买家 / The buyer 只有一个风险中性买家，效用 \(u(q,\theta)-t\)，准线性、无收入效应（脚注 20.1：因准线性，由 f.o.c. 可见消费品需求只取决于相对价格、对货币无收入效应）。\(q\) 为购买量：单位需求 \(Q=\{0,1\}\)，更一般 \(Q=[0,\overline q]\)；\(t\) 为转移支付；\(\theta\in\Theta=[\underline\theta,\overline\theta]\) 为买家类型（口味/估值）。\(\theta\) 的累积分布 \(F(\theta)\)、密度 \(f(\theta)\)。对 \(u(q,\theta)\) 的假设：关于 \(q\) 递增且凹；\(u_{q\theta}>0\)、\(u_\theta\ge0\)，且 \(u_\theta\) 在 \(\Theta\) 上有界。由买家最优化的 f.o.c. 得需求曲线 \(p=u_q(q,\theta)\)（\(q>0\)，\(p\) 为单价）；\(u_{q\theta}>0\) 意味对 \(\theta>\theta'\) 有 \(u_q(q,\theta)>u_q(q,\theta')\)，即高类型需求曲线更高，需求曲线按类型 \(\theta\) 整齐排序。There is only one risk-neutral buyer with utility \(u(q,\theta)-t\), quasi-linear with no income effect (footnote 20.1: by quasi-linearity, the f.o.c. shows the demand for the consumption good depends only on relative prices, with no income effect for money). \(q\) is the quantity purchased: unit demand \(Q=\{0,1\}\), more generally \(Q=[0,\overline q]\); \(t\) is the transfer; \(\theta\in\Theta=[\underline\theta,\overline\theta]\) is the buyer's type (taste/valuation). The c.d.f. of \(\theta\) is \(F(\theta)\) with density \(f(\theta)\). Assumptions on \(u(q,\theta)\): increasing and concave in \(q\); \(u_{q\theta}>0\), \(u_\theta\ge0\), and \(u_\theta\) is bounded on \(\Theta\). From the buyer's optimization f.o.c. we get the demand curve \(p=u_q(q,\theta)\) (\(q>0\), \(p\) the unit price); \(u_{q\theta}>0\) implies that for \(\theta>\theta'\), \(u_q(q,\theta)>u_q(q,\theta')\), i.e. higher types have higher demand curves, and demand curves are nicely ordered by type \(\theta\).

卖家 / The seller (20.1.2) 只有一个风险中性卖家，收入为 \(t\)，生产 \(q\) 的成本 \(c(\cdot)\) 关于 \(q\) 递增且凸，欲最大化利润 \(t(q)-c(q)\)。定义第一最优产量 \(q^{FB}(\theta)=\arg\max_{q\in Q}u(q,\theta)-c(q)\)。There is only one risk-neutral seller, with revenue \(t\) and cost \(c(\cdot)\) of producing \(q\) increasing and convex in \(q\), wanting to maximize profit \(t(q)-c(q)\). Define the first-best outcome quantity \(q^{FB}(\theta)=\arg\max_{q\in Q}u(q,\theta)-c(q)\).

时序与注 20.1 / Timing and Remark 20.1 (20.1.3) 时序：1. 自然按 \(F(\theta)\) 选 \(\theta\in\Theta\)，买家私下得知 \(\theta\)（卖家不知）；2. 卖家给买家一个"机制"（即一个博弈）；3. 买家拒绝该机制，或接受并玩该机制。注 20.1：时序顺序很重要。若把第 2、3 步提到第 1 步之前（先给机制、再接受/玩、最后自然选 \(\theta\)），则可定义第一最优利润 \(\pi^{FB}=\mathbb{E}_\theta[\max_q u(q,\theta)-c(q)]\)；卖家可在买家知道类型前以 \(\pi^{FB}\) 把企业卖给买家（第 3 步现在才发生），买家随后在类型揭示后选第一最优——卖家事前在期望意义上拿走买家的全部利润。为使博弈非平凡，我们保留原始时序。Timing: 1. Nature chooses \(\theta\in\Theta\) according to \(F(\theta)\), the buyer privately learns \(\theta\) (unknown to the seller); 2. the seller offers the buyer a "mechanism" (a game); 3. the buyer rejects the mechanism, or accepts and plays it. Remark 20.1: the order of the timing matters. If we move steps 2 and 3 ahead of step 1 (seller offers mechanism → buyer rejects/plays → nature chooses \(\theta\)), then we can define the first-best profit \(\pi^{FB}=\mathbb{E}_\theta[\max_q u(q,\theta)-c(q)]\); the seller can sell the firm to the buyer at \(\pi^{FB}\) before the buyer knows his type (step 3 now), and the buyer later chooses the first best after the type is revealed — the seller takes away all the buyer's profits ex ante in expectation. To make the game non-trivial, we keep the original order.

