Bilateral Trade 组导读 / Bilateral Trade group overview 「双边交易」组（Ch 28–29）研究一个买家与一个卖家、双方均有私人信息时的交易机制：Ch 28 证明 Myerson–Satterthwaite (1983) 不可能性定理——不存在同时满足事后有效、预算平衡、激励相容、个人理性的交易机制；Ch 29 给出在放松事后有效（容许无效率）后的第二好机制设计。

本章导读 本章把单方私人信息的筛选/拍卖框架推广到双边私人信息：买家与卖家各自只知道自己的估值。§28.1 框架：买家估值 \(\theta_B\sim(F_B,f_B)\) 支撑 \([\underline\theta_B,\overline\theta_B]\)、卖家估值 \(\theta_S\sim(F_S,f_S)\) 支撑 \([\underline\theta_S,\overline\theta_S]\)，二者支撑严格重叠；交易机制 \(\{\varphi(\hat\theta_B,\hat\theta_S),t(\hat\theta_B,\hat\theta_S)\}\)（\(\varphi\) 为交易概率、\(t\) 为买家付给卖家的转移）；定义 28.1 事后预算平衡、定义 28.2 事后有效（\(\theta_B\ge\theta_S\) 才交易）、定义 28.3 激励相容 (IC)、定义 28.4 个人理性 (IR，外部选项 \(0\))。§28.2 定理 28.1 给出 IC 的充要条件（\(\bar\varphi_B\) 递增、\(\bar\varphi_S\) 递减，加上两条效用积分式），命题 28.1 用分部积分把期望效用写成虚拟剩余形式。§28.3 定理 28.2（IC 蕴含的"剩余 = 效用"恒等式 (28.15)）与其逆命题定理 28.3（满足该恒等式的 \(\varphi\) 必可补上 \(t\) 使机制 IC 且 IR）。§28.4 定理 28.4：Myerson–Satterthwaite 不可能性定理——支撑严格重叠且密度严格为正时，没有任何交易机制能同时事后有效、预算平衡、IC、IR。本章无外部图。

定义 28.1（事后预算平衡 / Ex-post budget balance） 交易机制 \(\{\varphi,t\}\) 满足事后预算平衡，若买家支付的钱恰等于卖家收到的钱 \(t(\hat\theta_B,\hat\theta_S)\)，没有第三方注资或销毁货币。A trading mechanism \(\{\varphi,t\}\) satisfies ex-post budget balance if the money the buyer pays equals exactly the money \(t(\hat\theta_B,\hat\theta_S)\) the seller receives, with no outside party injecting or burning money.

定义 28.2（事后有效 / Ex-post efficiency） 交易机制是事后有效的，若当且仅当买家估值不低于卖家估值时交易：\(\varphi(\theta_B,\theta_S)=1\) 若 \(\theta_B\ge\theta_S\)，否则 \(\varphi(\theta_B,\theta_S)=0\)。A trading mechanism is ex-post efficient if trade occurs exactly when the buyer values the object at least as much as the seller: \(\varphi(\theta_B,\theta_S)=1\) if \(\theta_B\ge\theta_S\), and \(\varphi(\theta_B,\theta_S)=0\) otherwise.

定义 28.3（激励相容 / Incentive compatibility, IC） 机制 \(\{\varphi,t\}\) 是激励相容的，若双方均如实报告其类型为最优。即对买家所有 \(\theta_B,\hat\theta_B\)：\(U_B(\theta_B)\ge\theta_B\,\bar\varphi_B(\hat\theta_B)-\bar t_B(\hat\theta_B)\)；对卖家所有 \(\theta_S,\hat\theta_S\)：\(U_S(\theta_S)\ge\bar t_S(\hat\theta_S)-\theta_S\,\bar\varphi_S(\hat\theta_S)\)。A mechanism \(\{\varphi,t\}\) is incentive compatible if truthful reporting is optimal for both sides. That is, for the buyer all \(\theta_B,\hat\theta_B\): \(U_B(\theta_B)\ge\theta_B\,\bar\varphi_B(\hat\theta_B)-\bar t_B(\hat\theta_B)\); for the seller all \(\theta_S,\hat\theta_S\): \(U_S(\theta_S)\ge\bar t_S(\hat\theta_S)-\theta_S\,\bar\varphi_S(\hat\theta_S)\).

