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.

中文翻译：

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简单机制之间的收入差距很小

我们考虑微观经济学中的一个基本问题：将单个商品出售给多个潜在买家，这些买家的价值来自已知的独立和规则（不一定相同）分布。文献中有四种广泛使用和广泛研究的机制：{\sf Myerson Auction}~({\sf OPT})、{\sf Sequential Posted-Pricing}~({\sf SPM})、{\sf具有匿名储备的二价拍卖}~({\sf AR}) 和 {\sf 匿名定价}~({\sf AP})。{\sf OPT} 是收益最优但复杂的，在实践中也遇到了公平性等几个问题；{\sf AP} 是最简单的机制，但也是这四种机制中收益最低的；{\sf SPM} 和 {\sf AR} 具有中等复杂性和收入。我们探索这些机制之间的收入差距，每个都被定义为来自一对机制的收入之间的最大比率。我们建立了两个严格的界限和一个改进的界限： 1. {\sf SPM} vs.\ {\sf AP}：这个比率研究了定价方案中歧视的力量。我们得到了 $\mathcal{C^*} \approx 2.62$ 的紧比，缩小了之前 $\big[\frac{e}{e - 1}, e\big]$ 之间的差距。2. {\sf AR} vs.\ {\sf AP}：这个比率衡量拍卖方案与定价方案的相对力量，当不允许歧视时。我们达到了 $\frac{\pi^2}{6} \approx 1.64$ 的紧比率，关闭​​了先前已知的界限 $\big[\frac{e}{e - 1}, e\big]$。3. {\sf OPT} vs.\ {\sf AR}：这个比率量化了拍卖方案中歧视的能力，之前已知介于 $\big[2, e\big]$ 之间。Hartline 和 Roughgarden（2009 年）以及 Alaei 等人（2015 年）推测 2 美元的下限很紧。我们获得了更好的 $2.15$ 下限，从而反驳了这个猜想。