AND Testing and Robust Judgement Aggregation

TL;DR

This paper characterizes functions that approximately satisfy AND-homomorphism properties, showing they are close to constant or AND functions, with implications for judgment aggregation and social choice theory.

Contribution

It proves that approximate AND-homomorphisms are close to constant or AND functions, improving previous bounds that depended on the number of variables.

Findings

Approximate solutions are close to exact solutions.

Functions close to satisfying the eigenvalue equation are near exact solutions.

Results apply to judgment aggregation, showing closeness to oligarchies.

Abstract

A function $f\colon\{0,1\}^n\to \{0,1\}$ is called an approximate AND-homomorphism if choosing ${\bf x},{\bf y}\in\{0,1\}^n$ randomly, we have that $f({\bf x}\land {\bf y}) = f({\bf x})\land f({\bf y})$ with probability at least $1-\epsilon$, where $x\land y = (x_1\land y_1,\ldots,x_n\land y_n)$. We prove that if $f\colon \{0,1\}^n \to \{0,1\}$ is an approximate AND-homomorphism, then $f$ is $\delta$-close to either a constant function or an AND function, where $\delta(\epsilon) \to 0$ as $\epsilon\to0$. This improves on a result of Nehama, who proved a similar statement in which $\delta$ depends on $n$. Our theorem implies a strong result on judgement aggregation in computational social choice. In the language of social choice, our result shows that if $f$ is $\epsilon$-close to satisfying judgement aggregation, then it is $\delta(\epsilon)$-close to an oligarchy (the name for the AND function in social choice theory). This improves on Nehama's result, in which $\delta$ decays polynomially with $n$. Our result follows from a more general one, in which we characterize approximate solutions to the eigenvalue equation $\mathrm T f = \lambda g$, where $\mathrm T$ is the downwards noise operator $\mathrm T f(x) = \mathbb{E}_{{\bf y}}[f(x \land {\bf y})]$, $f$ is $[0,1]$-valued, and $g$ is $\{0,1\}$-valued. We identify all exact solutions to this equation, and show that any approximate solution in which $\mathrm T f$ and $\lambda g$ are close is close to an exact solution.