  We propose a uniform approach for the design and analysis of prior-free
competitive auctions and online auctions. Our philosophy is to view the
benchmark function as a variable parameter of the model and study a broad class
of functions instead of a individual target benchmark. We consider a multitude
of well-studied auction settings, and improve upon a few previous results.
  (1) Multi-unit auctions. Given a $\beta$-competitive unlimited supply
auction, the best previously known multi-unit auction is $2\beta$-competitive.
We design a $(1+\beta)$-competitive auction reducing the ratio from $4.84$ to
$3.24$. These results carry over to matroid and position auctions.
  (2) General downward-closed environments. We design a $6.5$-competitive
auction improving upon the ratio of $7.5$. Our auction is noticeably simpler
than the previous best one.
  (3) Unlimited supply online auctions. Our analysis yields an auction with a
competitive ratio of $4.12$, which significantly narrows the margin of
$[4,4.84]$ previously known for this problem.
  A particularly important tool in our analysis is a simple decomposition
lemma, which allows us to bound the competitive ratio against a sum of
benchmark functions. We use this lemma in a "divide and conquer" fashion by
dividing the target benchmark into the sum of simpler functions.