定义 28.4（个人理性 / Individual rationality, IR） 机制 \(\{\varphi,t\}\) 是个人理性的（事后/事中意义、外部选项为 \(0\)），若每个类型参与的期望效用非负：\(U_B(\theta_B)\ge0\) 对所有 \(\theta_B\)，\(U_S(\theta_S)\ge0\) 对所有 \(\theta_S\)。卖家不参与则保留物品得 \(0\)，买家不参与得 \(0\)。A mechanism \(\{\varphi,t\}\) is individually rational (interim sense, outside option \(0\)) if every type's expected utility from participating is non-negative: \(U_B(\theta_B)\ge0\) for all \(\theta_B\) and \(U_S(\theta_S)\ge0\) for all \(\theta_S\). A non-participating seller keeps the object and gets \(0\); a non-participating buyer gets \(0\).

注 28.1（有效 = 最大社会剩余 / Remark 28.1: efficiency = maximal social surplus） 事后有效等价于最大化社会剩余 \(\mathbb{E}[\varphi(\theta_B,\theta_S)(\theta_B-\theta_S)]\)：当 \(\theta_B\ge\theta_S\) 交易产生正剩余 \(\theta_B-\theta_S\)，故应交易；当 \(\theta_B<\theta_S\) 交易产生负剩余，故不应交易。转移 \(t\) 只在双方间再分配剩余，不影响总剩余。Ex-post efficiency is equivalent to maximizing social surplus \(\mathbb{E}[\varphi(\theta_B,\theta_S)(\theta_B-\theta_S)]\): when \(\theta_B\ge\theta_S\) trade creates positive surplus \(\theta_B-\theta_S\) so trade should occur; when \(\theta_B<\theta_S\) trade creates negative surplus so it should not. The transfer \(t\) only redistributes surplus between the two parties without affecting total surplus.

定理 28.1（IC 的充要条件 / N&S conditions for IC） 交易机制 \(\{\varphi,t\}\) 激励相容，当且仅当：(1) 单调性——\(\bar\varphi_B(\cdot)\) 关于 \(\theta_B\) 不减、\(\bar\varphi_S(\cdot)\) 关于 \(\theta_S\) 不增；(2) 效用积分式——\(U_B(\theta_B)=U_B(\underline\theta_B)+\int_{\underline\theta_B}^{\theta_B}\bar\varphi_B(x)\,dx\) 且 \(U_S(\theta_S)=U_S(\overline\theta_S)+\int_{\theta_S}^{\overline\theta_S}\bar\varphi_S(x)\,dx\)。A trading mechanism \(\{\varphi,t\}\) is incentive compatible if and only if: (1) monotonicity — \(\bar\varphi_B(\cdot)\) is non-decreasing in \(\theta_B\) and \(\bar\varphi_S(\cdot)\) is non-increasing in \(\theta_S\); (2) utility-integral identities — \(U_B(\theta_B)=U_B(\underline\theta_B)+\int_{\underline\theta_B}^{\theta_B}\bar\varphi_B(x)\,dx\) and \(U_S(\theta_S)=U_S(\overline\theta_S)+\int_{\theta_S}^{\overline\theta_S}\bar\varphi_S(x)\,dx\).

