Socially Efficient Mechanism Design 组导读 / Socially Efficient Mechanism Design group overview 「社会有效机制设计」组（仅 Ch 30，本书最后一章）把前面的单方/双方机制推广到多人（\(n>2\)）一般社会状态设定，讨论如何设计实现有效结果的机制。核心：VCG/Groves（DSIC + 事后有效 + IR，但预算盈余故不完全有效）；AGV 期望外部性机制（BIC + 预算平衡 + 事后有效 + IR，在事前 IR 下实现第一好）；可能性定理 30.2 给出 BIC + 预算平衡 + 事后有效 + 事中 IR 机制存在的充要条件——并由例 30.1/30.2 揭示事中 IR 比事前 IR 更强、恰恢复 §28 的 Myerson–Satterthwaite 不可能性。

本章导读 §30.1 框架：社会状态 \(x\in\mathcal X=\{x_1,\dots,x_k\}\) 完全由机制设计者控制；\(n\) 个参与人，参与人 \(i\) 有私人信号 \(\theta_i\in\Theta_i\)、拟线性效用 \(U_i=u_i(x,\theta_i)-t_i\)（\(u_i\) 关于 \(x\) 凹、\(t_i\) 上缴的税）；直接机制 \(\{\phi(x\mid\hat\theta),t_1(\hat\theta),\dots,t_n(\hat\theta)\}\)。定义 30.1 事后有效配置 \(\hat x(\theta)=\arg\max_x\sum_i u_i(x,\theta_i)\)；命题 30.1 拟线性下 \(\hat x\) 是唯一帕累托有效配置；定义 30.2 DSIC（占优策略激励相容）；注 30.1–30.3 比较 DSIC 与 BIC（DSIC 更强、更稳健）；§30.1.3 把框架映射到拍卖/双边交易/公共品。§30.2 用 DSIC 机制实现事后有效：Groves (1973) 机制（定义 30.3）DSIC（命题 30.2）；VCG/Clarke 枢轴机制（定义 30.4）DSIC（命题 30.3）、税非负（命题 30.4）、枢轴（定义 30.5）、税 = 对他人的负外部性（注 30.4–30.6）、运行预算盈余（命题 30.5）且 IR（命题 30.6）——故事后有效但因盈余而无效率。§30.3 用 BIC 机制实现事后有效：定理 30.1（任何运行期望盈余的 BIC 机制可改造为预算平衡的 BIC 机制、且各类型不更差）；AGV 期望外部性机制（定义 30.6）；推论 30.1 存在事后有效 + BIC + IR + 预算平衡机制；例 30.1 双边交易完整算例（均匀 $[0,1]\(，VCG 与 AGV 全部验证，AGV 实现第一好剩余 \)\frac16$）。§30.4 可能性定理：区分事前 IR 与事中 IR（定义 30.7）、最小补贴（定义 30.8）、IR-VCG 机制（定义 30.9）；推论 30.2 与定理 30.2（连续独立分布下 BIC + 预算平衡 + 事后有效 + 事中 IR 存在的充要条件 (30.11)）；例 30.2 重访例 30.1 算出 (30.11) 左侧 \(=-\frac23<0\) → 事中 IR 下不存在该机制（注 30.9：事中 IR 强于事前 IR，恢复不可能性）。图无。

定义 30.1（事后有效配置 / Ex-post efficient allocation） 称 \(\hat x(\theta_1,\dots,\theta_n)\) 为事后有效配置，当且仅当它最大化全体效用之和（不妨设 \(\arg\max\) 唯一）：\(\hat x(\theta_1,\dots,\theta_n)=\arg\max_{x\in\mathcal X}\sum_{i=1}^n u_i(x,\theta_i)\)。We call \(\hat x(\theta_1,\dots,\theta_n)\) an ex-post efficient allocation if and only if it maximizes the sum of all utilities (WLOG assume the \(\arg\max\) is unique): \(\hat x(\theta_1,\dots,\theta_n)=\arg\max_{x\in\mathcal X}\sum_{i=1}^n u_i(x,\theta_i)\).

命题 30.1（\(\hat x\) 是唯一帕累托有效配置 / \(\hat x\) is the unique Pareto-efficient allocation） 在拟线性效用 \(U_i=u_i(x,\theta_i)-t_i\) 下，定义 30.1 的 \(\hat x(\theta_1,\dots,\theta_n)\) 是唯一的帕累托有效配置。Under quasi-linear utility \(U_i=u_i(x,\theta_i)-t_i\), the \(\hat x(\theta_1,\dots,\theta_n)\) in Definition 30.1 is the unique Pareto-efficient allocation.

证明 / Proof 先证任意 \(\tilde x(\theta_1,\dots,\theta_n)\ne\hat x(\theta_1,\dots,\theta_n)\) 都可被帕累托改进。考虑转移 \(t_i=u_i(\hat x,\theta_i)-u_i(\tilde x,\theta_i)-\frac1n\sum_{j=1}^n(u_j(\hat x,\theta_j)-u_j(\tilde x,\theta_j))\)。则在 \(\hat x\) 下First show any \(\tilde x(\theta_1,\dots,\theta_n)\ne\hat x(\theta_1,\dots,\theta_n)\) can be Pareto improved. Consider the transfer \(t_i=u_i(\hat x,\theta_i)-u_i(\tilde x,\theta_i)-\frac1n\sum_{j=1}^n(u_j(\hat x,\theta_j)-u_j(\tilde x,\theta_j))\). Then under \(\hat x\),

$$U_i=u_i(\hat x,\theta_i)-t_i=u_i(\tilde x,\theta_i)+\frac1n\sum_{j=1}^n\big(u_j(\hat x,\theta_j)-u_j(\tilde x,\theta_j)\big)>u_i(\tilde x,\theta_i)$$

最后一步因 \(\hat x\) 最大化效用之和、\(\sum_j(u_j(\hat x,\theta_j)-u_j(\tilde x,\theta_j))>0\)。又 \(\sum_{i=1}^n t_i=\sum_i(u_i(\hat x,\theta_i)-u_i(\tilde x,\theta_i))-\sum_j(u_j(\hat x,\theta_j)-u_j(\tilde x,\theta_j))=0\)，是可行的自筹资金（零和）转移。故所有人在 \(\hat x\) 下都严格更好——\(\tilde x\) 被帕累托支配。最后，\(\hat x\) 因已最大化效用之和，任何人效用上升必使他人下降，故 \(\hat x\) 帕累托有效；由 \(\arg\max\) 唯一，\(\hat x\) 是唯一帕累托有效结果。\(\blacksquare\)The last step holds because \(\hat x\) maximizes the sum of utilities, so \(\sum_j(u_j(\hat x,\theta_j)-u_j(\tilde x,\theta_j))>0\). Also \(\sum_{i=1}^n t_i=\sum_i(u_i(\hat x,\theta_i)-u_i(\tilde x,\theta_i))-\sum_j(u_j(\hat x,\theta_j)-u_j(\tilde x,\theta_j))=0\), a feasible self-financing (zero-sum) transfer. So everyone is strictly better off under \(\hat x\) — \(\tilde x\) is Pareto dominated. Finally, since \(\hat x\) already maximizes the sum of utilities, raising anyone's utility must lower someone else's, so \(\hat x\) is Pareto efficient; and by uniqueness of the \(\arg\max\), \(\hat x\) is the unique Pareto-efficient outcome. \(\blacksquare\)

定义 30.2（占优策略激励相容 / Dominant strategy incentive compatibility, DSIC） 直接机制 \(\{\phi(x\mid\hat\theta),t_1(\hat\theta),\dots,t_n(\hat\theta)\}\) 是占优策略激励相容的，当且仅当对每个 \(i\)、对所有 \(\hat\theta_i,\theta_i,\hat\theta_{-i}\)：\(\sum_{x\in\mathcal X}\phi(x\mid\theta_i,\hat\theta_{-i})u_i(x,\theta_i)-t_i(\theta_i,\hat\theta_{-i})\ge\sum_{x\in\mathcal X}\phi(x\mid\hat\theta_i,\hat\theta_{-i})u_i(x,\theta_i)-t_i(\hat\theta_i,\hat\theta_{-i})\)。即无论他人如何报告，讲真话都最优。A direct mechanism \(\{\phi(x\mid\hat\theta),t_1(\hat\theta),\dots,t_n(\hat\theta)\}\) is dominant-strategy incentive compatible iff for each \(i\) and for all \(\hat\theta_i,\theta_i,\hat\theta_{-i}\): \(\sum_{x\in\mathcal X}\phi(x\mid\theta_i,\hat\theta_{-i})u_i(x,\theta_i)-t_i(\theta_i,\hat\theta_{-i})\ge\sum_{x\in\mathcal X}\phi(x\mid\hat\theta_i,\hat\theta_{-i})u_i(x,\theta_i)-t_i(\hat\theta_i,\hat\theta_{-i})\). I.e. truth-telling is optimal no matter what others report.

