推导说明 / Derivation note 上式倒数第二步用 \(\hat t_i(\theta_i)=\hat\phi_i(\theta_i)\theta_i-U_i(\theta_i)\)；最后一步假设 \(\sum_i\hat\phi_i(\theta_i)=1\)。若 \(\sum_i\hat\phi_i(\theta_i)=p<1\)（如卖家不售时为 0），则需再减一项 \((1-p)\theta_0\)；因 \(p\) 由卖家选定、为常数，不出现在最大化决策中，可忽略。In the above, the second-to-last step uses \(\hat t_i(\theta_i)=\hat\phi_i(\theta_i)\theta_i-U_i(\theta_i)\); the last step assumes \(\sum_i\hat\phi_i(\theta_i)=1\). If \(\sum_i\hat\phi_i(\theta_i)=p<1\) (e.g. 0 when the seller doesn't sell), then we subtract one more term \((1-p)\theta_0\); since \(p\) is picked by the seller and is a constant, it won't appear in the maximization decisions, so it can be ignored.

最优分配 / The optimal allocation 由 (27.1)，拍卖人最优地把概率 1（即 \(\phi_i(\theta_i,\theta_{-i})=1\)）放在 \(J_i(\theta_i)-\theta_0\) 最高且 \(\ge0\) 的主体 \(i\) 上、其余主体放 0（\(\phi_j=0,j\ne i\)）。同时最优拍卖中 \(U_i(\underline\theta_i)=0\) 对 \(\forall i\)。By (27.1), the auctioneer optimally puts probability 1 (i.e. \(\phi_i(\theta_i,\theta_{-i})=1\)) on the agent \(i\) whose \(J_i(\theta_i)-\theta_0\) is highest and \(\ge0\), and 0 on everyone else (\(\phi_j=0,j\ne i\)). Also, in the optimal auction \(U_i(\underline\theta_i)=0\) for \(\forall i\).

定理 27.1（最优拍卖，Myerson 1981）/ Theorem 27.1 (Optimal auction, Myerson 1981) 最优拍卖满足：(1) \(U_i(\underline\theta_i)=0\) 对 \(\forall i\)；(2) 当 \(J_i(\theta_i)>\max_{j\ne i}J_j(\theta_j)\) 且 \(J_i(\theta_i)\ge\theta_0\) 时 \(\phi_i=1\)；当 \(J_i(\theta_i)<\max_{j\ne i}J_j(\theta_j)\) 或 \(J_i(\theta_i)<\theta_0\) 时 \(\phi_i=0\)；当 \(J_i(\theta_i)=\max_{j\ne i}J_j(\theta_j)\ge\theta_0\) 且最高虚拟类型有 \(K\) 个平局时 \(\phi_i=\tfrac1K\)。证明直接由 (27.1) 得出：事前 \(J_i(\theta_i)\) 拍卖人不知，但仍可写下总把物品给 \(J_i(\theta_i)\) 最高（且不低于 \(\theta_0\)）者的最优机制，使期望收入最大。\(\blacksquare\)An optimal auction satisfies: (1) \(U_i(\underline\theta_i)=0\) for \(\forall i\); (2) \(\phi_i(\theta_i,\theta_{-i})=\begin{cases}1 & J_i(\theta_i)>\max_{j\ne i}J_j(\theta_j)\text{ and }J_i(\theta_i)\ge\theta_0\\ 0 & J_i(\theta_i)<\max_{j\ne i}J_j(\theta_j)\text{ or }J_i(\theta_i)<\theta_0\\ \tfrac1K & J_i(\theta_i)=\max_{j\ne i}J_j(\theta_j)\ge\theta_0\text{, and there are }K\text{ ties in highest virtual types}\end{cases}\). The proof is directly by (27.1): ex-ante \(J_i(\theta_i)\) is not known by the auctioneer, but he can still write down the optimal mechanism that always awards the good to the agent whose \(J_i(\theta_i)\) is maximum (and no less than \(\theta_0\)), maximizing the expected revenue. \(\blacksquare\)

