AND Testing and Robust Judgement Aggregation

A function $fcolon{0,1}^nto {0,1}$ 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 $xland y = (x_1land y_1,ldots,x_nland y_n)$. We prove that if $fcolon {0,1}^n to {0,1}$ 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 $epsilonto0$. 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 Nehamas 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.