注 30.1–30.3：DSIC vs BIC / Remarks 30.1–30.3: DSIC vs BIC 注 30.1：回忆常规激励相容（此后称 BIC，Bayesian IC）的约束写成 \(\sum_x\mathbb{E}_{\theta_{-i}}[\phi(x\mid\theta_i,\theta_{-i})]u_i(x,\theta_i)-\mathbb{E}_{\theta_{-i}}[t_i(\theta_i,\theta_{-i})]\ge\sum_x\mathbb{E}_{\theta_{-i}}[\phi(x\mid\hat\theta_i,\theta_{-i})]u_i(x,\theta_i)-\mathbb{E}_{\theta_{-i}}[t_i(\hat\theta_i,\theta_{-i})]\)，即更熟悉的 \(U_i(\theta_i)\ge U_i(\hat\theta_i\mid\theta_i)\)。BIC 只在对 \(\theta_{-i}\) 取平均的意义上施加约束，而 DSIC 对每个 \(\theta_{-i}\) 值都施加约束。故 DSIC 蕴含 BIC（两边对 \(\theta_{-i}\) 取期望即得），DSIC 是更强的约束。注 30.2：之前已证二价拍卖中讲真话是弱占优策略（无论他人如何出价），故二价拍卖讲真话是 DSIC；而一价拍卖的均衡竞价函数只在期望意义上激励相容，若 \(\theta_{-i}\) 被 \(i\) 观测到则不再 IC，故一价拍卖均衡竞价是 BIC 而非 DSIC。注 30.3：DSIC 机制有时称为稳健机制，因为现实中参与人难以算出对 \(\theta_{-i}\) 的正确期望（BIC 难以遵循），而 DSIC 是普通人在日常中能想明白并遵循的，故现实中更稳健。Remark 30.1: recall the regular incentive compatibility (called BIC, Bayesian IC, from now on) constraint written as \(\sum_x\mathbb{E}_{\theta_{-i}}[\phi(x\mid\theta_i,\theta_{-i})]u_i(x,\theta_i)-\mathbb{E}_{\theta_{-i}}[t_i(\theta_i,\theta_{-i})]\ge\sum_x\mathbb{E}_{\theta_{-i}}[\phi(x\mid\hat\theta_i,\theta_{-i})]u_i(x,\theta_i)-\mathbb{E}_{\theta_{-i}}[t_i(\hat\theta_i,\theta_{-i})]\), i.e. the more familiar \(U_i(\theta_i)\ge U_i(\hat\theta_i\mid\theta_i)\). BIC imposes a restriction only on the average over \(\theta_{-i}\), whereas DSIC imposes a restriction for each value of \(\theta_{-i}\). So DSIC implies BIC (take the expectation over \(\theta_{-i}\) on both sides), and DSIC is a more strict constraint. Remark 30.2: we have shown that bidding the true valuation is a weakly dominant strategy in a second-price auction (regardless of others' bids), so truth-telling in a second-price auction is DSIC; the equilibrium bidding function of a first-price auction is incentive compatible only in expectation, and is no longer IC once \(\theta_{-i}\) is observed by \(i\), so first-price equilibrium bidding is BIC but not DSIC. Remark 30.3: a DSIC mechanism is sometimes called a robust mechanism, because in practice agents find it hard to figure out the correct expectation over \(\theta_{-i}\) (BIC is hard to follow), but DSIC is what normal people can figure out and follow in daily practice, so it is robust.

三个例子 / Three examples 拍卖：社会状态 \(\{x_1,\dots,x_n,x_0\}\) 中 \(x_i\)（\(i\in\{1,\dots,n\}\)）表示买家 \(i\) 赢得物品、\(x_0\) 表示卖家保留物品。双边交易：社会状态变为二元 \(x\in\{0,1\}\)，\(x=0\) 不交易、\(x=1\) 交易。公共品：社会状态 \(x\in\{0,1,2,\dots,m\}\)，\(x=0\) 不生产公共品、\(x=i\)（\(i=1,\dots,m\)）表示选择第 \(i\) 个公共品。这说明一般社会状态记号可方便地映射到各具体问题，本节结论也适用于具体问题。Auctions: among social states \(\{x_1,\dots,x_n,x_0\}\), \(x_i\) (\(i\in\{1,\dots,n\}\)) means bidder \(i\) wins the good and \(x_0\) means the seller keeps the good. Bilateral trade: the social state becomes binary \(x\in\{0,1\}\), \(x=0\) no trade and \(x=1\) trade. Public goods: the social state \(x\in\{0,1,2,\dots,m\}\), \(x=0\) no public good is produced and \(x=i\) (\(i=1,\dots,m\)) means public good \(i\) is chosen. So the general social-state notation maps conveniently onto each particular problem, and the results here apply to specific problems as well.

定义 30.3（Groves 机制 / Groves mechanism） Groves 机制是 \(\{\hat x(\hat\theta_1,\dots,\hat\theta_n),\,t_1^G(\hat\theta),\dots,t_n^G(\hat\theta)\}\)，其中 \(\hat x(\cdot)\) 是事后有效配置，且The Groves mechanism is \(\{\hat x(\hat\theta_1,\dots,\hat\theta_n),\,t_1^G(\hat\theta),\dots,t_n^G(\hat\theta)\}\), where \(\hat x(\cdot)\) is the ex-post efficient allocation and

$$t_i^G(\hat\theta_1,\dots,\hat\theta_n)\equiv\Big(-\sum_{j\ne i}u_j(\hat x(\hat\theta_i,\hat\theta_{-i}),\hat\theta_j)\Big)+h_i(\hat\theta_{-i})$$

\(h_i(\cdot)\) 为任意与 \(\hat\theta_i\) 无关的函数。where \(h_i(\cdot)\) is an arbitrary function not related to \(\hat\theta_i\).

命题 30.2（Groves 机制是 DSIC / Groves is DSIC） Groves 机制 \(\{\hat x(\hat\theta),t_1^G(\hat\theta),\dots,t_n^G(\hat\theta)\}\) 是 DSIC。The Groves mechanism \(\{\hat x(\hat\theta),t_1^G(\hat\theta),\dots,t_n^G(\hat\theta)\}\) is DSIC.

证明 / Proof 记参与人 \(i\) 真实类型 \(\theta_i\)、他人报告 \(\hat\theta_{-i}\) 时报告 \(\hat\theta_i\) 的效用为Denote player \(i\)'s utility from reporting \(\hat\theta_i\) given true type \(\theta_i\) and others' reports \(\hat\theta_{-i}\) by

$$U_i(\hat\theta_i\mid\theta_i,\hat\theta_{-i})\equiv u_i(\hat x(\hat\theta_i,\hat\theta_{-i}),\theta_i)-t_i^G(\hat\theta)=\underbrace{u_i(\hat x(\hat\theta_i,\hat\theta_{-i}),\theta_i)+\sum_{j\ne i}u_j(\hat x(\hat\theta_i,\hat\theta_{-i}),\hat\theta_j)}_{\text{Part 1}}-\underbrace{h_i(\hat\theta_{-i})}_{\text{Part 2}}$$

Part 1 是类型组合 \((\theta_i,\hat\theta_{-i})\) 下的总剩余。由事后有效 \(\hat x\) 的定义，Part 1 在 \(\hat\theta_i=\theta_i\) 时最大化（\(i\) 不能改变 \(\theta_i\)、但能改变 \(\hat\theta_i\)）。Part 2 与 \(\hat\theta_i\) 无关。故 \(U_i(\hat\theta_i\mid\theta_i,\hat\theta_{-i})\) 在 \(\hat\theta_i=\theta_i\) 时最大化，无论 \(\hat\theta_{-i}\) 如何，即 \(U_i(\theta_i\mid\theta_i,\hat\theta_{-i})\ge U_i(\hat\theta_i\mid\theta_i,\hat\theta_{-i})\)。故 Groves 机制 DSIC。\(\blacksquare\)Part 1 is the total surplus to the type profile \((\theta_i,\hat\theta_{-i})\). By the definition of the ex-post efficient \(\hat x\), Part 1 is maximized at \(\hat\theta_i=\theta_i\) (\(i\) cannot change \(\theta_i\) but can change \(\hat\theta_i\)). Part 2 is unrelated to \(\hat\theta_i\). So \(U_i(\hat\theta_i\mid\theta_i,\hat\theta_{-i})\) is maximized at \(\hat\theta_i=\theta_i\) regardless of \(\hat\theta_{-i}\), i.e. \(U_i(\theta_i\mid\theta_i,\hat\theta_{-i})\ge U_i(\hat\theta_i\mid\theta_i,\hat\theta_{-i})\). So the Groves mechanism is DSIC. \(\blacksquare\)

Green–Laffont (1979) 的两个论断 / Two arguments of Green–Laffont (1979) 论断 1：若支付关于货币线性，且偏好空间 \(\mathcal U=\{u_i(\cdot,\theta_i)\}_{\theta_i\in\Theta_i}\) 足够丰富（每个从 \(\mathcal X\) 到 \(\mathbb R\) 的函数形式都可能），则任何事后有效的 DSIC 机制必属 Groves 族。论断 2：在论断 1 同样条件下，任何事后有效的 DSIC 机制（从而 Groves 族）都违反预算平衡。完整证明见 Green & Laffont (1979)。下面我们以 VCG（Vickrey–Clarke–Groves）这个特例证明论断 2，VCG 是 Groves 机制中 \(h_i(\hat\theta_{-i})\) 取特定形式的特例。Argument 1: if payoffs are linear in money and the space of preferences \(\mathcal U=\{u_i(\cdot,\theta_i)\}_{\theta_i\in\Theta_i}\) is sufficiently rich (every functional form mapping \(\mathcal X\) to \(\mathbb R\) is possible), then any ex-post efficient DSIC mechanism must be of the Groves form. Argument 2: under the same conditions, any ex-post efficient DSIC mechanism (and thus the Groves family) violates budget balance. See the full proof in Green & Laffont (1979). Below we prove a special case of Argument 2 with the VCG (Vickrey–Clarke–Groves) mechanism, which is the special case of Groves imposing a specific functional form of \(h_i(\hat\theta_{-i})\).

