Tight Revenue Gaps among Simple Mechanisms

We consider a fundamental problem in microeconomics: selling a single item to a number of potential buyers, whose values are drawn from known independent and regular (not necessarily identical) distributions. There are four widely-used and widely-studied mechanisms in the literature: {\sf Myerson Auction}~({\sf OPT}), {\sf Sequential Posted-Pricing}~({\sf SPM}), {\sf Second-Price Auction with Anonymous Reserve}~({\sf AR}), and {\sf Anonymous Pricing}~({\sf AP}). {\sf OPT} is revenue-optimal but complicated, which also experiences several issues in practice such as fairness; {\sf AP} is the simplest mechanism, but also generates the lowest revenue among these four mechanisms; {\sf SPM} and {\sf AR} are of intermediate complexity and revenue. We explore revenue gaps among these mechanisms, each of which is defined as the largest ratio between revenues from a pair of mechanisms. We establish two tight bounds and one improved bound: 1. {\sf SPM} vs.\ {\sf AP}: this ratio studies the power of discrimination in pricing schemes. We obtain the tight ratio of $\mathcal{C^*} \approx 2.62$, closing the gap between $\big[\frac{e}{e - 1}, e\big]$ left before. 2. {\sf AR} vs.\ {\sf AP}: this ratio measures the relative power of auction scheme vs.\ pricing scheme, when no discrimination is allowed. We attain the tight ratio of $\frac{\pi^2}{6} \approx 1.64$, closing the previously known bounds $\big[\frac{e}{e - 1}, e\big]$. 3. {\sf OPT} vs.\ {\sf AR}: this ratio quantifies the power of discrimination in auction schemes, and is previously known to be somewhere between $\big[2, e\big]$. The lower-bound of $2$ was conjectured to be tight by Hartline and Roughgarden (2009) and Alaei et al.\ (2015). We acquire a better lower-bound of $2.15$, and thus disprove this conjecture.