  The Generalized Second Price (GSP) auction is the primary auction used for
monetizing the use of the Internet. It is well-known that truthtelling is not a
dominant strategy in this auction and that inefficient equilibria can arise. In
this paper we study the space of equilibria in GSP, and quantify the efficiency
loss that can arise in equilibria under a wide range of sources of uncertainty,
as well as in the full information setting. The traditional Bayesian game
models uncertainty in the valuations (types) of the participants. The
Generalized Second Price (GSP) auction gives rise to a further form of
uncertainty: the selection of quality factors resulting in uncertainty about
the behavior of the underlying ad allocation algorithm. The bounds we obtain
apply to both forms of uncertainty, and are robust in the sense that they apply
under various perturbations of the solution concept, extending to models with
information asymmetries and bounded rationality in the form of learning
strategies.
  We present a constant bound (2.927) on the factor of the efficiency loss
(\emph{price of anarchy}) of the corresponding game for the Bayesian model of
partial information about other participants and about ad quality factors. For
the full information setting, we prove a surprisingly low upper bound of 1.282
on the price of anarchy over pure Nash equilibria, nearly matching a lower
bound of 1.259 for the case of three advertisers. Further, we do not require
that the system reaches equilibrium, and give similarly low bounds also on the
quality degradation for any no-regret learning outcome. Our conclusion is that
the number of advertisers in the auction has almost no impact on the price of
anarchy, and that the efficiency of GSP is very robust with respect to the
belief and rationality assumptions imposed on the participants.
