Iterated majority

Definition

Let [math]n = 3^k[/math]. The iterated majority function [math]f:\{-1,1\}^n \to \{-1,1\}[/math] is a recursively-defined variation on the majority function:

[math] f(x) = \begin{cases} \textrm{maj}(x), & \textrm{if}~ n = 3 \\ \textrm{maj}(f(x^{(1)}),f(x^{(2)}),f(x^{(3)}) & \textrm{otherwise}, \end{cases}[/math]

where [math]x^{(1)} = (x_1,x_2\ldots x_{n/3})[/math], [math]x^{(2)} = (x_{n/3+1},\ldots x_{2n/3})[/math] and [math]x^{(3)} = (x_{2n/3+1},\ldots x_{n})[/math].

Properties

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• TODO: add influence.