AND Testing and Robust Judgement Aggregation
Abstract
A function $f\colon\{0,1\}^n\to \{0,1\}$ is called an approximate ANDhomomorphism 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 ANDhomomorphism, 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2019
 arXiv:
 arXiv:1911.00159
 Bibcode:
 2019arXiv191100159F
 Keywords:

 Computer Science  Discrete Mathematics;
 Computer Science  Computer Science and Game Theory;
 Mathematics  Combinatorics
 EPrint:
 43 pages