注 28.2（等价的转移条件 / Remark 28.2: equivalent transfer conditions） 把 \(U_B,U_S\) 的定义代入 (2)，可得对 \(\bar t_B,\bar t_S\) 的等价条件（即正文 (28.1)、(28.2)）：买家的期望转移 \(\bar t_B(\theta_B)=\theta_B\bar\varphi_B(\theta_B)-U_B(\underline\theta_B)-\int_{\underline\theta_B}^{\theta_B}\bar\varphi_B(x)dx\)，卖家的期望转移 \(\bar t_S(\theta_S)=\theta_S\bar\varphi_S(\theta_S)+U_S(\overline\theta_S)+\int_{\theta_S}^{\overline\theta_S}\bar\varphi_S(x)dx\)。直觉与单方机制相同：信息租金由边际类型的"分配规则斜率"积分决定。Substituting the definitions of \(U_B,U_S\) into (2) gives the equivalent conditions on \(\bar t_B,\bar t_S\) (the text's (28.1), (28.2)): the buyer's expected transfer \(\bar t_B(\theta_B)=\theta_B\bar\varphi_B(\theta_B)-U_B(\underline\theta_B)-\int_{\underline\theta_B}^{\theta_B}\bar\varphi_B(x)dx\), and the seller's expected transfer \(\bar t_S(\theta_S)=\theta_S\bar\varphi_S(\theta_S)+U_S(\overline\theta_S)+\int_{\theta_S}^{\overline\theta_S}\bar\varphi_S(x)dx\). The intuition is the same as in one-sided mechanisms: information rents are determined by integrating the slope of the allocation rule over marginal types.

证明 / Proof 必要性。 设机制 IC。固定买家、考虑两类型 \(\theta_B,\theta_B'\)。讲真话不低于谎报为 \(\theta_B'\)：Necessity. Suppose the mechanism is IC. Fix the buyer and consider two types \(\theta_B,\theta_B'\). Truth-telling weakly dominates misreporting \(\theta_B'\):

$$U_B(\theta_B)\ge\theta_B\bar\varphi_B(\theta_B')-\bar t_B(\theta_B')=U_B(\theta_B')+(\theta_B-\theta_B')\bar\varphi_B(\theta_B')$$

对称地交换角色：\(U_B(\theta_B')\ge U_B(\theta_B)+(\theta_B'-\theta_B)\bar\varphi_B(\theta_B)\)。两式相加整理得Swapping roles symmetrically: \(U_B(\theta_B')\ge U_B(\theta_B)+(\theta_B'-\theta_B)\bar\varphi_B(\theta_B)\). Adding and rearranging the two inequalities gives

$$(\theta_B-\theta_B')\big(\bar\varphi_B(\theta_B)-\bar\varphi_B(\theta_B')\big)\ge0$$

即 \(\bar\varphi_B\) 不减。由 \(U_B(\theta_B)=\max_{\hat\theta_B}\{\theta_B\bar\varphi_B(\hat\theta_B)-\bar t_B(\hat\theta_B)\}\) 是仿射函数族的上包络，\(U_B\) 凸且几乎处处可导，由包络定理 \(U_B'(\theta_B)=\bar\varphi_B(\theta_B)\)，积分得 \(U_B(\theta_B)=U_B(\underline\theta_B)+\int_{\underline\theta_B}^{\theta_B}\bar\varphi_B(x)dx\)。i.e. \(\bar\varphi_B\) is non-decreasing. Since \(U_B(\theta_B)=\max_{\hat\theta_B}\{\theta_B\bar\varphi_B(\hat\theta_B)-\bar t_B(\hat\theta_B)\}\) is the upper envelope of a family of affine functions, \(U_B\) is convex and differentiable a.e.; by the envelope theorem \(U_B'(\theta_B)=\bar\varphi_B(\theta_B)\), and integrating gives \(U_B(\theta_B)=U_B(\underline\theta_B)+\int_{\underline\theta_B}^{\theta_B}\bar\varphi_B(x)dx\).