定义 20.1（机制）/ Definition 20.1 (Mechanism) 机制是一个扩展型博弈 \(\Gamma\)，其中卖家的策略预先承诺。注 20.2：卖家设计博弈并对每个决策节点的行动作出承诺；在只有一个买家的情形下，买家清楚地知道对手在每个节点会做什么，故扩展型博弈退化为单人（买家）决策树。A mechanism is an extensive-form game \(\Gamma\) in which the seller's strategy is pre-committed. Remark 20.2: the seller designs the game and commits to actions at each decision node; with only one buyer, the buyer clearly knows what the counterpart would do at each node, so the extensive-form game reduces to a single-person (buyer) decision tree.

四个机制例子 / Four examples of mechanisms 例 20.1（单位需求）：\(u(q,\theta)=\theta\cdot q\)，\(q\in\{0,1\}\)。卖家张贴单一价格 \(p\)；买家可买 1 单位或离开，支付为 \(u(1,\theta)-p\)（\(q=1\)）或 \(u(0,\theta)-0\)（\(q=0\)）。例 20.2（价格函数）：机制可为卖家张贴的价格函数 \(t=q\times p(q)\)，买家选 \(q\) 的支付 \(u(q,\theta)-q\times p(q)\)。例 20.3（彩票菜单）：机制可为价格 \(p(\phi)\) 的彩票菜单 \(\phi(q,t)\)，买家期望支付 \(\mathbb{E}_\phi[u(q,\theta)-t]=\int_Q\int_\mathbb{R}(u(q,\theta)-t-p(\phi))\phi(q,t)\,dt\,dq\)。例 20.4（多轮博弈）：机制可为多轮博弈，买家经数步最终得到价格 \(p\)、数量 \(q\) 的报价。Example 20.1 (Unit demand): \(u(q,\theta)=\theta\cdot q\), \(q\in\{0,1\}\). The seller posts a single price \(p\); the buyer can buy 1 unit or leave, with payoff \(u(1,\theta)-p\) (\(q=1\)) or \(u(0,\theta)-0\) (\(q=0\)). Example 20.2 (Price function): the mechanism could be a price function \(t=q\times p(q)\) posted by the seller, the buyer's payoff for choosing \(q\) being \(u(q,\theta)-q\times p(q)\). Example 20.3 (Menu of lotteries): the mechanism could be a menu of lotteries \(\phi(q,t)\) at price \(p(\phi)\), the buyer's expected payoff \(\mathbb{E}_\phi[u(q,\theta)-t]=\int_Q\int_\mathbb{R}(u(q,\theta)-t-p(\phi))\phi(q,t)\,dt\,dq\). Example 20.4 (Multi-round game): the mechanism could be a multi-round game where the buyer goes through several steps to finally obtain an offer of price \(p\) and quantity \(q\).

定义 20.2（直接机制）/ Definition 20.2 (Direct mechanism) 直接机制 \(\hat\Gamma\) 含两步：(1) 买家向卖家报告类型 \(\hat\theta\in\Theta\)（讲真话或撒谎）；(2) 卖家基于报告 \(\hat\theta\) 给买家一个 \((q,t)\) 的分布 \(\{\hat\phi(q,t\mid\hat\theta)\}_{\hat\theta\in\Theta}\)（无额外成本）。若最终结果是类型的确定性函数，则该分布退化为 \(\{\hat q(\hat\theta),\hat t(\hat\theta)\}_{\hat\theta\in\Theta}\)。（脚注 20.2：此后记报告类型为 \(\hat\theta\)、真实类型为 \(\theta\)。）注 20.3：\(\hat\Gamma\) 之所以"直接"，是因为任何机制最终都由到达末端节点时 \((q,t)\) 的分布刻画；在 \(\hat\Gamma\) 中卖家直接给出该末端分布，故名"直接"。A direct mechanism \(\hat\Gamma\) involves two steps: (1) the buyer reports a type \(\hat\theta\in\Theta\) to the seller (truthfully or lying); (2) based on the reported \(\hat\theta\), the seller offers the buyer a distribution \(\{\hat\phi(q,t\mid\hat\theta)\}_{\hat\theta\in\Theta}\) of \((q,t)\) (at no additional cost). If the final outcomes are a deterministic function of types, the distribution degenerates to \(\{\hat q(\hat\theta),\hat t(\hat\theta)\}_{\hat\theta\in\Theta}\). (footnote 20.2: hereafter denote reported type by \(\hat\theta\) and true type by \(\theta\).) Remark 20.3: \(\hat\Gamma\) is called "direct" because any mechanism is finally characterized by a distribution of \((q,t)\) when the ending nodes are reached; in \(\hat\Gamma\) the seller directly offers that final distribution, which justifies the name.