定义 30.4（VCG 机制 / VCG mechanism） VCG 机制是 \(\{\hat x(\hat\theta),t_1^{\text{VCG}}(\hat\theta),\dots,t_n^{\text{VCG}}(\hat\theta)\}\)，其中 \(\hat x(\cdot)\) 是事后有效配置，且The VCG mechanism is \(\{\hat x(\hat\theta),t_1^{\text{VCG}}(\hat\theta),\dots,t_n^{\text{VCG}}(\hat\theta)\}\), where \(\hat x(\cdot)\) is the ex-post efficient allocation and

$$t_i^{\text{VCG}}(\hat\theta_1,\dots,\hat\theta_n)\equiv\underbrace{\Big(-\sum_{j\ne i}u_j(\hat x(\hat\theta_i,\hat\theta_{-i}),\hat\theta_j)\Big)}_{\text{first term}}+\underbrace{\Big(\sum_{j\ne i}u_j(\hat x_{-i}(\hat\theta_{-i}),\hat\theta_j)\Big)}_{\text{second term}}\tag{30.2}$$

命题 30.3、30.4 / Propositions 30.3, 30.4 命题 30.3：VCG 机制是 DSIC（由命题 30.2，代入 \(h_i(\hat\theta_{-i})=\sum_{j\ne i}u_j(\hat x_{-i}(\hat\theta_{-i}),\hat\theta_j)\) 与 \(\hat\theta_i\) 无关即得）。命题 30.4：VCG 机制中 \(t_i^{\text{VCG}}(\hat\theta_1,\dots,\hat\theta_n)\ge0\) 对所有 \(i\)。Proposition 30.3: the VCG mechanism is DSIC (from Proposition 30.2, since \(h_i(\hat\theta_{-i})=\sum_{j\ne i}u_j(\hat x_{-i}(\hat\theta_{-i}),\hat\theta_j)\) is unrelated to \(\hat\theta_i\)). Proposition 30.4: in a VCG mechanism \(t_i^{\text{VCG}}(\hat\theta_1,\dots,\hat\theta_n)\ge0\) for all \(i\).

命题 30.4 证明 / Proof of Proposition 30.4 在 (30.2) 中，第二项是当仅为 \(i\) 之外的所有人选取最优状态 \(\hat x_{-i}\) 时除 \(i\) 外所有人的效用——按定义这是最大值；第一项是当为包括 \(i\) 在内所有人选取最优状态 \(\hat x\) 时除 \(i\) 外所有人的效用——此处 \(\hat x\) 未必最大化除 \(i\) 外的效用之和。故第一项（取负前）\(\le\) 第二项，即 \(t_i^{\text{VCG}}\ge0\)。\(\blacksquare\)In (30.2), the second term is the utility of all agents but \(i\) when the optimal state \(\hat x_{-i}\) is chosen for all agents but \(i\) — by definition this is the maximum; the first term is the utility of all agents but \(i\) when the optimal state \(\hat x\) is chosen for all agents including \(i\) — here \(\hat x\) might not maximize the sum of utilities of all but \(i\). So the first term (before negation) \(\le\) the second term, i.e. \(t_i^{\text{VCG}}\ge0\). \(\blacksquare\)

定义 30.5（枢轴 / Pivotal） 称参与人 \(i\) 的报告 \(\hat\theta_i\) 是枢轴的 (pivotal)，当且仅当 \(t_i^{\text{VCG}}(\hat\theta_1,\dots,\hat\theta_n)>0\)。We say player \(i\)'s report \(\hat\theta_i\) is pivotal if and only if \(t_i^{\text{VCG}}(\hat\theta_1,\dots,\hat\theta_n)>0\).

注 30.4–30.6：枢轴、负外部性与二价拍卖 / Remarks 30.4–30.6: pivotality, negative externality, second-price auction 注 30.4：\(i\) 枢轴（\(t_i^{\text{VCG}}>0\)）等价于 \(\hat x_{-i}(\hat\theta_{-i})\ne\hat x(\hat\theta_i,\hat\theta_{-i})\)——即 \(i\) 的出现改变了社会选择。注 30.5：VCG 机制也被 Clarke 称为枢轴机制 (Pivot mechanism)。注 30.6：\(t_i^{\text{VCG}}=(\text{first term})+(\text{second term})\) 恰是 \(i\) 的报告对其他所有人造成的成本（负效用），即 \(i\) 出现导致的负外部性。类似地，二价拍卖的支付可看作 VCG 支付：赢家支付第二高估值（在价值竞价均衡里），正是他赢得物品对整个社会（本应享有物品的第二高估值者）造成的负效用。Remark 30.4: \(i\) being pivotal (\(t_i^{\text{VCG}}>0\)) is equivalent to \(\hat x_{-i}(\hat\theta_{-i})\ne\hat x(\hat\theta_i,\hat\theta_{-i})\) — i.e. \(i\)'s presence changes the social choice. Remark 30.5: the VCG mechanism is also called the Pivot mechanism by Clarke. Remark 30.6: \(t_i^{\text{VCG}}=(\text{first term})+(\text{second term})\) is exactly the cost (disutility) to all other agents caused by \(i\)'s report, i.e. the negative externality caused by \(i\)'s showing up. Analogously, the second-price auction payment can be regarded as a VCG payment: the winner pays the second-highest valuation (in the value-bidding equilibrium), which is the disutility to the whole society (the second-highest-valuation agent who should have enjoyed the good) caused by the winning agent's bidding.

命题 30.5、30.6 / Propositions 30.5, 30.6 命题 30.5：任何 VCG 机制运行（弱）预算盈余 \(\sum_{i=1}^n t_i^{\text{VCG}}\ge0\)，一般情形下严格盈余 \(\sum_i t_i^{\text{VCG}}>0\)（直接由命题 30.4）。命题 30.6：任何 VCG 机制是个人理性 (IR) 的，即所有参与人自愿参与。Proposition 30.5: any VCG mechanism runs a (weak) budget surplus \(\sum_{i=1}^n t_i^{\text{VCG}}\ge0\), and in general a strict surplus \(\sum_i t_i^{\text{VCG}}>0\) (directly from Proposition 30.4). Proposition 30.6: any VCG mechanism is individually rational (IR), i.e. all agents voluntarily participate.

命题 30.6 证明 / Proof of Proposition 30.6 对参与人 \(i\)，不参与的支付（由 DSIC）为 \(U^{\text{not participate}}\equiv u_i(\hat x_{-i}(\theta_{-i}),\theta_i)\cdot1\) (30.3)。参与的支付为For agent \(i\), the payoff from not participating (by DSIC) is \(U^{\text{not participate}}\equiv u_i(\hat x_{-i}(\theta_{-i}),\theta_i)\cdot1\) (30.3). The payoff from participating is

$$U^{\text{participate}}\equiv u_i(\hat x(\theta_i,\theta_{-i}),\theta_i)\cdot1-t_i^{\text{VCG}}=u_i(\hat x(\theta_i,\theta_{-i}),\theta_i)+\sum_{j\ne i}u_j(\hat x(\theta_i,\theta_{-i}),\theta_j)-\sum_{j\ne i}u_j(\hat x_{-i}(\theta_{-i}),\theta_j)\tag{30.4}$$

两者之差为 \(U^{\text{participate}}-U^{\text{not participate}}=\sum_{j=1}^n u_j(\hat x(\theta_i,\theta_{-i}),\theta_j)-\sum_{j=1}^n u_j(\hat x_{-i}(\theta_{-i}),\theta_j)\ge0\)，因 \(\hat x(\theta_i,\theta_{-i})\) 最大化全体 \(\sum_{j=1}^n u_j\) 而 \(\hat x_{-i}(\theta_{-i})\) 未必。故 IR 对每个参与人成立。\(\blacksquare\)The difference is \(U^{\text{participate}}-U^{\text{not participate}}=\sum_{j=1}^n u_j(\hat x(\theta_i,\theta_{-i}),\theta_j)-\sum_{j=1}^n u_j(\hat x_{-i}(\theta_{-i}),\theta_j)\ge0\), since \(\hat x(\theta_i,\theta_{-i})\) maximizes the full \(\sum_{j=1}^n u_j\) while \(\hat x_{-i}(\theta_{-i})\) might not. So IR holds for every agent. \(\blacksquare\)

小结 / Summary 综上，VCG 机制是 DSIC、事后有效、IR，但运行预算盈余，因而（作为"有效机制 = 事后有效 + 预算平衡"的标准）是无效率的。In conclusion, the VCG mechanism is DSIC, ex-post efficient, and IR, but runs a budget surplus, so (by the standard "efficient mechanism = ex-post efficient + budget balanced") it is inefficient.