\(MR_i(\theta_i)=J_i(\theta_i)\) / Marginal revenue equals virtual type 在价格 \(p=\theta_i\) 处求边际收入即得 \(MR_i(\theta_i)=J_i(\theta_i)\)。它表示把 \(q\) 从 0 增到 1（以概率 1 把物品卖给类型 \(\theta_i\) 的主体 \(i\)）所产生的收入；该 \(MR\) 低于 \(\theta_i\)，因为是价格 \(\theta_i\) 减去"对那些实际类型更高的情形把价格降到 \(\theta_i\) 所造成的损失"。要让拍卖人确定地把物品卖给 \(i\)，其产生的边际收入至少应达到边际成本 \(\theta_0\)；又因只有一件物品，拍卖人会卖给产生最高边际收入的人。Evaluating the marginal revenue at price \(p=\theta_i\) gives \(MR_i(\theta_i)=J_i(\theta_i)\). It is the revenue generated by increasing \(q\) from 0 to 1 (selling the good to agent \(i\) with type \(\theta_i\) with probability 1); this \(MR\) is lower than \(\theta_i\) because it is the price \(\theta_i\) minus the loss from reducing the price to \(\theta_i\) for cases where agent \(i\) actually has a higher type. To make the auctioneer sell to agent \(i\) for sure, the marginal revenue generated should be at least the marginal cost \(\theta_0\); and since there is only one unit, the auctioneer sells to the agent generating the highest marginal revenue.

命题 27.1 / Proposition 27.1 \(1-F_i(\theta_i)\) 的对数凹性（即 \(\ln(1-F_i(\theta_i))\) 凹）等价于单调风险率条件，二者都蕴含正则性条件 \(J_i(\theta_i)\) 关于 \(\theta_i\) 递增。Log-concavity of \(1-F_i(\theta_i)\) (i.e. concavity of \(\ln(1-F_i(\theta_i))\)) is equivalent to the monotone hazard rate condition, both of which imply the regularity condition that \(J_i(\theta_i)\) increases in \(\theta_i\).

证明 / Proof \(\ln(1-F_i(\theta_i))\) 的二阶导：\(\frac{d}{d\theta_i}\ln(1-F_i)=\frac{-f_i}{1-F_i}\)，\(\frac{d^2}{d\theta_i^2}\ln(1-F_i)=\frac{d}{d\theta_i}\frac{-f_i}{1-F_i}\)。对数凹性蕴含 \(\frac{d}{d\theta_i}\frac{-f_i}{1-F_i}<0\Leftrightarrow\frac{d}{d\theta_i}\frac{f_i}{1-F_i}>0\Leftrightarrow\frac{d}{d\theta_i}\frac{1-F_i}{f_i}<0\)，恰为单调风险率条件。又 \(J_i'(\theta_i)=1-\frac{d}{d\theta_i}\frac{1-F_i}{f_i}\)，由对数凹（或 MHRC）\(\frac{d}{d\theta_i}\frac{1-F_i}{f_i}<0\)，故 \(J_i'(\theta_i)>0\)。\(\blacksquare\)The second derivative of \(\ln(1-F_i(\theta_i))\): \(\frac{d}{d\theta_i}\ln(1-F_i)=\frac{-f_i}{1-F_i}\), \(\frac{d^2}{d\theta_i^2}\ln(1-F_i)=\frac{d}{d\theta_i}\frac{-f_i}{1-F_i}\). Log-concavity implies \(\frac{d}{d\theta_i}\frac{-f_i}{1-F_i}<0\Leftrightarrow\frac{d}{d\theta_i}\frac{f_i}{1-F_i}>0\Leftrightarrow\frac{d}{d\theta_i}\frac{1-F_i}{f_i}<0\), exactly the monotone hazard rate condition. And \(J_i'(\theta_i)=1-\frac{d}{d\theta_i}\frac{1-F_i}{f_i}\); by log-concavity (or MHRC) \(\frac{d}{d\theta_i}\frac{1-F_i}{f_i}<0\), so \(J_i'(\theta_i)>0\). \(\blacksquare\)

