Competitive analysis via benchmark decomposition

TL;DR

This paper introduces a unified framework for designing and analyzing prior-free competitive auctions across various settings, improving competitive ratios and simplifying analysis through a benchmark decomposition approach.

Contribution

It presents a novel decomposition lemma for bounding competitive ratios and applies it to enhance auction performance in multiple environments.

Findings

Multi-unit auction competitiveness improved from 4.84 to 3.24

Simpler auction design for downward-closed environments with ratio 6.5

Online auction competitive ratio narrowed to 4.12

Abstract

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.