定理 30.1（盈余 BIC 机制 ⟹ 预算平衡 BIC 机制 / surplus BIC ⟹ budget-balanced BIC） 设 \(\{\phi(x\mid\cdot),t_1(\cdot),\dots,t_n(\cdot)\}\) 是一个运行期望预算盈余的 BIC 机制，即 \(\mathbb{E}_{\{\theta_i\}_{i=1}^n}[\sum_{i=1}^n t_i(\theta_1,\dots,\theta_n)]\ge0\)。则存在一个 BIC 机制 \(\{\phi(x\mid\cdot),\tilde t_1(\cdot),\dots,\tilde t_n(\cdot)\}\)，使 \(\sum_{i=1}^n\tilde t_i(\theta_1,\dots,\theta_n)=0\)（预算平衡）且 \(\tilde U_i(\theta_i)\ge U_i(\theta_i)\) 对所有 \(i,\theta_i\)。Let \(\{\phi(x\mid\cdot),t_1(\cdot),\dots,t_n(\cdot)\}\) be a BIC mechanism running an expected budget surplus, i.e. \(\mathbb{E}_{\{\theta_i\}_{i=1}^n}[\sum_{i=1}^n t_i(\theta_1,\dots,\theta_n)]\ge0\). Then there exists a BIC mechanism \(\{\phi(x\mid\cdot),\tilde t_1(\cdot),\dots,\tilde t_n(\cdot)\}\) such that \(\sum_{i=1}^n\tilde t_i(\theta_1,\dots,\theta_n)=0\) (budget balanced) and \(\tilde U_i(\theta_i)\ge U_i(\theta_i)\) for all \(i,\theta_i\).

证明（构造转移 / constructing the transfer） 记 \(\bar t_i(\theta_i)\equiv\mathbb{E}_{\theta_{-i}}[t_i(\theta_i,\theta_{-i})]\)、\(\bar t_i\equiv\mathbb{E}_{\theta_i}[\bar t_i(\theta_i)]=\mathbb{E}_{\{\theta_i\}}[t_i(\theta_i,\theta_{-i})]\)。构造（对 \(\forall iLet \(\bar t_i(\theta_i)\equiv\mathbb{E}_{\theta_{-i}}[t_i(\theta_i,\theta_{-i})]\) and \(\bar t_i\equiv\mathbb{E}_{\theta_i}[\bar t_i(\theta_i)]=\mathbb{E}_{\{\theta_i\}}[t_i(\theta_i,\theta_{-i})]\). Construct (for all \(i

$$\sum_{i=1}^n\tilde t_i(\theta)=\underbrace{\Big(\sum_{i=1}^n\bar t_i(\theta_i)-\sum_{i=1}^n\bar t_{i+1}(\theta_{i+1})\Big)}_{=0}+\underbrace{\Big(\sum_{i=1}^n\bar t_{i+1}-\sum_{j=1}^n\bar t_j\Big)}_{=0}=0$$

（下标按循环 \(n+1\equiv1\)；两组求和各自抵消。）故新机制预算平衡。再证 BIC 与 \(\tilde U_i(\theta_i)\ge U_i(\theta_i)\)。回忆(indices cyclic with \(n+1\equiv1\); each sum cancels.) So the new mechanism is budget balanced. Next show BIC and \(\tilde U_i(\theta_i)\ge U_i(\theta_i)\). Recall

$$U_i(\hat\theta_i\mid\theta_i)\equiv\Big(\sum_{x\in\mathcal X}\bar\phi_i(x\mid\hat\theta_i)u_i(x,\theta_i)\Big)-\bar t_i(\hat\theta_i)$$

则新机制下Then under the new mechanism

$$\tilde U_i(\hat\theta_i\mid\theta_i)=U_i(\hat\theta_i\mid\theta_i)+\bar t_i(\hat\theta_i)-\mathbb{E}_{\theta_{-i}}\Big[\bar t_i(\hat\theta_i)+\underbrace{(\bar t_{i+1}-\bar t_{i+1}(\theta_{i+1}))}_{\text{expectation is }0}-\tfrac1n\sum_{j=1}^n\bar t_j\Big]=U_i(\hat\theta_i\mid\theta_i)+\frac1n\sum_{j=1}^n\bar t_j\tag{30.5}$$

即新旧效用只相差一个与 \(\hat\theta_i\) 无关的常数 \(\frac1n\sum_j\bar t_j\ge0\)（由期望盈余假设）。由 \(\{\phi,t\}\) BIC，\(U_i(\hat\theta_i\mid\theta_i)\) 在 \(\hat\theta_i=\theta_i\) 最大化，故 \(\tilde U_i(\hat\theta_i\mid\theta_i)\) 也在 \(\hat\theta_i=\theta_i\) 最大化，新机制 BIC；且 \(\tilde U_i(\theta_i)=U_i(\theta_i)+\frac1n\sum_j\bar t_j\ge U_i(\theta_i)\)。即任何运行期望盈余的 BIC 机制都对应一个使所有人（弱）更满意的预算平衡 BIC 机制。\(\blacksquare\)i.e. the new and old utilities differ only by a constant unrelated to \(\hat\theta_i\), namely \(\frac1n\sum_j\bar t_j\ge0\) (by the expected-surplus assumption). Since \(\{\phi,t\}\) is BIC, \(U_i(\hat\theta_i\mid\theta_i)\) is maximized at \(\hat\theta_i=\theta_i\), so \(\tilde U_i(\hat\theta_i\mid\theta_i)\) is also maximized at \(\hat\theta_i=\theta_i\) and the new mechanism is BIC; and \(\tilde U_i(\theta_i)=U_i(\theta_i)+\frac1n\sum_j\bar t_j\ge U_i(\theta_i)\). So any budget-surplus BIC mechanism corresponds to a budget-balanced BIC mechanism in which all agents become equally or more happy. \(\blacksquare\)

定义 30.6（期望外部性机制 / Expected externality (AGV) mechanism） 期望外部性机制（又名 d'Aspremont & Gérard-Varet (AGV) 机制）定义为 \(\{\hat x(\cdot),t_1^{\text{EE}}(\cdot),\dots,t_n^{\text{EE}}(\cdot)\}\)，其中 \(\hat x(\cdot)\) 是事后有效配置，且The expected externality mechanism (also called the d'Aspremont & Gérard-Varet (AGV) mechanism) is defined by \(\{\hat x(\cdot),t_1^{\text{EE}}(\cdot),\dots,t_n^{\text{EE}}(\cdot)\}\), where \(\hat x(\cdot)\) is the ex-post efficient allocation and

$$t_i^{\text{EE}}(\hat\theta_1,\dots,\hat\theta_n)=\bar t_i^{\text{VCG}}(\hat\theta_i)+\big(\bar t_{i+1}^{\text{VCG}}-\bar t_{i+1}^{\text{VCG}}(\hat\theta_{i+1})\big)-\frac1n\sum_{j=1}^n\bar t_j^{\text{VCG}}\tag{30.6}$$

其中 \(\bar t_i^{\text{VCG}}(\hat\theta_i)\equiv\mathbb{E}_{\theta_{-i}}[t_i^{\text{VCG}}(\hat\theta_i,\theta_{-i})]\)、\(\bar t_i^{\text{VCG}}\equiv\mathbb{E}_{\theta_i}[\bar t_i^{\text{VCG}}(\theta_i)]\)，\(t_i^{\text{VCG}}\) 同定义 30.4。where \(\bar t_i^{\text{VCG}}(\hat\theta_i)\equiv\mathbb{E}_{\theta_{-i}}[t_i^{\text{VCG}}(\hat\theta_i,\theta_{-i})]\), \(\bar t_i^{\text{VCG}}\equiv\mathbb{E}_{\theta_i}[\bar t_i^{\text{VCG}}(\theta_i)]\), and \(t_i^{\text{VCG}}\) is as in Definition 30.4.

推论 30.1（对应定理 30.1 / To Theorem 30.1） 存在一个事后有效、BIC 且 IR 的预算平衡机制。There exists an ex-post efficient, BIC, and IR mechanism that is budget balanced.

