  We study the price of anarchy of the generalized second-price auction where
bidders are value maximizers (i.e., autobidders). We show that in general the
price of anarchy can be as bad as $0$. For comparison, the price of anarchy of
running VCG is $1/2$ in the autobidding world. We further show a fined-grained
price of anarchy with respect to the discount factors (i.e., the ratios of
click probabilities between lower slots and the highest slot in each auction)
in the generalized second-price auction, which highlights the qualitative
relation between the smoothness of the discount factors and the efficiency of
the generalized second-price auction.
