  This paper studies equilibrium quality of semi-separable position auctions
(known as the Ad Types setting) with greedy or optimal allocation combined with
generalized second-price (GSP) or Vickrey-Clarke-Groves (VCG) pricing. We make
three contributions: first, we give upper and lower bounds on the Price of
Anarchy (PoA) for auctions which use greedy allocation with GSP pricing, greedy
allocations with VCG pricing, and optimal allocation with GSP pricing. Second,
we give Bayes-Nash equilibrium characterizations for two-player, two-slot
instances (for all auction formats) and show that there exists both a revenue
hierarchy and revenue equivalence across some formats. Finally, we use
no-regret learning algorithms and bidding data from a large online advertising
platform and no-regret learning algorithms to evaluate the performance of the
mechanisms under semi-realistic conditions. For welfare, we find that the
optimal-to-realized welfare ratio (an empirical PoA analogue) is broadly better
than our upper bounds on PoA; For revenue, we find that the hierarchy in
practice may sometimes agree with simple theory, but generally appears
sensitive to the underlying distribution of bidder valuations.