保留类型 / Reserve type 拍卖人的保留价（成本）\(\theta_0\) 对应每个主体 \(i\) 的一个保留类型 \(r_i^\star\)，满足 \(J_i(r_i^\star)=\theta_0\)。由正则性 \(J_i'(\theta_i)>0\)，\(J_i(\theta_i)\ge\theta_0\Leftrightarrow\theta_i\ge r_i^\star\)。于是中标概率函数可重写为：当 \(\theta_i>\max_{j\ne i}\theta_j\) 且 \(\theta_i\ge r_i^\star\) 时 \(\phi_i=1\)；当 \(\theta_i<\max_{j\ne i}\theta_j\) 或 \(\theta_iThe auctioneer's reserve price (cost) \(\theta_0\) corresponds to a reserve type \(r_i^\star\) for each agent \(i\), satisfying \(J_i(r_i^\star)=\theta_0\). By regularity \(J_i'(\theta_i)>0\), \(J_i(\theta_i)\ge\theta_0\Leftrightarrow\theta_i\ge r_i^\star\). So the winning-probability function can be rewritten as: \(\phi_i=1\) when \(\theta_i>\max_{j\ne i}\theta_j\) and \(\theta_i\ge r_i^\star\); \(\phi_i=0\) when \(\theta_i<\max_{j\ne i}\theta_j\) or \(\theta_i

命题 27.2 / Proposition 27.2 在正则性 \(J_i'(\theta_i)>0\)（\(\forall i\)）与对称分布 \(F_i=F_j\) 下，记拍卖人保留价（成本）\(\theta_0\)，则只要正确设定保留类型 \(r^\star\) 使 \(J(r^\star)=r^\star-\frac{1-F(r^\star)}{f(r^\star)}=\theta_0\)，一价拍卖与二价拍卖都是最优的。Under regularity \(J_i'(\theta_i)>0\) (\(\forall i\)) and symmetric distribution \(F_i=F_j\), denoting the auctioneer's reserve price (cost) \(\theta_0\), both the first-price and second-price auction are optimal if the reserve type \(r^\star\) is set correctly such that \(J(r^\star)=r^\star-\frac{1-F(r^\star)}{f(r^\star)}=\theta_0\).

证明 / Proof §24.3 证均衡竞价 \(\bar b(\theta)=\theta-\int_0^\theta(F(x)/F(\theta))^{n-1}dx\) 关于 \(\theta\) 严格递增，故对称一价拍卖以概率 1 把物品分给最高类型、其余 0；§24.4 二价拍卖出真实类型也严格递增，同样把物品给最高类型。两者（讲真话）中最低类型剩余 \(U(\underline\theta)=0\)、中标概率为零、不付款。只要 \(r^\star\) 正确设为 \(J(r^\star)=\theta_0\)，由定理 27.1 二者皆最优。\(\blacksquare\)§24.3 showed the equilibrium bid \(\bar b(\theta)=\theta-\int_0^\theta(F(x)/F(\theta))^{n-1}dx\) is strictly increasing in \(\theta\), so the symmetric first-price auction awards the good to the highest type with probability 1 and 0 to others; §24.4 the second-price auction with truthful bidding is also strictly increasing, awarding the good to the highest type. In both (truth-telling), the lowest type's surplus \(U(\underline\theta)=0\), with zero winning probability and no payment. As long as \(r^\star\) is set correctly as \(J(r^\star)=\theta_0\), by Theorem 27.1 both are optimal. \(\blacksquare\)

