AND Testing and Robust Judgement Aggregation
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A function f{0,1}^n→{0,1} is called an approximate AND-homomorphism if choosing x, y∈{0,1}^n randomly, we have that f( x y) = f( x) f( y) with probability at least 1-ϵ, where x y = (x_1 y_1,...,x_n y_n). We prove that if f{0,1}^n →{0,1} is an approximate AND-homomorphism, then f is δ-close to either a constant function or an AND function, where δ(ϵ) → 0 as ϵ→0. 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 T f = λ g, where T is the downwards noise operator T f(x) = E_ y[f(x 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 T f and λ g are close is close to an exact solution.
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