  We prove new positive and negative results concerning the existence of
truthful and individually rational mechanisms for purchasing private data from
individuals with unbounded and sensitive privacy preferences. We strengthen the
impossibility results of Ghosh and Roth (EC 2011) by extending it to a much
wider class of privacy valuations. In particular, these include privacy
valuations that are based on ({\epsilon}, {\delta})-differentially private
mechanisms for non-zero {\delta}, ones where the privacy costs are measured in
a per-database manner (rather than taking the worst case), and ones that do not
depend on the payments made to players (which might not be observable to an
adversary). To bypass this impossibility result, we study a natural special
setting where individuals have mono- tonic privacy valuations, which captures
common contexts where certain values for private data are expected to lead to
higher valuations for privacy (e.g. having a particular disease). We give new
mech- anisms that are individually rational for all players with monotonic
privacy valuations, truthful for all players whose privacy valuations are not
too large, and accurate if there are not too many players with too-large
privacy valuations. We also prove matching lower bounds showing that in some
respects our mechanism cannot be improved significantly.