例 27.1（最优拍卖的自然扭曲）/ Example 27.1 (Natural distortion) 两买家，\(\theta_1\sim\text{Unif}[0,2]\)、\(\theta_2\sim\text{Unif}[1,3]\)，拍卖人保留价 \(\theta_0=0\)。社会最优：卖家总售出（\(\theta_1\ge\theta_0\)、\(\theta_2>\theta_0\) 概率 1）；\(\theta_1>\theta_2\) 时卖给买家 1、\(\theta_1<\theta_2\) 时卖给买家 2。最优拍卖：\(J_1(\theta_1)=\theta_1-\frac{1-\theta_1/2}{1/2}=2\theta_1-2\)，\(r_1^\star\)：\(J_1(r_1^\star)=0\Rightarrow r_1^\star=1\)；\(J_2(\theta_2)=\theta_2-\frac{1-(\theta_2-1)/2}{1/2}=2\theta_2-3\)，\(r_2^\star\)：\(J_2(r_2^\star)=0\Rightarrow r_2^\star=3/2\)。故 \(0<\theta_1<1\) 且 \(1<\theta_2<3/2\) 时两者都比拍卖人更想要物品，但拍卖人不售（第一类扭曲）。卖给买家 1 当 \(J_1>J_2\)：\(2\theta_1-2>2\theta_2-3\Leftrightarrow\theta_1+\tfrac12>\theta_2\)，不等价于社会最优（\(\theta_1>\theta_2\)）；对 \(\theta_1<\theta_2<\theta_1+\tfrac12\)，社会最优应给买家 2，但最优拍卖给买家 1（第二类扭曲）。Two bidders, \(\theta_1\sim\text{Unif}[0,2]\), \(\theta_2\sim\text{Unif}[1,3]\), auctioneer reserve \(\theta_0=0\). Socially optimal: the seller always sells (\(\theta_1\ge\theta_0\), \(\theta_2>\theta_0\) with probability 1); sell to bidder 1 if \(\theta_1>\theta_2\), to bidder 2 if \(\theta_1<\theta_2\). Optimal auction: \(J_1(\theta_1)=\theta_1-\frac{1-\theta_1/2}{1/2}=2\theta_1-2\), \(r_1^\star\): \(J_1(r_1^\star)=0\Rightarrow r_1^\star=1\); \(J_2(\theta_2)=\theta_2-\frac{1-(\theta_2-1)/2}{1/2}=2\theta_2-3\), \(r_2^\star\): \(J_2(r_2^\star)=0\Rightarrow r_2^\star=3/2\). So when \(0<\theta_1<1\) and \(1<\theta_2<3/2\), both want the good more than the auctioneer, but the auctioneer won't sell (first distortion). Sell to bidder 1 when \(J_1>J_2\): \(2\theta_1-2>2\theta_2-3\Leftrightarrow\theta_1+\tfrac12>\theta_2\), not equivalent to socially optimal (\(\theta_1>\theta_2\)); for \(\theta_1<\theta_2<\theta_1+\tfrac12\), socially optimally bidder 2 should get the good, but the optimal auction awards it to bidder 1 (second distortion).

图 29 / Figure 29（社会最优与最优拍卖结果的分歧，已转述 / Divergence, paraphrased） 在 \(\theta_1\)（横轴 $[0,2]$）–\(\theta_2\)（纵轴 $[1,3]$）平面上，两条直线 \(\theta_2=\theta_1+\tfrac12\)（红）与 \(\theta_2=\theta_1\)（黑）及虚线 \(\theta_1=1\)、\(\theta_2=\tfrac32\) 把区域分为 A、B、C、D。社会最优：所有区域都应售出、按 \(\theta_1\) 与 \(\theta_2\) 大小分配；最优拍卖：在区域 B（\(\theta_1<1\) 且 \(\theta_2<\tfrac32\)）不售；社会最优在区域 D 卖给买家 1、在 A/B/C 卖给买家 2，而最优拍卖在 C、D 卖给买家 1、在 A 卖给买家 2。In the \(\theta_1\) (horizontal $[0,2]$)–\(\theta_2\) (vertical $[1,3]\() plane, the two lines \)\theta_2=\theta_1+\tfrac12$ (red) and \(\theta_2=\theta_1\) (black), and the dashed lines \(\theta_1=1\), \(\theta_2=\tfrac32\), partition the area into A, B, C, D. Socially optimal: sell in all regions, allocated by the comparison of \(\theta_1\) and \(\theta_2\); optimal auction: doesn't sell in region B (\(\theta_1<1\) and \(\theta_2<\tfrac32\)); socially optimally sell to bidder 1 in region D and to bidder 2 in A/B/C, but the optimal auction sells to bidder 1 in C, D and to bidder 2 in A.

