  Firms and statistical agencies must protect the privacy of the individuals
whose data they collect, analyze, and publish. Increasingly, these
organizations do so by using publication mechanisms that satisfy differential
privacy. We consider the problem of choosing such a mechanism so as to maximize
the value of its output to end users. We show that mechanisms which add noise
to the statistic of interest--like most of those used in practice--are
generally not optimal when the statistic is a sum or average of magnitude data
(e.g., income). However, we also show that adding noise is always optimal when
the statistic is a count of data entries with a certain characteristic, and the
underlying database is drawn from a symmetric distribution (e.g., if
individuals' data are i.i.d.). When, in addition, data users have supermodular
payoffs, we show that the simple geometric mechanism is always optimal by using
a novel comparative static that ranks information structures according to their
usefulness in supermodular decision problems.
