Three candidate plurality is stablest for small correlations

Using the calculus of variations, we prove the following structure theorem for noise-stable partitions: a partition of n-dimensional Euclidean space into m disjoint sets of fixed Gaussian volumes that maximise their noise stability must be $(m-1)$-dimensional, if $m-1\leq n$. In particular, the maxi...

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Bibliographic Details
Published in:Forum of mathematics. Sigma 2021-01, Vol.9, Article e65
Main Authors: Heilman, Steven, Tarter, Alex
Format: Article
Language:English
Subjects:
49Q05
53A10
60E15
Algorithms
Approximation
Calculus of variations
Corruption
Dimensional stability
Elections
Euclidean geometry
Semidefinite programming
Theorems
Theoretical Computer Science
Voter behavior
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Summary:Using the calculus of variations, we prove the following structure theorem for noise-stable partitions: a partition of n-dimensional Euclidean space into m disjoint sets of fixed Gaussian volumes that maximise their noise stability must be $(m-1)$-dimensional, if $m-1\leq n$. In particular, the maximum noise stability of a partition of m sets in $\mathbb {R}^{n}$ of fixed Gaussian volumes is constant for all n satisfying $n\geq m-1$. From this result, we obtain: (i)A proof of the plurality is stablest conjecture for three candidate elections, for all correlation parameters $\rho $ satisfying $0
ISSN:2050-5094
2050-5094
DOI:10.1017/fms.2021.56