例 27.2（风险厌恶买家的一价/二价拍卖，JR 习题 9.4）/ Example 27.2 (FPA and SPA with risk-averse bidders, JR Exercise 9.4) \(n\) 个买家参与一价拍卖，估值独立抽自 $[0,1]$、连续严格正密度 \(f\)，vNM 效用 \((\theta_i-b_i)^{1/\alpha}\)（\(\alpha\ge1\) 共同常数，\(\alpha=1\) 风险中性、\(\alpha>1\) 风险厌恶）。记对称均衡 \(\bar b_\alpha(\theta)\)。(a) 记报告 \(\hat\theta\)、真实 \(\theta\) 的期望效用 \(U(\hat\theta\mid\theta)=F(\hat\theta)^{n-1}(\theta-\bar b_\alpha(\hat\theta))^{1/\alpha}\)（中标概率 \(F(\hat\theta)^{n-1}\) 由独立性），由 IC 须在 \(\hat\theta=\theta\) 处最大。(b) 因 \(\alpha\ge1\)，\(U(\hat\theta\mid\theta)^\alpha=F(\hat\theta)^{\alpha(n-1)}(\theta-\bar b_\alpha(\hat\theta))\) 是 \(U\) 的严格增变换，也在 \(\hat\theta=\theta\) 处最大。(c) 故"\(n\) 个风险厌恶买家、估值分布 \(F\)"的一价拍卖等价于"\(n\) 个风险中性买家、估值分布 \(\hat F=F^\alpha\)"的一价拍卖；用 §24.3 风险中性解得 \(\bar b_\alpha(\theta)=\theta-\int_0^\theta(F(x)/F(\theta))^{\alpha(n-1)}\,dx\)。(d) \(\frac{\partial\bar b_\alpha(\theta)}{\partial\alpha}=\int_0^\theta(n-1)(F(x)/F(\theta))^{\alpha(n-1)}(-\ln(F(x)/F(\theta)))\,dx>0\)，故 \(\bar b_\alpha\) 关于 \(\alpha\) 严格递增——买家越风险厌恶越偏好"更确定但价值更低"的结果（提高中标概率、降低中标剩余），卖家一价收入随风险厌恶上升。(e) 二价拍卖中讲真话仍占优，故 \(ER^{II}_{ra}=ER^{II}_{rn}\)；由 §24.4 \(ER^{II}_{rn}=ER^{I}_{rn}\)，又由 (d) \(ER^{I}_{ra}>ER^{I}_{rn}\)，故 \(ER^{I}_{ra}>ER^{II}_{ra}\)——风险厌恶下一价拍卖收入高于二价，两标准拍卖不再等收入。(f) \(\alpha\to\infty\)（无限风险厌恶）时 \(\lim_{\alpha\to\infty}\bar b_\alpha(\theta)=\theta\)，所有人出真实类型、赢家付自己出价，期望收入 \(\mathbb{E}[\max\{\theta_1,\dots,\theta_n\}]\)——第一最优，估值最高者得物品、付出全部估值、零剩余。\(n\) bidders in a first-price auction, values independently drawn from $[0,1]$ with continuous strictly positive density \(f\), vNM utility \((\theta_i-b_i)^{1/\alpha}\) (\(\alpha\ge1\) a common constant, \(\alpha=1\) risk-neutral, \(\alpha>1\) risk-averse). Denote the symmetric equilibrium \(\bar b_\alpha(\theta)\). (a) The expected utility of reporting \(\hat\theta\) at true \(\theta\) is \(U(\hat\theta\mid\theta)=F(\hat\theta)^{n-1}(\theta-\bar b_\alpha(\hat\theta))^{1/\alpha}\) (winning probability \(F(\hat\theta)^{n-1}\) by independence), maximized at \(\hat\theta=\theta\) by IC. (b) Since \(\alpha\ge1\), \(U(\hat\theta\mid\theta)^\alpha=F(\hat\theta)^{\alpha(n-1)}(\theta-\bar b_\alpha(\hat\theta))\) is a strictly increasing transformation of \(U\), also maximized at \(\hat\theta=\theta\). (c) So a first-price auction with \(n\) risk-averse bidders whose values are distributed \(F\) is equivalent to one with \(n\) risk-neutral bidders distributed \(\hat F=F^\alpha\); using the §24.3 risk-neutral solution, \(\bar b_\alpha(\theta)=\theta-\int_0^\theta(F(x)/F(\theta))^{\alpha(n-1)}\,dx\). (d) \(\frac{\partial\bar b_\alpha(\theta)}{\partial\alpha}=\int_0^\theta(n-1)(F(x)/F(\theta))^{\alpha(n-1)}(-\ln(F(x)/F(\theta)))\,dx>0\), so \(\bar b_\alpha\) is strictly increasing in \(\alpha\) — more risk-averse bidders prefer a "more certain but lower-value" outcome (higher winning probability, lower surplus if winning), and the seller's first-price revenue rises with risk aversion. (e) Truth-telling remains dominant in the second-price auction, so \(ER^{II}_{ra}=ER^{II}_{rn}\); by §24.4 \(ER^{II}_{rn}=ER^{I}_{rn}\), and by (d) \(ER^{I}_{ra}>ER^{I}_{rn}\), so \(ER^{I}_{ra}>ER^{II}_{ra}\) — under risk aversion the first-price auction raises more than the second-price, and the two standard auctions no longer generate the same revenue. (f) As \(\alpha\to\infty\) (infinite risk aversion), \(\lim_{\alpha\to\infty}\bar b_\alpha(\theta)=\theta\), all bid the true type and the winner pays his own bid, with expected revenue \(\mathbb{E}[\max\{\theta_1,\dots,\theta_n\}]\) — the first best, where the agent valuing the good most gets it, pays his entire valuation, and is left with zero surplus.

