  We study the following computational problem: for which values of $k$, the
majority of $n$ bits $\text{MAJ}_n$ can be computed with a depth two formula
whose each gate computes a majority function of at most $k$ bits? The
corresponding computational model is denoted by $\text{MAJ}_k \circ
\text{MAJ}_k$. We observe that the minimum value of $k$ for which there exists
a $\text{MAJ}_k \circ \text{MAJ}_k$ circuit that has high correlation with the
majority of $n$ bits is equal to $\Theta(n^{1/2})$. We then show that for a
randomized $\text{MAJ}_k \circ \text{MAJ}_k$ circuit computing the majority of
$n$ input bits with high probability for every input, the minimum value of $k$
is equal to $n^{2/3+o(1)}$. We show a worst case lower bound: if a
$\text{MAJ}_k \circ \text{MAJ}_k$ circuit computes the majority of $n$ bits
correctly on all inputs, then $k\geq n^{13/19+o(1)}$. This lower bound exceeds
the optimal value for randomized circuits and thus is unreachable for pure
randomized techniques. For depth $3$ circuits we show that a circuit with $k=
O(n^{2/3})$ can compute $\text{MAJ}_n$ correctly on all inputs.