卖家对称：因卖家估值越高越不愿交易，谎报不优条件给出 \(\bar\varphi_S\) 不增、\(U_S'(\theta_S)=-\bar\varphi_S(\theta_S)\)，从上端 \(\overline\theta_S\) 向下积分得 \(U_S(\theta_S)=U_S(\overline\theta_S)+\int_{\theta_S}^{\overline\theta_S}\bar\varphi_S(x)dx\)。The seller is symmetric: since a higher-valued seller is less willing to trade, the no-misreporting conditions give \(\bar\varphi_S\) non-increasing and \(U_S'(\theta_S)=-\bar\varphi_S(\theta_S)\); integrating downward from the top \(\overline\theta_S\) gives \(U_S(\theta_S)=U_S(\overline\theta_S)+\int_{\theta_S}^{\overline\theta_S}\bar\varphi_S(x)dx\).

充分性。 设 (1)(2) 成立。对买家，需验证对任意 \(\theta_B,\hat\theta_B\) 有 \(U_B(\theta_B)\ge\theta_B\bar\varphi_B(\hat\theta_B)-\bar t_B(\hat\theta_B)=U_B(\hat\theta_B)+(\theta_B-\hat\theta_B)\bar\varphi_B(\hat\theta_B)\)。由 (2)，\(U_B(\theta_B)-U_B(\hat\theta_B)=\int_{\hat\theta_B}^{\theta_B}\bar\varphi_B(x)dx\)。于是只需Sufficiency. Suppose (1)(2) hold. For the buyer we must verify that for all \(\theta_B,\hat\theta_B\), \(U_B(\theta_B)\ge\theta_B\bar\varphi_B(\hat\theta_B)-\bar t_B(\hat\theta_B)=U_B(\hat\theta_B)+(\theta_B-\hat\theta_B)\bar\varphi_B(\hat\theta_B)\). By (2), \(U_B(\theta_B)-U_B(\hat\theta_B)=\int_{\hat\theta_B}^{\theta_B}\bar\varphi_B(x)dx\). So it suffices that

$$\int_{\hat\theta_B}^{\theta_B}\bar\varphi_B(x)\,dx\ge(\theta_B-\hat\theta_B)\bar\varphi_B(\hat\theta_B)$$

当 \(\theta_B>\hat\theta_B\)，由 \(\bar\varphi_B\) 不减，\(\int_{\hat\theta_B}^{\theta_B}\bar\varphi_B(x)dx\ge\int_{\hat\theta_B}^{\theta_B}\bar\varphi_B(\hat\theta_B)dx=(\theta_B-\hat\theta_B)\bar\varphi_B(\hat\theta_B)\)；当 \(\theta_B<\hat\theta_B\) 同理（两侧符号同时翻转）成立。故买家 IC 成立。卖家由 \(\bar\varphi_S\) 不增对称验证。\(\blacksquare\)When \(\theta_B>\hat\theta_B\), since \(\bar\varphi_B\) is non-decreasing, \(\int_{\hat\theta_B}^{\theta_B}\bar\varphi_B(x)dx\ge\int_{\hat\theta_B}^{\theta_B}\bar\varphi_B(\hat\theta_B)dx=(\theta_B-\hat\theta_B)\bar\varphi_B(\hat\theta_B)\); when \(\theta_B<\hat\theta_B\) the same holds (both sides flip sign together). So buyer IC holds. The seller follows symmetrically from \(\bar\varphi_S\) non-increasing. \(\blacksquare\)

命题 28.1（期望效用的虚拟剩余表示 / Virtual-surplus form of expected utilities） 在 IC 下，把定理 28.1 的效用积分式对 \(\theta_B\)（resp. \(\theta_S\)）取期望并分部积分，得Under IC, taking the expectation of Theorem 28.1's utility-integral identities over \(\theta_B\) (resp. \(\theta_S\)) and integrating by parts gives

$$\mathbb{E}[U_B(\theta_B)]=U_B(\underline\theta_B)+\mathbb{E}\!\left[\varphi(\theta_B,\theta_S)\,\frac{1-F_B(\theta_B)}{f_B(\theta_B)}\right]$$

$$\mathbb{E}[U_S(\theta_S)]=U_S(\overline\theta_S)+\mathbb{E}\!\left[\varphi(\theta_B,\theta_S)\,\frac{F_S(\theta_S)}{f_S(\theta_S)}\right]$$