定理 27.2 / Theorem 27.2 \(n+1\) 个买家、无保留价二价拍卖的期望收入不小于 \(n\) 个买家的最优拍卖，即 \(ER^{SPA}_{n+1}\ge ER^\star_n\)。The expected revenue in a second-price auction without reserve prices with \(n+1\) bidders is larger than or equal to that of the optimal auction with \(n\) bidders, i.e. \(ER^{SPA}_{n+1}\ge ER^\star_n\).

证明 / Proof \(\max\{MR(\theta_1),\dots,MR(\theta_n),MR(\theta_{n+1})\}\) 关于 \(MR(\theta_{n+1})\) 凸，由 Jensen 不等式 \(\mathbb{E}_{\theta_{n+1}}[\max\{\cdots,MR(\theta_{n+1})\}]\ge\max\{\cdots,\mathbb{E}_{\theta_{n+1}}[MR(\theta_{n+1})]\}\) (27.3)。又 \(\mathbb{E}_{\theta_i}[MR(\theta_i)]=\underline\theta\)，因为：\(\max\{MR(\theta_1),\dots,MR(\theta_n),MR(\theta_{n+1})\}\) is convex in \(MR(\theta_{n+1})\), so by Jensen's inequality \(\mathbb{E}_{\theta_{n+1}}[\max\{\cdots,MR(\theta_{n+1})\}]\ge\max\{\cdots,\mathbb{E}_{\theta_{n+1}}[MR(\theta_{n+1})]\}\) (27.3). And \(\mathbb{E}_{\theta_i}[MR(\theta_i)]=\underline\theta\), because:

$$\mathbb{E}_{\theta_i}[MR(\theta_i)]=\int_{\underline\theta}^{\overline\theta}\Big(\theta_i f(\theta_i)-(1-F(\theta_i))\Big)d\theta_i=-\theta_i(1-F(\theta_i))\Big|_{\underline\theta}^{\overline\theta}=-\overline\theta\cdot0+\underline\theta\cdot1=\underline\theta$$