证明 / Proof 由命题 30.3、30.4、30.6，VCG 机制 DSIC、IR、事后有效且运行预算盈余。因 DSIC 蕴含 BIC，VCG 也 BIC。于是定理 30.1 告诉我们期望外部性机制 \(\{\hat x(\cdot),t_1^{\text{EE}}(\cdot),\dots,t_n^{\text{EE}}(\cdot)\}\) 预算平衡且 BIC。由于期望外部性机制按构造与 VCG 有相同的 \(\hat x(\cdot)\)，它也事后有效。又由定理 30.1，每个人在新机制下（弱）更满意，故它也 IR。综上，期望外部性机制事后有效、BIC、IR、预算平衡。\(\blacksquare\)By Propositions 30.3, 30.4, 30.6, the VCG mechanism is DSIC, IR, ex-post efficient, and runs a budget surplus. Since DSIC implies BIC, VCG is also BIC. Then Theorem 30.1 tells us the expected externality mechanism \(\{\hat x(\cdot),t_1^{\text{EE}}(\cdot),\dots,t_n^{\text{EE}}(\cdot)\}\) is budget balanced and BIC. Since the expected externality mechanism has the same \(\hat x(\cdot)\) as VCG by construction, it is also ex-post efficient. And by Theorem 30.1 every agent is at least equally happy under the new mechanism, so it is also IR. In conclusion, the expected externality mechanism is ex-post efficient, BIC, IR, and budget balanced. \(\blacksquare\)

例 30.1（双边交易 / Bilateral trade） 设买家类型 \(\theta_B\)、卖家类型 \(\theta_S\) 独立同服从 $[0,1]$ 均匀分布。社会状态 \(x\in\{0,1\}\) 为交易量（成交一单位或不交易），效用 \(u_B(x,\theta_B)=\theta_B\cdot x\)、\(u_S(x,\theta_S)=-\theta_S\cdot x\)。则（不以对方报告为条件的）总效用 \(U_B(\hat\theta_B\mid\theta_B)=\sum_{x=0}^1 u_B(x,\theta_B)\mathbb{E}_{\theta_S}[\phi(x\mid\hat\theta_B,\theta_S)]-\mathbb{E}_{\theta_S}[t_B(\hat\theta_B,\theta_S)]\)，\(U_S\) 类似；以对方报告为条件的总效用 \(U_B(\hat\theta_B\mid\theta_B,\hat\theta_S)=\sum_x u_B(x,\theta_B)\phi(x\mid\hat\theta_B,\hat\theta_S)-t_B(\hat\theta_B,\hat\theta_S)\)。下面验证 VCG 与期望外部性机制的性质。Let the buyer's type \(\theta_B\) and seller's type \(\theta_S\) be i.i.d. uniform on $[0,1]\(. The social state \)x\in{0,1}$ is the quantity traded (one unit or no trade), with utilities \(u_B(x,\theta_B)=\theta_B\cdot x\) and \(u_S(x,\theta_S)=-\theta_S\cdot x\). The total utility (unconditional on the other side's report) is \(U_B(\hat\theta_B\mid\theta_B)=\sum_{x=0}^1 u_B(x,\theta_B)\mathbb{E}_{\theta_S}[\phi(x\mid\hat\theta_B,\theta_S)]-\mathbb{E}_{\theta_S}[t_B(\hat\theta_B,\theta_S)]\), similarly for \(U_S\); the total utility conditional on the other side's report is \(U_B(\hat\theta_B\mid\theta_B,\hat\theta_S)=\sum_x u_B(x,\theta_B)\phi(x\mid\hat\theta_B,\hat\theta_S)-t_B(\hat\theta_B,\hat\theta_S)\). Below we verify the properties of the VCG and expected-externality mechanisms.

VCG 机制：转移、DSIC、IR、非预算平衡 / VCG mechanism: transfers, DSIC, IR, not budget balanced 转移（按 (30.2)）：注意此设定下 \(\hat x_{-B}(\hat\theta_S)=0\)（买家缺席则事后有效机制不让卖家生产，因生产对卖家有成本），\(\hat x_{-S}(\hat\theta_B)=1\)（卖家缺席则消费对社会无成本，故事后有效机制总让 \(x=1\)）。于是对买家Transfers (per (30.2)): note that here \(\hat x_{-B}(\hat\theta_S)=0\) (when the buyer is absent the ex-post efficient mechanism won't let the seller produce, since producing is costly to the seller), and \(\hat x_{-S}(\hat\theta_B)=1\) (when the seller is absent consumption is costless to society, so the ex-post efficient mechanism always assigns \(x=1\)). So for the buyer

$$t_B^{\text{VCG}}(\hat\theta_B,\hat\theta_S)=u_S(\hat x_{-B}(\hat\theta_S),\hat\theta_S)-u_S(\hat x(\hat\theta_B,\hat\theta_S),\hat\theta_S)=u_S(0,\hat\theta_S)+\hat x(\hat\theta_B,\hat\theta_S)\cdot\hat\theta_S=\hat x(\hat\theta_B,\hat\theta_S)\cdot\hat\theta_S\tag{30.7}$$

对卖家and for the seller

$$t_S^{\text{VCG}}(\hat\theta_B,\hat\theta_S)=u_B(\hat x_{-S}(\hat\theta_B),\hat\theta_B)-u_B(\hat x(\hat\theta_B,\hat\theta_S),\hat\theta_B)=u_B(1,\hat\theta_B)-\hat x(\hat\theta_B,\hat\theta_S)\cdot\hat\theta_B=\hat\theta_B\cdot\big(1-\hat x(\hat\theta_B,\hat\theta_S)\big)\tag{30.8}$$

买家 DSIC：事后有效 \(\phi(x\mid\hat\theta_B,\hat\theta_S)=1\) 当 \(x=\hat x(\hat\theta_B,\hat\theta_S)\)。买家以对方报告为条件的总效用 \(U_B^{\text{VCG}}(\hat\theta_B\mid\theta_B,\hat\theta_S)=u_B(\hat x(\hat\theta_B,\hat\theta_S),\theta_B)-t_B^{\text{VCG}}=\hat x(\hat\theta_B,\hat\theta_S)\theta_B-\hat x(\hat\theta_B,\hat\theta_S)\hat\theta_S=\hat x(\hat\theta_B,\hat\theta_S)(\theta_B-\hat\theta_S)\)，即（伪）社会剩余；由事后有效 \(\hat x\) 定义，报告 \(\hat\theta_B=\theta_B\) 最大化之，无论 \(\hat\theta_S\)，故 DSIC。买家 IR：不参与对应 \(\hat x_{-B}(\hat\theta_S)=0\)、外部选项效用 \(0\)；\(U_B^{\text{VCG}}=\hat x(\theta_B,\hat\theta_S)(\theta_B-\hat\theta_S)\) 总可取零，故 \(\ge0\)，IR。卖家 DSIC：\(U_S^{\text{VCG}}(\hat\theta_S\mid\theta_B,\theta_S)=u_S(\hat x(\hat\theta_B,\hat\theta_S),\theta_S)-t_S^{\text{VCG}}=-\hat x\theta_S-\hat\theta_B(1-\hat x)=-\hat\theta_B+\hat x(\hat\theta_B,\hat\theta_S)(\hat\theta_B-\theta_S)\)；\(-\hat\theta_B\) 是与 \(\hat\theta_S\) 无关常数可略，剩 \(\hat x(\hat\theta_B,\hat\theta_S)(\hat\theta_B-\theta_S)\) 在 \(\hat\theta_S=\theta_S\) 时最大化，故 DSIC。卖家 IR：不参与对应 \(\hat x_{-S}(\hat\theta_B)=1\)、外部选项效用 \(-\theta_S\)（卖家不参与仍被要求生产 \(x=1\)）；\(U_S^{\text{VCG}}-(-\theta_S)=(\hat x(\hat\theta_B,\theta_S)-1)(\hat\theta_B-\theta_S)\ge0\)（\(\hat\theta_B\ge\theta_S\) 时 \(\hat x=1\Rightarrow0\)；\(\hat\theta_B<\theta_S\) 时 \(\hat x=0\Rightarrow-(\hat\theta_B-\theta_S)>0\)），故 IR。非预算平衡：\(t_B^{\text{VCG}}+t_S^{\text{VCG}}=\hat x\hat\theta_S+\hat\theta_B(1-\hat x)=\hat x(\hat\theta_S-\hat\theta_B)+\hat\theta_B\)；由 \(\hat x(\hat\theta_B,\hat\theta_S)=\mathbf 1\{\hat\theta_B\ge\hat\theta_S\}\) (30.9)，得 \(=\hat\theta_S\)（当 \(\hat\theta_B\ge\hat\theta_S\)，\(\ge0\)）或 \(=\hat\theta_B\)（当 \(\hat\theta_B<\hat\theta_S\)，\(\ge0\)），以概率 \(1\) 严格为正，故 VCG 运行预算盈余、非预算平衡。\(\blacksquare\)Buyer DSIC: ex-post efficiency gives \(\phi(x\mid\hat\theta_B,\hat\theta_S)=1\) when \(x=\hat x(\hat\theta_B,\hat\theta_S)\). The buyer's total utility conditional on the other's report is \(U_B^{\text{VCG}}(\hat\theta_B\mid\theta_B,\hat\theta_S)=u_B(\hat x(\hat\theta_B,\hat\theta_S),\theta_B)-t_B^{\text{VCG}}=\hat x(\hat\theta_B,\hat\theta_S)\theta_B-\hat x(\hat\theta_B,\hat\theta_S)\hat\theta_S=\hat x(\hat\theta_B,\hat\theta_S)(\theta_B-\hat\theta_S)\), the (pseudo) social surplus; by the definition of the ex-post efficient \(\hat x\), reporting \(\hat\theta_B=\theta_B\) maximizes it regardless of \(\hat\theta_S\), so DSIC. Buyer IR: not participating corresponds to \(\hat x_{-B}(\hat\theta_S)=0\) with outside option utility \(0\); \(U_B^{\text{VCG}}=\hat x(\theta_B,\hat\theta_S)(\theta_B-\hat\theta_S)\) can always be zero, so \(\ge0\), IR. Seller DSIC: \(U_S^{\text{VCG}}(\hat\theta_S\mid\theta_B,\theta_S)=u_S(\hat x(\hat\theta_B,\hat\theta_S),\theta_S)-t_S^{\text{VCG}}=-\hat x\theta_S-\hat\theta_B(1-\hat x)=-\hat\theta_B+\hat x(\hat\theta_B,\hat\theta_S)(\hat\theta_B-\theta_S)\); \(-\hat\theta_B\) is a constant not involving \(\hat\theta_S\) and can be ignored, leaving \(\hat x(\hat\theta_B,\hat\theta_S)(\hat\theta_B-\theta_S)\) maximized at \(\hat\theta_S=\theta_S\), so DSIC. Seller IR: not participating corresponds to \(\hat x_{-S}(\hat\theta_B)=1\) with outside option utility \(-\theta_S\) (a non-participating seller is still required to produce \(x=1\)); \(U_S^{\text{VCG}}-(-\theta_S)=(\hat x(\hat\theta_B,\theta_S)-1)(\hat\theta_B-\theta_S)\ge0\) (\(\hat\theta_B\ge\theta_S\Rightarrow\hat x=1\Rightarrow0\); \(\hat\theta_B<\theta_S\Rightarrow\hat x=0\Rightarrow-(\hat\theta_B-\theta_S)>0\)), so IR. Not budget balanced: \(t_B^{\text{VCG}}+t_S^{\text{VCG}}=\hat x\hat\theta_S+\hat\theta_B(1-\hat x)=\hat x(\hat\theta_S-\hat\theta_B)+\hat\theta_B\); by \(\hat x(\hat\theta_B,\hat\theta_S)=\mathbf 1\{\hat\theta_B\ge\hat\theta_S\}\) (30.9) this equals \(\hat\theta_S\) (when \(\hat\theta_B\ge\hat\theta_S\), \(\ge0\)) or \(\hat\theta_B\) (when \(\hat\theta_B<\hat\theta_S\), \(\ge0\)), strictly positive with probability \(1\), so VCG runs a budget surplus and is not budget balanced. \(\blacksquare\)

