  One of the major drawbacks of the celebrated VCG auction is its low (or zero)
revenue even when the agents have high value for the goods and a {\em
competitive} outcome could have generated a significant revenue. A competitive
outcome is one for which it is impossible for the seller and a subset of buyers
to `block' the auction by defecting and negotiating an outcome with higher
payoffs for themselves. This corresponds to the well-known concept of {\em
core} in cooperative game theory.
  In particular, VCG revenue is known to be not competitive when the goods
being sold have complementarities. A bottleneck here is an impossibility result
showing that there is no auction that simultaneously achieves competitive
prices (a core outcome) and incentive-compatibility.
  In this paper we try to overcome the above impossibility result by asking the
following natural question: is it possible to design an incentive-compatible
auction whose revenue is comparable (even if less) to a competitive outcome?
Towards this, we define a notion of {\em core-competitive} auctions. We say
that an incentive-compatible auction is $\alpha$-core-competitive if its
revenue is at least $1/\alpha$ fraction of the minimum revenue of a
core-outcome. We study the Text-and-Image setting. In this setting, there is an
ad slot which can be filled with either a single image ad or $k$ text ads. We
design an $O(\ln \ln k)$ core-competitive randomized auction and an
$O(\sqrt{\ln(k)})$ competitive deterministic auction for the Text-and-Image
setting. We also show that both factors are tight.
