Before revision:  Along the way we shed light on the solutions of optimization problems over distributions subject to a mean-preserving contraction constraint and additional constraints which might be of independent interest  .
After revision:  Along the way we shed light on the solutions of optimization problems over distributions subject to a mean-preserving contraction constraint and additional constraints which might be of independent interest . Finally, we establish that public signals in general achieve only a 1/n share of the optimal value for the designer .