期望外部性机制：转移、预算平衡、BIC、IR、第一好 / Expected externality mechanism: transfers, budget balance, BIC, IR, first best 转移（按 (30.6)，\(n=2\)）：对买家 \(t_B^{\text{EE}}(\hat\theta_B,\hat\theta_S)=\bar t_B^{\text{VCG}}(\hat\theta_B)+(\bar t_S^{\text{VCG}}-\bar t_S^{\text{VCG}}(\hat\theta_S))-\frac{\bar t_B^{\text{VCG}}+\bar t_S^{\text{VCG}}}{2}\) (30.10)。用均匀分布与 (30.7)、(30.8)、(30.9)：\(\bar t_B^{\text{VCG}}(\hat\theta_B)=\mathbb{E}_{\theta_S}[\hat x(\hat\theta_B,\theta_S)\theta_S]=\int_0^{\hat\theta_B}\theta_S\,d\theta_S=\frac{\hat\theta_B^2}{2}\)，\(\bar t_S^{\text{VCG}}(\hat\theta_S)=\mathbb{E}_{\theta_B}[\theta_B(1-\hat x(\theta_B,\hat\theta_S))]=\int_0^{\hat\theta_S}\theta_B\,d\theta_B=\frac{\hat\theta_S^2}{2}\)，\(\bar t_B^{\text{VCG}}=\mathbb{E}_{\theta_B}[\frac{\theta_B^2}{2}]=\frac16\)，\(\bar t_S^{\text{VCG}}=\frac16\)。代回 (30.10)：\(t_B^{\text{EE}}=\frac{\hat\theta_B^2}{2}+(\frac16-\frac{\hat\theta_S^2}{2})-\frac{\frac16+\frac16}{2}=\frac{\hat\theta_B^2}{2}-\frac{\hat\theta_S^2}{2}\)，对卖家 \(t_S^{\text{EE}}=\frac{\hat\theta_S^2}{2}-\frac{\hat\theta_B^2}{2}\)。预算平衡：\(t_B^{\text{EE}}+t_S^{\text{EE}}=0\)。✓ 买家 BIC & IR：\(U_B^{\text{EE}}(\hat\theta_B\mid\theta_B)=\mathbb{P}(\theta_S\le\hat\theta_B)\theta_B+\mathbb{P}(\theta_S>\hat\theta_B)\cdot0-\int_0^1(\frac{\hat\theta_B^2}{2}-\frac{\theta_S^2}{2})d\theta_S=\hat\theta_B\theta_B-\frac{\hat\theta_B^2}{2}+\frac16\)，关于 \(\hat\theta_B\) 全局凹，FOC \(\theta_B-\hat\theta_B=0\Rightarrow\hat\theta_B=\theta_B\)，BIC；\(U_B^{\text{EE}}(\theta_B)=\frac{\theta_B^2}{2}+\frac16>0\)（外部选项 \(0\)），IR。卖家 BIC & IR：\(U_S^{\text{EE}}(\hat\theta_S\mid\theta_S)=(1-\hat\theta_S)(-\theta_S)-\frac{\hat\theta_S^2}{2}+\frac16\)，关于 \(\hat\theta_S\) 全局凹，FOC \(\theta_S-\hat\theta_S=0\Rightarrow\hat\theta_S=\theta_S\)，BIC；\(U_S^{\text{EE}}(\theta_S)=-(1-\theta_S)\theta_S-\frac{\theta_S^2}{2}+\frac16=\frac{\theta_S^2}{2}-\theta_S+\frac16>-\theta_S\)（外部选项 \(-\theta_S\)），IR。第一好：期望总剩余 \(\mathbb{E}[U_B^{\text{EE}}]+\mathbb{E}[U_S^{\text{EE}}]=\int_0^1(\frac{\theta_B^2}{2}+\frac16)d\theta_B+\int_0^1(\frac{\theta_S^2}{2}+\frac16-\theta_S)d\theta_S=\frac13+(-\frac16)=\frac16\)；而事前社会福利最大化（\(\theta_B>\theta_S\) 时交易）的期望总剩余 \(\mathbb{E}[\max\{\theta_B-\theta_S,0\}]=\int_0^1\int_0^{\theta_B}(\theta_B-\theta_S)d\theta_S d\theta_B=\int_0^1\frac{\theta_B^2}{2}d\theta_B=\frac16\)，二者相等。故期望外部性机制实现第一好。\(\blacksquare\)Transfers (per (30.6), \(n=2\)): for the buyer \(t_B^{\text{EE}}(\hat\theta_B,\hat\theta_S)=\bar t_B^{\text{VCG}}(\hat\theta_B)+(\bar t_S^{\text{VCG}}-\bar t_S^{\text{VCG}}(\hat\theta_S))-\frac{\bar t_B^{\text{VCG}}+\bar t_S^{\text{VCG}}}{2}\) (30.10). Using the uniform distribution and (30.7), (30.8), (30.9): \(\bar t_B^{\text{VCG}}(\hat\theta_B)=\mathbb{E}_{\theta_S}[\hat x(\hat\theta_B,\theta_S)\theta_S]=\int_0^{\hat\theta_B}\theta_S\,d\theta_S=\frac{\hat\theta_B^2}{2}\), \(\bar t_S^{\text{VCG}}(\hat\theta_S)=\mathbb{E}_{\theta_B}[\theta_B(1-\hat x(\theta_B,\hat\theta_S))]=\int_0^{\hat\theta_S}\theta_B\,d\theta_B=\frac{\hat\theta_S^2}{2}\), \(\bar t_B^{\text{VCG}}=\mathbb{E}_{\theta_B}[\frac{\theta_B^2}{2}]=\frac16\), \(\bar t_S^{\text{VCG}}=\frac16\). Plugging into (30.10): \(t_B^{\text{EE}}=\frac{\hat\theta_B^2}{2}+(\frac16-\frac{\hat\theta_S^2}{2})-\frac{\frac16+\frac16}{2}=\frac{\hat\theta_B^2}{2}-\frac{\hat\theta_S^2}{2}\), and for the seller \(t_S^{\text{EE}}=\frac{\hat\theta_S^2}{2}-\frac{\hat\theta_B^2}{2}\). Budget balance: \(t_B^{\text{EE}}+t_S^{\text{EE}}=0\). ✓ Buyer BIC & IR: \(U_B^{\text{EE}}(\hat\theta_B\mid\theta_B)=\mathbb{P}(\theta_S\le\hat\theta_B)\theta_B+\mathbb{P}(\theta_S>\hat\theta_B)\cdot0-\int_0^1(\frac{\hat\theta_B^2}{2}-\frac{\theta_S^2}{2})d\theta_S=\hat\theta_B\theta_B-\frac{\hat\theta_B^2}{2}+\frac16\), globally concave in \(\hat\theta_B\), FOC \(\theta_B-\hat\theta_B=0\Rightarrow\hat\theta_B=\theta_B\), BIC; \(U_B^{\text{EE}}(\theta_B)=\frac{\theta_B^2}{2}+\frac16>0\) (outside option \(0\)), IR. Seller BIC & IR: \(U_S^{\text{EE}}(\hat\theta_S\mid\theta_S)=(1-\hat\theta_S)(-\theta_S)-\frac{\hat\theta_S^2}{2}+\frac16\), globally concave in \(\hat\theta_S\), FOC \(\theta_S-\hat\theta_S=0\Rightarrow\hat\theta_S=\theta_S\), BIC; \(U_S^{\text{EE}}(\theta_S)=-(1-\theta_S)\theta_S-\frac{\theta_S^2}{2}+\frac16=\frac{\theta_S^2}{2}-\theta_S+\frac16>-\theta_S\) (outside option \(-\theta_S\)), IR. First best: expected total surplus \(\mathbb{E}[U_B^{\text{EE}}]+\mathbb{E}[U_S^{\text{EE}}]=\int_0^1(\frac{\theta_B^2}{2}+\frac16)d\theta_B+\int_0^1(\frac{\theta_S^2}{2}+\frac16-\theta_S)d\theta_S=\frac13+(-\frac16)=\frac16\); and the expected total surplus of the ex-ante welfare-maximizing rule (trade when \(\theta_B>\theta_S\)) is \(\mathbb{E}[\max\{\theta_B-\theta_S,0\}]=\int_0^1\int_0^{\theta_B}(\theta_B-\theta_S)d\theta_S d\theta_B=\int_0^1\frac{\theta_B^2}{2}d\theta_B=\frac16\), which is equal. So the expected externality mechanism achieves the first best. \(\blacksquare\)

