  Designing revenue optimal auctions for selling an item to $n$ symmetric
bidders is a fundamental problem in mechanism design. Myerson (1981) shows that
the second price auction with an appropriate reserve price is optimal when
bidders' values are drawn i.i.d. from a known regular distribution. A
cornerstone in the prior-independent revenue maximization literature is a
result by Bulow and Klemperer (1996) showing that the second price auction
without a reserve achieves $(n-1)/n$ of the optimal revenue in the worst case.
  We construct a randomized mechanism that strictly outperforms the second
price auction in this setting. Our mechanism inflates the second highest bid
with a probability that varies with $n$. For two bidders we improve the
performance guarantee from $0.5$ to $0.512$ of the optimal revenue. We also
resolve a question in the design of revenue optimal mechanisms that have access
to a single sample from an unknown distribution. We show that a randomized
mechanism strictly outperforms all deterministic mechanisms in terms of worst
case guarantee.
