "normal pairs" of functions on the Hamming cube, which we analyze with Fourier analysis and elementary combinatorics. We obtain judgment aggregation results in the special case of "symbol-complete" agendas and compare them with existing theorems in the literature. Amusingly, the non-dictatorial classes of functions that arise are precisely the classical logical functions OR, AND, and XOR.