  The Generalized Second Price auction (GSP) has been widely used by search
engines to sell ad slots. Previous studies have shown that the pure Price Of
Anarchy (POA) of GSP is 1.25 when there are two ad slots and 1.259 when three
ad slots. For the cases with more than three ad slots, however, only some
untight upper bounds of the pure POA were obtained. In this work, we improve
previous results in two aspects: (1) We prove that the pure POA for GSP is
1.259 when there are four ad slots, and (2) We show that the pure POA for GSP
with more than four ad slots is also 1.259 given the bidders are ranked
according to a particular permutation.