证明 / Proof 对买家：\(\mathbb{E}[U_B]=\int_{\underline\theta_B}^{\overline\theta_B}U_B(\theta_B)f_B(\theta_B)d\theta_B=U_B(\underline\theta_B)+\int_{\underline\theta_B}^{\overline\theta_B}\Big(\int_{\underline\theta_B}^{\theta_B}\bar\varphi_B(x)dx\Big)f_B(\theta_B)d\theta_B\)。交换积分次序（或分部积分，\(\int u\,dF\) 取 \(u=\int_{\underline\theta_B}^{\theta_B}\bar\varphi_B\)）：For the buyer: \(\mathbb{E}[U_B]=\int_{\underline\theta_B}^{\overline\theta_B}U_B(\theta_B)f_B(\theta_B)d\theta_B=U_B(\underline\theta_B)+\int_{\underline\theta_B}^{\overline\theta_B}\Big(\int_{\underline\theta_B}^{\theta_B}\bar\varphi_B(x)dx\Big)f_B(\theta_B)d\theta_B\). Swapping the order of integration (or integrating by parts, \(\int u\,dF\) with \(u=\int_{\underline\theta_B}^{\theta_B}\bar\varphi_B\)):

$$\int_{\underline\theta_B}^{\overline\theta_B}\!\Big(\int_{\underline\theta_B}^{\theta_B}\bar\varphi_B(x)dx\Big)f_B(\theta_B)d\theta_B=\int_{\underline\theta_B}^{\overline\theta_B}\bar\varphi_B(x)\big(1-F_B(x)\big)dx=\mathbb{E}\!\left[\bar\varphi_B(\theta_B)\frac{1-F_B(\theta_B)}{f_B(\theta_B)}\right]$$

再由 \(\bar\varphi_B(\theta_B)=\mathbb{E}_{\theta_S}[\varphi(\theta_B,\theta_S)]\) 把内层期望放回，即得第一式。卖家对称：从 \(\overline\theta_S\) 向下积分、权重变为 \(F_S(\theta_S)\)，得第二式。\(\blacksquare\)Then putting the inner expectation back via \(\bar\varphi_B(\theta_B)=\mathbb{E}_{\theta_S}[\varphi(\theta_B,\theta_S)]\) yields the first identity. The seller is symmetric: integrating downward from \(\overline\theta_S\), the weight becomes \(F_S(\theta_S)\), giving the second identity. \(\blacksquare\)

定理 28.2（IC + 预算平衡蕴含的恒等式 / Identity implied by IC + budget balance） 若交易机制 \(\{\varphi,t\}\) 激励相容且事后预算平衡，则If a trading mechanism \(\{\varphi,t\}\) is incentive compatible and ex-post budget balanced, then

$$\mathbb{E}\!\left[\varphi(\theta_B,\theta_S)\Big(\big(\theta_B-\tfrac{1-F_B(\theta_B)}{f_B(\theta_B)}\big)-\big(\theta_S+\tfrac{F_S(\theta_S)}{f_S(\theta_S)}\big)\Big)\right]=U_B(\underline\theta_B)+U_S(\overline\theta_S)\tag{28.15}$$

证明 / Proof 预算平衡下买家付的等于卖家收的，故总期望剩余等于双方期望效用之和：\(\mathbb{E}[\varphi(\theta_B-\theta_S)]=\mathbb{E}[U_B(\theta_B)]+\mathbb{E}[U_S(\theta_S)]\)（左侧 \(=\mathbb{E}[\theta_B\bar\varphi_B-\bar t_B]+\mathbb{E}[\bar t_S-\theta_S\bar\varphi_S]\)，而 \(\mathbb{E}[\bar t_B]=\mathbb{E}[t]=\mathbb{E}[\bar t_S]\) 相消）。代入命题 28.1 的两式：Under budget balance what the buyer pays equals what the seller receives, so the total expected surplus equals the sum of expected utilities: \(\mathbb{E}[\varphi(\theta_B-\theta_S)]=\mathbb{E}[U_B(\theta_B)]+\mathbb{E}[U_S(\theta_S)]\) (the LHS \(=\mathbb{E}[\theta_B\bar\varphi_B-\bar t_B]+\mathbb{E}[\bar t_S-\theta_S\bar\varphi_S]\), and \(\mathbb{E}[\bar t_B]=\mathbb{E}[t]=\mathbb{E}[\bar t_S]\) cancel). Substituting the two identities of Proposition 28.1:

