AND testing and robust judgement aggregation
Name
3357713.3384254.pdf
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1.01 MB
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Adobe PDF
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12c5419117b727b4ab95548eb8c4b02f
Author(s)
Filmus, Yuval
Lifshitz, Noam
Minzer, Dor
Mossel, Elchanan
Date Issued
2020
Journal
Proceedings of the Annual ACM Symposium on Theory of Computing
Publisher
ACM
Citation
Filmus, Yuval, Lifshitz, Noam, Minzer, Dor and Mossel, Elchanan. 2020. "AND testing and robust judgement aggregation." Proceedings of the Annual ACM Symposium on Theory of Computing.
Version
Final published version
Abstract
© 2020 ACM. A function fg¶{0,1}n→ {0,1} is called an approximate AND-homomorphism if choosing x,ygn uniformly at random, we have that f(xg§ y) = f(x)g§ f(y) with probability at least 1-ϵ, where xg§ y = (x1g§ y1,...,xng§ yn). We prove that if fg¶ {0,1}n → {0,1} is an approximate AND-homomorphism, then f is -close to either a constant function or an AND function, where (ϵ) → 0 as ϵ→ 0. This improves on a result of Nehama, who proved a similar statement in which δdepends 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 (ϵ)-close to an oligarchy (the name for the AND function in social choice theory). This improves on Nehama's result, in which δdecays polynomially with n. Our result follows from a more general one, in which we characterize approximate solutions to the eigenvalue equation f = λ g, where is the downwards noise operator f(x) = y[f(x g§ 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 f and λ g are close is close to an exact solution.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Massachusetts Institute of Technology. Institute for Data, Systems, and Society
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DOI of Published Version
https://doi.org/10.1145/3357713.3384254
