Competitive analysis via benchmark decomposition

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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 β-competitive unlimited supply auction, the best previously known multi-unit auction is 2β-competitive. We design a (1+β)-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.