$$\mathbb{E}[\varphi(\theta_B-\theta_S)]=U_B(\underline\theta_B)+\mathbb{E}\!\left[\varphi\tfrac{1-F_B}{f_B}\right]+U_S(\overline\theta_S)+\mathbb{E}\!\left[\varphi\tfrac{F_S}{f_S}\right]$$

将两个期望项移到左边、合并到 \(\varphi\) 内即得 (28.15)。\(\blacksquare\)Moving the two expectation terms to the left and combining them inside \(\varphi\) yields (28.15). \(\blacksquare\)

解读 / Interpretation 左侧 \(\theta_B-\frac{1-F_B}{f_B}\) 是买家的虚拟估值（边际收入），\(\theta_S+\frac{F_S}{f_S}\) 是卖家的虚拟成本（边际成本）。恒等式说：IC + 预算平衡机制下，期望"虚拟剩余"被双方边界类型的租金完全占用。要 IR（即 \(U_B(\underline\theta_B),U_S(\overline\theta_S)\ge0\)），就要求左侧虚拟剩余 \(\ge0\)。On the left, \(\theta_B-\frac{1-F_B}{f_B}\) is the buyer's virtual valuation (marginal revenue) and \(\theta_S+\frac{F_S}{f_S}\) is the seller's virtual cost (marginal cost). The identity says: under an IC + budget-balanced mechanism, expected "virtual surplus" is entirely absorbed by the boundary types' rents. IR (i.e. \(U_B(\underline\theta_B),U_S(\overline\theta_S)\ge0\)) then requires the left-hand virtual surplus to be \(\ge0\).

定理 28.3（逆命题：可实现性 / Converse: implementability） 设分配规则 \(\varphi(\theta_B,\theta_S)\) 满足 \(\bar\varphi_B(\cdot)\) 不减、\(\bar\varphi_S(\cdot)\) 不增，且Suppose an allocation rule \(\varphi(\theta_B,\theta_S)\) has \(\bar\varphi_B(\cdot)\) non-decreasing, \(\bar\varphi_S(\cdot)\) non-increasing, and satisfies

$$\mathbb{E}\!\left[\varphi(\theta_B,\theta_S)\Big(\big(\theta_B-\tfrac{1-F_B(\theta_B)}{f_B(\theta_B)}\big)-\big(\theta_S+\tfrac{F_S(\theta_S)}{f_S(\theta_S)}\big)\Big)\right]\ge0\tag{28.16}$$

则存在转移规则 \(t(\theta_B,\theta_S)\)，使机制 \(\{\varphi,t\}\) 同时激励相容、事后预算平衡且个人理性。then there exists a transfer rule \(t(\theta_B,\theta_S)\) such that the mechanism \(\{\varphi,t\}\) is simultaneously incentive compatible, ex-post budget balanced, and individually rational.

