AND Testing and Robust Judgement Aggregation

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 $1-ε$, 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 $δ$-close to either a constant function or an AND function, where $δ(ε) \to 0$ as $ε\to0$. This improves on a result of Nehama, who proved a similar statement in which $δ$ 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 $ε$-close to satisfying judgement aggregation, then it is $δ(ε)$-close to an oligarchy (the name for the AND function in social choice theory). This improves on Nehama's result, in which $δ$ decays polynomially with $n$. Our result follows from a more general one, in which we characterize approximate solutions to the eigenvalue equation $\mathrm T f = λg$, 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 $λg$ are close is close to an exact solution.