定义 30.7（事中个人理性 / Interim individual rationality, interim IR） 机制 \(\{\phi(x\mid\cdot),t_1(\cdot),\dots,t_n(\cdot)\}\) 是事中个人理性 (interim IR) 的，当且仅当 \(U_i(\theta_i)\equiv U_i(\theta_i\mid\theta_i)\ge\underline U_i(\theta_i)\) 对所有 \(i,\theta_i\) 成立。A mechanism \(\{\phi(x\mid\cdot),t_1(\cdot),\dots,t_n(\cdot)\}\) is interim individually rational (interim IR) if and only if \(U_i(\theta_i)\equiv U_i(\theta_i\mid\theta_i)\ge\underline U_i(\theta_i)\) for all \(i,\theta_i\).

注 30.7（事中 IR vs 事前 IR / Remark 30.7: interim vs ex-ante IR） 例 30.1 用的 IR 是事前 IR约束——问参与人在了解自身类型之前是否愿意参与；而事中 IR问的是参与人了解自身类型之后是否愿意参与。不可能性定理 28.4 用的是事中 IR，比事前 IR 更难满足。事实上，§30 之前所有讨论用的 IR 都是这种事中 IR（总要保证最差类型仍愿参与），只是此前一直隐式使用，现在把这一区分讲明。The IR used in Example 30.1 is an ex-ante IR constraint — asking whether an agent wants to participate before learning his type; whereas interim IR asks whether an agent wants to participate after learning his type. The impossibility theorem 28.4 uses interim IR, which is more difficult to satisfy than ex-ante IR. In fact, the IR used in all discussion before section 30 was this interim IR (always guaranteeing the worst type still participates); we had been using it implicitly all along, and now make the distinction explicit.

定义 30.8、30.9（最小补贴、IR-VCG 机制 / Minimum subsidy, IR-VCG mechanism） 定义 30.8（最小补贴）：给参与人 \(i\) 的补贴记为 \(\varphi_i\)，满足 \(U_i^{\text{VCG}}(\theta_i)+\varphi_i\ge\underline U_i(\theta_i)\)。定义最小补贴 \(\varphi_i^\star=\max_{\theta_i}\big(\underline U_i(\theta_i)-U_i^{\text{VCG}}(\theta_i)\big)\)，即使 \(U_i^{\text{VCG}}(\theta_i)+\varphi_i\ge\underline U_i(\theta_i)\) 对所有 \(\theta_i\) 成立的最低补贴值。定义 30.9（IR-VCG 机制）：IR-VCG 机制定义为 \(\{\hat x(\cdot),t_1^{\text{IR-VCG}}(\cdot),\dots,t_n^{\text{IR-VCG}}(\cdot)\}\)，其中 \(\hat x(\cdot)\) 事后有效，\(t_i^{\text{IR-VCG}}(\theta_1,\dots,\theta_n)=t_i^{\text{VCG}}(\theta_1,\dots,\theta_n)-\varphi_i^\star\)。Definition 30.8 (minimum subsidy): a subsidy for agent \(i\) is denoted \(\varphi_i\) such that \(U_i^{\text{VCG}}(\theta_i)+\varphi_i\ge\underline U_i(\theta_i)\). Define the minimum subsidy \(\varphi_i^\star=\max_{\theta_i}\big(\underline U_i(\theta_i)-U_i^{\text{VCG}}(\theta_i)\big)\), the lowest subsidy value such that \(U_i^{\text{VCG}}(\theta_i)+\varphi_i\ge\underline U_i(\theta_i)\) holds for all \(\theta_i\). Definition 30.9 (IR-VCG mechanism): the IR-VCG mechanism is defined by \(\{\hat x(\cdot),t_1^{\text{IR-VCG}}(\cdot),\dots,t_n^{\text{IR-VCG}}(\cdot)\}\), where \(\hat x(\cdot)\) is ex-post efficient and \(t_i^{\text{IR-VCG}}(\theta_1,\dots,\theta_n)=t_i^{\text{VCG}}(\theta_1,\dots,\theta_n)-\varphi_i^\star\).

命题 30.7、注 30.8 / Proposition 30.7, Remark 30.8 命题 30.7：IR-VCG 机制是 DSIC（它与 VCG 只差一个对每个 \(i\) 而言的常数补贴 \(\varphi_i^\star\)，不影响讲真话；由命题 30.3 即得）。注 30.8：IR-VCG 虽仍 DSIC，但未必像 VCG 那样运行期望盈余（因从每人支付中减去补贴），故各税之和可能不再非负。Proposition 30.7: the IR-VCG mechanism is DSIC (it differs from VCG only by a constant subsidy \(\varphi_i^\star\) for each \(i\), which doesn't affect truth-telling; immediate from Proposition 30.3). Remark 30.8: although the IR-VCG mechanism is still DSIC, it does not necessarily run an expected surplus like VCG (since a subsidy is subtracted from each agent's payment), so the taxes may no longer add up to be positive.

推论 30.2（可能性，对应定理 30.1 / Possibility, to Theorem 30.1） 若 IR-VCG 机制运行期望盈余，即 \(\mathbb{E}_{\{\theta_i\}}[\sum_{i=1}^n t_i^{\text{IR-VCG}}(\theta_1,\dots,\theta_n)]=\mathbb{E}_{\{\theta_i\}}[\sum_{i=1}^n(t_i^{\text{VCG}}(\theta_1,\dots,\theta_n)-\varphi_i^\star)]\ge0\) (30.11)，则存在一个 BIC、预算平衡且事中 IR 的机制。If the IR-VCG mechanism runs an expected surplus, i.e. \(\mathbb{E}_{\{\theta_i\}}[\sum_{i=1}^n t_i^{\text{IR-VCG}}(\theta_1,\dots,\theta_n)]=\mathbb{E}_{\{\theta_i\}}[\sum_{i=1}^n(t_i^{\text{VCG}}(\theta_1,\dots,\theta_n)-\varphi_i^\star)]\ge0\) (30.11), then there exists a BIC, budget balanced and interim IR mechanism.

推论 30.2 证明 / Proof of Corollary 30.2 由命题 30.7，IR-VCG 是 DSIC（从而 BIC）。若 (30.11) 成立，由定理 30.1 存在与之对应的预算平衡 BIC 机制。又 IR-VCG 按构造事中 IR（最小补贴恰保证 \(U_i^{\text{VCG}}+\varphi_i^\star\ge\underline U_i\)），而定理 30.1 说新机制中各人至少同样满意，故新机制也事中 IR。\(\blacksquare\)By Proposition 30.7, IR-VCG is DSIC (and thus BIC). If (30.11) holds, by Theorem 30.1 there exists a corresponding budget-balanced BIC mechanism. Also IR-VCG is interim IR by construction (the minimum subsidy exactly guarantees \(U_i^{\text{VCG}}+\varphi_i^\star\ge\underline U_i\)), and Theorem 30.1 says agents are at least equally happy in the new mechanism, so the new mechanism is also interim IR. \(\blacksquare\)

定理 30.2（可能性定理 / Possibility theorem） 若对所有 \(i\)，\(\theta_i\) 在 \([\underline\theta_i,\overline\theta_i]\) 上连续分布且各参与人分布独立，则 (30.11) 是存在 BIC、预算平衡、事后有效且事中 IR 机制的充要条件。If for all \(i\), \(\theta_i\) is continuously distributed on \([\underline\theta_i,\overline\theta_i]\) and the distributions across agents are independent, then (30.11) is necessary and sufficient for the existence of a BIC, budget balanced, ex-post efficient and interim IR mechanism.