证明（构造转移 / constructing the transfer） 取 \(U_B(\underline\theta_B)=U_S(\overline\theta_S)=\frac12\,\mathbb{E}[\varphi((\theta_B-\frac{1-F_B}{f_B})-(\theta_S+\frac{F_S}{f_S}))]\ge0\)（由 (28.16) 非负，故 IR 在边界类型处成立；单调性 + 积分式保证所有类型 \(U_B,U_S\ge0\)）。给定单调 \(\varphi\) 与这两个边界效用，由注 28.2 可定出期望转移 \(\bar t_B(\theta_B),\bar t_S(\theta_S)\) 使 IC 成立。最后只需找一个事后转移 \(t(\theta_B,\theta_S)\) 同时满足两个边际条件 \(\mathbb{E}_{\theta_S}[t]=\bar t_B(\theta_B)\)、\(\mathbb{E}_{\theta_B}[t]=\bar t_S(\theta_S)\)。一种可行构造为Set \(U_B(\underline\theta_B)=U_S(\overline\theta_S)=\frac12\,\mathbb{E}[\varphi((\theta_B-\frac{1-F_B}{f_B})-(\theta_S+\frac{F_S}{f_S}))]\ge0\) (non-negative by (28.16), so IR holds at the boundary types; monotonicity + the integral identities then ensure \(U_B,U_S\ge0\) for all types). Given the monotone \(\varphi\) and these two boundary utilities, Remark 28.2 pins down the expected transfers \(\bar t_B(\theta_B),\bar t_S(\theta_S)\) that make IC hold. It remains to find an ex-post transfer \(t(\theta_B,\theta_S)\) satisfying the two marginal conditions \(\mathbb{E}_{\theta_S}[t]=\bar t_B(\theta_B)\) and \(\mathbb{E}_{\theta_B}[t]=\bar t_S(\theta_S)\). One feasible construction is

$$t(\theta_B,\theta_S)=\bar t_B(\theta_B)+\bar t_S(\theta_S)-\mathbb{E}[t],\qquad \mathbb{E}[t]=\mathbb{E}[\bar t_B(\theta_B)]=\mathbb{E}[\bar t_S(\theta_S)]$$

直接验证 \(\mathbb{E}_{\theta_S}[t]=\bar t_B(\theta_B)\)、\(\mathbb{E}_{\theta_B}[t]=\bar t_S(\theta_S)\) 成立（用 \(\mathbb{E}[\bar t_B]=\mathbb{E}[\bar t_S]=\mathbb{E}[t]\)），且 \(t\) 事后即为买家所付、卖家所收（预算平衡）。故 \(\{\varphi,t\}\) 满足全部要求。\(\blacksquare\)One directly verifies \(\mathbb{E}_{\theta_S}[t]=\bar t_B(\theta_B)\) and \(\mathbb{E}_{\theta_B}[t]=\bar t_S(\theta_S)\) (using \(\mathbb{E}[\bar t_B]=\mathbb{E}[\bar t_S]=\mathbb{E}[t]\)), and \(t\) is ex post both what the buyer pays and what the seller receives (budget balance). Hence \(\{\varphi,t\}\) meets all requirements. \(\blacksquare\)

定理 28.4（Myerson–Satterthwaite 1983 不可能性定理 / Impossibility theorem） 设买家与卖家的支撑有非空交集（\(\overline\theta_S\ge\underline\theta_B\) 且 \(\underline\theta_S<\overline\theta_B\)）、且密度在各自支撑上严格为正。则不存在任何交易机制能同时满足：事后有效、事后预算平衡、激励相容、个人理性。Suppose the buyer's and seller's supports have a non-empty intersection (\(\overline\theta_S\ge\underline\theta_B\) and \(\underline\theta_S<\overline\theta_B\)) and the densities are strictly positive on their supports. Then no trading mechanism can be simultaneously ex-post efficient, ex-post budget balanced, incentive compatible, and individually rational.

证明 / Proof 设存在这样的机制。事后有效即 \(\varphi(\theta_B,\theta_S)=\mathbf 1\{\theta_B\ge\theta_S\}\)。代入 (28.15) 的左侧，先对 \(\theta_S\) 在 \([\underline\theta_S,\theta_B]\) 上积分（仅此区间交易）。买家虚拟项部分：Suppose such a mechanism exists. Ex-post efficiency means \(\varphi(\theta_B,\theta_S)=\mathbf 1\{\theta_B\ge\theta_S\}\). Substitute into the LHS of (28.15) and integrate over \(\theta_S\in[\underline\theta_S,\theta_B]\) (trade occurs only there). For the buyer's virtual part:

$$\mathbb{E}\!\left[\mathbf 1\{\theta_B\ge\theta_S\}\Big(\theta_B-\tfrac{1-F_B(\theta_B)}{f_B(\theta_B)}\Big)\right]=\int\Big(\theta_B-\tfrac{1-F_B(\theta_B)}{f_B(\theta_B)}\Big)F_S(\theta_B)f_B(\theta_B)\,d\theta_B$$

