  We consider a setting where an auctioneer sells a single item to $n$
potential agents with {\em interdependent values}. That is, each agent has her
own private signal, and the valuation of each agent is a known function of all
$n$ private signals. This captures settings such as valuations for artwork, oil
drilling rights, broadcast rights, and many more.
  In the interdependent value setting, all previous work has assumed a
so-called {\sl single-crossing condition}. Single-crossing means that the
impact of agent $i$'s private signal, $s_i$, on her own valuation is greater
than the impact of $s_i$ on the valuation of any other agent. It is known that
without the single-crossing condition an efficient outcome cannot be obtained.
We study welfare maximization for interdependent valuations through the lens of
approximation.
  We show that, in general, without the single-crossing condition, one cannot
hope to approximate the optimal social welfare any better than the
approximation given by assigning the item to a random bidder. Consequently, we
introduce a relaxed version of single-crossing, {\sl $c$-single-crossing},
parameterized by $c\geq 1$, which means that the impact of $s_i$ on the
valuation of agent $i$ is at least $1/c$ times the impact of $s_i$ on the
valuation of any other agent ($c=1$ is single-crossing). Using this
parameterized notion, we obtain a host of positive results.
  We propose a prior-free deterministic mechanism that gives an
$(n-1)c$-approximation guarantee to welfare. We then show that a random version
of the proposed mechanism gives a prior-free universally truthful
$2c$-approximation to the optimal welfare for any concave $c$-single crossing
setting (and a $2\sqrt{n}c^{3/2}$-approximation in the absence of concavity).
We extend this mechanism to a universally truthful mechanism that gives
$O(c^2)$-approximation to the optimal revenue.