证明（充分性 + 必要性 / sufficiency + necessity） 充分性：由推论 30.2 与定理 30.1 直接得到（其证明不依赖 \(\theta_i\) 的分布）。必要性：对任意 BIC、预算平衡、事后有效、事中 IR 机制（用 \(a\) 标记），BIC 蕴含 \(U_i^a(\theta_i)\ge U_i^a(\hat\theta_i\mid\theta_i)\)，其中 \(U_i^a(\theta_i)=\mathbb{E}_{\theta_{-i}}[u_i(\hat x(\theta_i,\theta_{-i}),\theta_i)]-\mathbb{E}_{\theta_{-i}}[t_i^a(\theta_i,\theta_{-i})]\)。利用双向偏离不等式 (30.12)、(30.13)，并对 \(\theta_i\to\hat\theta_i\) 取极限得包络条件，积分得 (30.15)：\(U_i^a(\theta_i)=\underbrace{U_i^a(\underline\theta_i)}_{\text{first term}}+\int_{\underline\theta_i}^{\theta_i}\mathbb{E}_{\theta_{-i}}[\frac{\partial u_i(\hat x(w,\theta_{-i}),\theta_i)}{\partial\theta_i}]dw\)，第二项与具体机制无关，故 \(U_i^a(\theta_i)\) 跨机制只通过 \(U_i^a(\underline\theta_i)\) 不同。由 IR-VCG 也事中 IR，结合 (30.16) 得：任何事中 IR 机制 \(a\) 满足 \(U_i^a(\underline\theta_i)\le U_i^{\text{IR-VCG}}(\underline\theta_i)\)，从而 \(\bar t_i^{\text{IR-VCG}}(\theta_i)\ge\bar t_i^a(\theta_i)\)（机制只通过 \(\bar t_i^a(\theta_i)\) 区别 \(U_i^a\)）。由机制 \(a\) 预算平衡 \(\mathbb{E}[\sum_i t_i^a]=0\)（独立性使 \(\mathbb{E}[\sum_i t_i^a]=\sum_i\mathbb{E}_{\theta_i}[\bar t_i^a(\theta_i)]\)），(30.16) 给出 \(\mathbb{E}_{\theta_i}[\sum_i t_i^{\text{IR-VCG}}]\ge\mathbb{E}_{\theta_i}[\sum_i t_i^a]=0\)，即 (30.11)。\(\blacksquare\)Sufficiency: follows directly from Corollary 30.2 and Theorem 30.1 (whose proofs do not depend on the distribution of \(\theta_i\)). Necessity: for any BIC, budget balanced, ex-post efficient, interim IR mechanism (indexed by \(a\)), BIC implies \(U_i^a(\theta_i)\ge U_i^a(\hat\theta_i\mid\theta_i)\), where \(U_i^a(\theta_i)=\mathbb{E}_{\theta_{-i}}[u_i(\hat x(\theta_i,\theta_{-i}),\theta_i)]-\mathbb{E}_{\theta_{-i}}[t_i^a(\theta_i,\theta_{-i})]\). Using the two-sided deviation inequalities (30.12), (30.13) and taking \(\theta_i\to\hat\theta_i\) to get an envelope condition, integration gives (30.15): \(U_i^a(\theta_i)=\underbrace{U_i^a(\underline\theta_i)}_{\text{first term}}+\int_{\underline\theta_i}^{\theta_i}\mathbb{E}_{\theta_{-i}}[\frac{\partial u_i(\hat x(w,\theta_{-i}),\theta_i)}{\partial\theta_i}]dw\); the second term is not mechanism-specific, so \(U_i^a(\theta_i)\) differs across mechanisms only through \(U_i^a(\underline\theta_i)\). Since IR-VCG is also interim IR, combined with (30.16) we get: any interim-IR mechanism \(a\) satisfies \(U_i^a(\underline\theta_i)\le U_i^{\text{IR-VCG}}(\underline\theta_i)\), hence \(\bar t_i^{\text{IR-VCG}}(\theta_i)\ge\bar t_i^a(\theta_i)\) (mechanisms differ in \(U_i^a\) only through \(\bar t_i^a(\theta_i)\)). Since mechanism \(a\) is budget balanced \(\mathbb{E}[\sum_i t_i^a]=0\) (independence gives \(\mathbb{E}[\sum_i t_i^a]=\sum_i\mathbb{E}_{\theta_i}[\bar t_i^a(\theta_i)]\)), (30.16) gives \(\mathbb{E}_{\theta_i}[\sum_i t_i^{\text{IR-VCG}}]\ge\mathbb{E}_{\theta_i}[\sum_i t_i^a]=0\), which is exactly (30.11). \(\blacksquare\)

例 30.2（重访例 30.1 / Revisit Example 30.1） 在例 30.1 中我们已算得 \(\bar t_B^{\text{VCG}}(\theta_B)=\frac{\theta_B^2}{2}\)、\(\bar t_S^{\text{VCG}}(\theta_S)=\frac{\theta_S^2}{2}\)。则 \(U_B^{\text{VCG}}(\theta_B)=\mathbb{E}_{\theta_S}[u_B(\hat x(\theta_B,\theta_S),\theta_B)]-\bar t_B^{\text{VCG}}(\theta_B)=\mathbb{P}(\theta_S\le\theta_B)\theta_B-\frac{\theta_B^2}{2}=\theta_B^2-\frac{\theta_B^2}{2}=\frac{\theta_B^2}{2}\)，\(U_S^{\text{VCG}}(\theta_S)=(1-\theta_S)(-\theta_S)\dots=-\theta_S+\frac{\theta_S^2}{2}\)。设双方外部选项为 \(0\)：\(\underline U_B(\theta_B)=0\)、\(\underline U_S(\theta_S)=0\)。由定义 30.8，\(\varphi_B^\star=\max_{\theta_B}(0-\frac{\theta_B^2}{2})=0\)、\(\varphi_S^\star=\max_{\theta_S}(0-(-\theta_S+\frac{\theta_S^2}{2}))=\max_{\theta_S}(\theta_S-\frac{\theta_S^2}{2})=\frac12\)（在 \(\theta_S=1\)）。于是 (30.11) 左侧In Example 30.1 we computed \(\bar t_B^{\text{VCG}}(\theta_B)=\frac{\theta_B^2}{2}\) and \(\bar t_S^{\text{VCG}}(\theta_S)=\frac{\theta_S^2}{2}\). Then \(U_B^{\text{VCG}}(\theta_B)=\mathbb{E}_{\theta_S}[u_B(\hat x(\theta_B,\theta_S),\theta_B)]-\bar t_B^{\text{VCG}}(\theta_B)=\mathbb{P}(\theta_S\le\theta_B)\theta_B-\frac{\theta_B^2}{2}=\theta_B^2-\frac{\theta_B^2}{2}=\frac{\theta_B^2}{2}\), and \(U_S^{\text{VCG}}(\theta_S)=(1-\theta_S)(-\theta_S)\dots=-\theta_S+\frac{\theta_S^2}{2}\). Suppose the outside options for both agents are \(0\): \(\underline U_B(\theta_B)=0\), \(\underline U_S(\theta_S)=0\). By Definition 30.8, \(\varphi_B^\star=\max_{\theta_B}(0-\frac{\theta_B^2}{2})=0\) and \(\varphi_S^\star=\max_{\theta_S}(0-(-\theta_S+\frac{\theta_S^2}{2}))=\max_{\theta_S}(\theta_S-\frac{\theta_S^2}{2})=\frac12\) (at \(\theta_S=1\)). Then the LHS of (30.11) is

$$\mathbb{E}[(t_B^{\text{VCG}}-\varphi_B^\star)+(t_S^{\text{VCG}}-\varphi_S^\star)]=\mathbb{E}[t_B^{\text{VCG}}]+\mathbb{E}[t_S^{\text{VCG}}]-\frac12=\frac16+\frac16-\frac12=-\frac23<0\tag{30.17}$$

由于此例中双方类型连续且独立分布于 $[0,1]\(，定理 30.2 适用；(30.17) 表明 (30.11) **不成立**，故**不存在**同时 BIC、预算平衡、事后有效且事中 IR 的机制。Since both types are continuously and independently distributed on \)[0,1]$, Theorem 30.2 applies; (30.17) shows that (30.11) fails, so there is no mechanism that is simultaneously BIC, budget balanced, ex-post efficient and interim IR.

注 30.9（事中 IR 强于事前 IR / Remark 30.9: interim IR is stronger than ex-ante IR） 在例 30.1 中，我们确有一个 BIC、预算平衡、事后有效且事前 IR 的机制（期望外部性机制）。但这里（例 30.2，完全相同的问题）把事前 IR 换成事中 IR 后，就找不到同时 BIC、预算平衡、事后有效且事中 IR 的机制。所以事中 IR 是比事前 IR 更强的约束——这正是 §28 Myerson–Satterthwaite 不可能性定理在多人一般设定下的体现。In Example 30.1 we do have a mechanism that is BIC, budget balanced, ex-post efficient and ex-ante IR (the expected externality mechanism). But here (Example 30.2, exactly the same problem), by switching ex-ante IR to interim IR, we cannot find such a mechanism that is BIC, budget balanced, ex-post efficient and interim IR. So interim IR is a stronger restriction than ex-ante IR — which is precisely the §28 Myerson–Satterthwaite impossibility theorem manifested in the general multi-player setting.