卖家虚拟项部分对称。把两项合并、对交易区域 \(\{\theta_B\ge\theta_S\}\) 重排并分部积分，经标准化简（Myerson–Satterthwaite 1983），(28.15) 的左侧化为The seller's virtual part is symmetric. Combining the two terms, rearranging over the trading region \(\{\theta_B\ge\theta_S\}\) and integrating by parts, standard simplification (Myerson–Satterthwaite 1983) reduces the LHS of (28.15) to

$$\mathbb{E}\!\left[\mathbf 1\{\theta_B\ge\theta_S\}\Big((\theta_B-\tfrac{1-F_B}{f_B})-(\theta_S+\tfrac{F_S}{f_S})\Big)\right]=-\int_{\underline\theta_B}^{\overline\theta_S}\big(1-F_B(\theta)\big)F_S(\theta)\,d\theta$$

在交集内部 \((\underline\theta_B,\overline\theta_S)\)，由密度严格为正有 \(00\)，积分严格为正，于是左侧严格为负。但由 (28.15)，左侧 \(=U_B(\underline\theta_B)+U_S(\overline\theta_S)\)，而 IR 要求二者均 \(\ge0\)，故左侧 \(\ge0\)。矛盾。因此这样的机制不存在。\(\blacksquare\)On the interior of the intersection \((\underline\theta_B,\overline\theta_S)\), strict positivity of the densities gives \(00\), the integral is strictly positive, and hence the LHS is strictly negative. But by (28.15) the LHS \(=U_B(\underline\theta_B)+U_S(\overline\theta_S)\), and IR requires both \(\ge0\), so the LHS \(\ge0\). Contradiction. Hence no such mechanism exists. \(\blacksquare\)

直觉 / Intuition 要事后有效就必须在所有 \(\theta_B\ge\theta_S\) 时交易，但双方都有动机谎报以攫取信息租金：买家想压低报价、卖家想抬高报价。补偿这两类信息租金所需的钱，超过了机制内部可再分配的剩余——除非有外部补贴打破预算平衡，否则有效交易、自愿参与、预算自足三者不可兼得。这是双边私人信息下交易必然存在无效率（错失部分本应发生的交易）的根本原因。Ex-post efficiency requires trade whenever \(\theta_B\ge\theta_S\), but both sides have an incentive to misreport to capture information rents: the buyer wants to understate, the seller to overstate. The money needed to compensate both kinds of information rent exceeds the surplus that can be redistributed within the mechanism — so unless an outside subsidy breaks budget balance, efficient trade, voluntary participation, and budget self-sufficiency cannot all hold at once. This is the fundamental reason why trade under two-sided private information is necessarily inefficient (some socially desirable trades are missed).

注 28.3（事后 IR vs 事前合约 / Remark 28.3: ex-post IR vs ex-ante contracting） 不可能性定理是相对事后 IR约束而言的——即双方在各自已私下得知类型后仍自愿参与。若考虑事前交易机制（例如双方在了解各自类型之前就签订交易合约），则可以实现第一好（社会福利/有效率意义上的最优），此时事前 IR约束不会被事后有效的交易机制破坏。The impossibility theorem is stated relative to the ex-post IR constraint — i.e. both agents still participate voluntarily after each has privately learned their type. If instead we consider an ex-ante trading mechanism (e.g. the two sides sign a trading contract before learning their types), then the first best (the optimum in terms of social welfare or efficiency) can be realized, and in that case the ex-ante IR constraint won't be violated by an ex-post efficient trading mechanism.