A Counterexample to the "Majority is Least Stable" Conjecture

In their document Real Analysis in Computer Science: A Collection of Open Problems, Filmus et al. [FHHMOSWW14] present the following “Majority is Least Stable” conjecture due to Benjamini, Kalai and Schramm [BKS99]: Conjecture. Let f : {−1, 1} → {−1, 1} be a linear threshold function for n odd. Then, for all ρ ∈ [0, 1], Stabρ[f ] ≥ Stabρ[Majn]. In this note, we provide a simple counterexample to this conjecture, even when n = 5. We begin by noting that for an unbiased linear threshold function f : {−1, 1} → {−1, 1}, the statement W [f ] < W 1[Majn] would disprove the conjecture. An example of such an LTF is provided by f : {−1, 1} → {−1, 1} given by f(x1, x2, x3, x4, x5) = sgn(2x1 + 2x2 + x3 + x4 + x5) To show this, we explicitly compute the degree-1 Fourier coefficients of f and of Maj 5 . Since both these functions are monotone, computing f̂(i) is the same as computing the influence Infi[f ]. Moreover, by symmetry, it suffices to compute Inf1[Maj5], Inf1[f ] and Inf3[f ]. Since coordinate 1 is influential for Maj 5 iff x2 + x3 + x4 + x5 = 0, it follows that