  We consider the single-item interdependent value setting, where there is a
monopolist, $n$ buyers, and each buyer has a private signal $s_i$ describing a
piece of information about the item. Each bidder $i$ also has a valuation
function $v_i(s_1,\ldots,s_n)$ mapping the (private) signals of all buyers to a
positive real number representing their value for the item. This setting
captures scenarios where the item's information is asymmetric or dispersed
among agents, such as in competitions for oil drilling rights, or in auctions
for art pieces. Due to the increased complexity of this model compared to
standard private values, it is generally assumed that each bidder's valuation
function $v_i$ is public knowledge. But in many situations, the seller may not
know how a bidder aggregates signals into a valuation. In this paper, we design
mechanisms that guarantee approximately-optimal social welfare while satisfying
ex-post incentive compatibility and individual rationality for the case where
the valuation functions are private to the bidders.
  When the valuations are public, it is possible for optimal social welfare to
be attained by a deterministic mechanism under a single-crossing condition. In
contrast, when the valuations are the bidders' private information, we show
that no finite bound can be achieved by any deterministic mechanism even under
single-crossing.
  Moreover, no randomized mechanism can guarantee better than an
$n$-approximation. We thus consider valuation functions that are submodular
over signals (SOS), introduced in the context of combinatorial auctions in a
recent breakthrough paper by Eden et al. [EC'19]. Our main result is an
$O(\log^2 n)$-approximation for buyers with private signals and valuations
under the SOS condition. We also give a tight $\Theta(k)$-approximation for the
case each agent's valuation depends on at most $k$ other signals even for
unknown $k$.
