简单经济几乎是最优的
Simple Economies are Almost Optimal
摘要：
考虑一个打算拍卖某件物品的卖家。卖方可以投入资金和精力在不同的细分市场上做广告，以便招募$n$投标人参加拍卖。或者，卖方可以有一个更便宜和集中的营销业务，并从一个单一的细分市场招募相同数量的投标人。卖家应该选择哪种营销方式？更正式地说，让$D=\{\mathcal D_1,\ldots, \mathcal D_n\}$是一组分布。我们的主要结果表明，总是有$\mathcal D_i\in D$，使得可以从$n$投标人中提取的收入，其中每个投标人的价值是独立地从$\mathcal D_i$中提取的，至少是任何可能的投标人组合可以获得的收入的$\frac 1 2 \cdot (1-\frac 1 e)$，其中每个投标人的价值是从属于$D$的一些（可能不同的）分布中提取的。接下来我们考虑拍卖人不能使用最优拍卖而必须使用第二价格拍卖的情况。我们表明，总是有$\mathcal D_i\in D$，这样，如果所有投标人的价值是独立地从$\mathcal D_i$然后运行第二价格拍卖保证了一个恒定的分数的收入，可以通过第二价格拍卖的任何可能的混合投标人。最后，我们表明，对于任何$\varepsilon>0$存在一个功能$f$，只依赖于$\varepsilon$（特别是，该功能不依赖于$n$或$D$的集合），这样，招聘$n$投标人有最多$f(\varepsilon)$不同的分布，所有从$D$，保证$(1-\varepsilon)$ -部分的收入，可以通过第二价格拍卖的任何可能的混合投标人。
英文摘要：
Consider a seller that intends to auction some item. The seller can invest money and effort in advertising in different market segments in order to recruit $n$ bidders to the auction. Alternatively, the seller can have a much cheaper and focused marketing operation and recruit the same number of bidders from a single market segment. Which marketing operation should the seller choose? More formally, let $D=\{\mathcal D_1,\ldots, \mathcal D_n\}$ be a set of distributions. Our main result shows that there is always $\mathcal D_i\in D$ such that the revenue that can be extracted from $n$ bidders, where the value of each is independently drawn from $\mathcal D_i$, is at least $\frac 1 2 \cdot (1-\frac 1 e)$ of the revenue that can be obtained by any possible mix of bidders, where the value of each bidder is drawn from some (possibly different) distribution that belongs to $D$. We next consider situations in which the auctioneer cannot use the optimal auction and is required to use a second price auction. We show that there is always $\mathcal D_i\in D$ such that if the value of all bidders is independently drawn from $\mathcal D_i$ then running a second price auction guarantees a constant fraction of the revenue that can be obtained by a second-price auction by any possible mix of bidders. Finally, we show that for any $\varepsilon>0$ there exists a function $f$ that depends only on $\varepsilon$ (in particular, the function does not depend on $n$ or on the set $D$), such that recruiting $n$ bidders which have at most $f(\varepsilon)$ different distributions, all from $D$, guarantees $(1-\varepsilon)$-fraction of the revenue that can be obtained by a second-price auction by any possible mix of bidders.
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