  Suppose $Y^{n}$ is obtained by observing a uniform Bernoulli random vector
$X^{n}$ through a binary symmetric channel. Courtade and Kumar asked how large
the mutual information between $Y^{n}$ and a Boolean function
$\mathsf{b}(X^{n})$ could be, and conjectured that the maximum is attained by a
dictator function. An equivalent formulation of this conjecture is that
dictator minimizes the prediction cost in a sequential prediction of $Y^{n}$
under logarithmic loss, given $\mathsf{b}(X^{n})$. In this paper, we study the
question of minimizing the sequential prediction cost under a different
(proper) loss function - the quadratic loss. In the noiseless case, we show
that majority asymptotically minimizes this prediction cost among all Boolean
functions. We further show that for weak noise, majority is better than
dictator, and that for strong noise dictator outperforms majority. We
conjecture that for quadratic loss, there is no single sequence of Boolean
functions that is simultaneously (asymptotically) optimal at all noise levels.