代入 (27.3)：\(ER^{SPA}_{n+1}\ge\mathbb{E}_{\{\theta_i\}_{i=1}^n}[\max\{MR(\theta_1),\dots,MR(\theta_n),\underline\theta\}]\ge\mathbb{E}_{\{\theta_i\}_{i=1}^n}[\max\{MR(\theta_1),\dots,MR(\theta_n),\theta_0\}]=ER^\star_n\)，最后一步由假设 \(\underline\theta\ge\theta_0\)。\(\blacksquare\)Plug into (27.3): \(ER^{SPA}_{n+1}\ge\mathbb{E}_{\{\theta_i\}_{i=1}^n}[\max\{MR(\theta_1),\dots,MR(\theta_n),\underline\theta\}]\ge\mathbb{E}_{\{\theta_i\}_{i=1}^n}[\max\{MR(\theta_1),\dots,MR(\theta_n),\theta_0\}]=ER^\star_n\), the last step by the assumption \(\underline\theta\ge\theta_0\). \(\blacksquare\)

注 27.1、27.2 与 §27.4.4 简单含义 / Remarks 27.1, 27.2 and the simple implication (§27.4.4) 注 27.1：定理 27.2 表明，多一个买家能帮拍卖人正确设定最优保留价，使期望收入不低于"无该买家但保留价最优"的情形——这便是额外买家带来的价值。§27.4.4 简单含义：只有一个买家（\(n=1\)）时，拍卖人知道保留价值 \(\theta_0\) 却不知如何设最优保留价 \(r^\star\)（\(J(r^\star)=\theta_0\)），因只观察到某另一买家事前真实报告的一个数据点 \(\hat\theta\)、无法恢复分布 \(F\)。但拍卖人可用一种"古怪二价拍卖"（两买家 Bob、Alice 都讲真话；Bob 出价更高则得物品付 Alice 的出价，Alice 更高则谁都不卖）保证至少一半的最优收入：该古怪二价等价于一个买家问题（\(ER=\tfrac12 ER^{SPA}_2\)），而由定理 27.2 \(ER^{SPA}_2\ge ER^\star_1\)，故 \(ER=\tfrac12 ER^{SPA}_2\ge\tfrac12 ER^\star_1\)。注 27.2：把保留价设为唯一可观察类型 \(\hat\theta\) 保证一半最优收入，仅在事前期望意义上成立。Remark 27.1: Theorem 27.2 implies an additional bidder can help the auctioneer correctly set up the optimal reserve prices such that the expected revenue is no less than the case without that bidder but with optimal reserve prices — the value brought by the additional bidder. §27.4.4 simple implication: with one bidder (\(n=1\)), the auctioneer knows the reservation value \(\theta_0\) but doesn't know how to set the optimal reserve \(r^\star\) (\(J(r^\star)=\theta_0\)), since he only observes one data point \(\hat\theta\) truthfully reported by some other bidder and cannot recover the distribution \(F\). But the auctioneer can use a "weird second-price auction" (two bidders Bob and Alice both bid truthfully; if Bob bids higher he gets the good and pays Alice's bid, if Alice is higher nobody gets it) to guarantee at least one-half of the optimal revenue: this weird second-price auction is equivalent to the one-bidder problem (\(ER=\tfrac12 ER^{SPA}_2\)), and by Theorem 27.2 \(ER^{SPA}_2\ge ER^\star_1\), so \(ER=\tfrac12 ER^{SPA}_2\ge\tfrac12 ER^\star_1\). Remark 27.2: setting the reserve price to the only observable type \(\hat\theta\) guarantees one-half of the optimal revenue only in expectation ex-ante.