  We initiate the study of the social welfare loss caused by corrupt
auctioneers, both in single-item and multi-unit auctions. In our model, the
auctioneer may collude with the winning bidders by letting them lower their
bids in exchange for a (possibly bidder-dependent) fraction $\gamma$ of the
surplus. We consider different corruption schemes. In the most basic one, all
winning bidders lower their bid to the highest losing bid. We show that this
setting is equivalent to a $\gamma$-hybrid auction in which the payments are a
convex combination of first-price and the second-price payments. More
generally, we consider corruption schemes that can be related to
$\gamma$-approximate first-price auctions ($\gamma$-FPA), where the payments
recover at least a $\gamma$-fraction of the first-price payments. Our goal is
to obtain a precise understanding of the robust price of anarchy (POA) of such
auctions. If no restrictions are imposed on the bids, we prove a bound on the
robust POA of $\gamma$-FPA which is tight (over the entire range of $\gamma$)
for the single-item and the multi-unit auction setting. On the other hand, if
the bids satisfy the no-overbidding assumption a more fine-grained landscape of
the price of anarchy emerges, depending on the auction setting and the
equilibrium notion. Albeit being more challenging, we derive (almost) tight
bounds for both auction settings and several equilibrium notions, basically
leaving open some (small) gaps for the coarse-correlated price of anarchy only.