例 20.5（重访单位需求）与注 20.4 / Example 20.5 (Unit demand revisited) and Remark 20.4 确定性直接机制：\(u(q,\theta)=\theta\cdot q\)、\(q\in\{0,1\}\)。第一步买家报告 \(\hat\theta\in\Theta\)；第二步卖家给出 \(\hat q(\hat\theta)=\{1\text{ if }\hat\theta\ge p;\ 0\text{ if }\hat\theta注 20.4：买家无激励偏离真实 \(\theta\)，真实类型被揭示，达到完全揭示。下面将证明任何均衡结果都可由一个达到完全揭示的直接机制实现。A deterministic direct mechanism: \(u(q,\theta)=\theta\cdot q\), \(q\in\{0,1\}\). Step 1 the buyer reports \(\hat\theta\in\Theta\); step 2 the seller offers \(\hat q(\hat\theta)=\{1\text{ if }\hat\theta\ge p;\ 0\text{ if }\hat\thetaRemark 20.4: the buyer has no incentive to deviate from reporting true \(\theta\), the true type is revealed, and full revelation is achieved. We next show any equilibrium outcome can be implemented with a direct mechanism achieving full revelation.

定理 20.1（确定性结果的显示原理）/ Theorem 20.1 (Revelation principle for deterministic outcomes) 对任何具有确定性均衡结果 \(\{q^\star(\theta),t^\star(\theta)\}_{\theta\in\Theta}\) 的机制 \(\Gamma\)，都存在一个直接机制 \(\hat\Gamma\)，使买家的最优策略是讲真话（\(\hat\theta^\star_{\hat\Gamma}(\theta)=\theta\)），且原机制的均衡结果被实现（\(\hat q(\hat\theta)=q^\star(\theta)\)、\(\hat t(\theta)=t^\star(\theta)\)）。For any mechanism \(\Gamma\) with deterministic equilibrium outcome \(\{q^\star(\theta),t^\star(\theta)\}_{\theta\in\Theta}\), there exists a direct mechanism \(\hat\Gamma\) such that the buyer's optimal strategy is telling the truth (\(\hat\theta^\star_{\hat\Gamma}(\theta)=\theta\)), and the equilibrium outcome in the original mechanism is implemented (\(\hat q(\hat\theta)=q^\star(\theta)\), \(\hat t(\theta)=t^\star(\theta)\)).

证明 / Proof 构造 \(\hat\Gamma\)：令 \(\hat q(\hat\theta)=q^\star(\hat\theta)\)、\(\hat t(\hat\theta)=t^\star(\hat\theta)\)，对任一报告 \(\hat\theta\in\Theta\) 复制原机制 \(\Gamma\) 的均衡结果。若买家讲真话 \(\hat\theta=\theta\)，支付为 \(u(\hat q(\theta),\theta)-\hat t(\theta)=u(q^\star(\theta),\theta)-t^\star(\theta)\)；若撒谎 \(\hat\theta\ne\theta\)，支付为 \(u(q^\star(\hat\theta),\theta)-t^\star(\hat\theta)\)。因 \(\{q^\star,t^\star\}\) 是原 \(\Gamma\) 的均衡结果，由显示偏好无有利偏离，故 \(u(q^\star(\theta),\theta)-t^\star(\theta)\ge u(q^\star(\hat\theta),\theta)-t^\star(\hat\theta)\)，即 \(u(\hat q(\theta),\theta)-\hat t(\theta)\ge u(\hat q(\hat\theta),\theta)-\hat t(\hat\theta)\)，买家讲真话。\(\blacksquare\)Construct \(\hat\Gamma\): let \(\hat q(\hat\theta)=q^\star(\hat\theta)\), \(\hat t(\hat\theta)=t^\star(\hat\theta)\), replicating the equilibrium outcome of \(\Gamma\) for any reported \(\hat\theta\in\Theta\). If the buyer tells the truth \(\hat\theta=\theta\), his payoff is \(u(\hat q(\theta),\theta)-\hat t(\theta)=u(q^\star(\theta),\theta)-t^\star(\theta)\); if he lies \(\hat\theta\ne\theta\), his payoff is \(u(q^\star(\hat\theta),\theta)-t^\star(\hat\theta)\). Since \(\{q^\star,t^\star\}\) is the equilibrium outcome of the original \(\Gamma\), by revealed preference there is no attractive deviation, so \(u(q^\star(\theta),\theta)-t^\star(\theta)\ge u(q^\star(\hat\theta),\theta)-t^\star(\hat\theta)\), i.e. \(u(\hat q(\theta),\theta)-\hat t(\theta)\ge u(\hat q(\hat\theta),\theta)-\hat t(\hat\theta)\), and the buyer tells the truth. \(\blacksquare\)

定理 20.2（随机结果的显示原理）/ Theorem 20.2 (Revelation principle for random outcomes) 对任何具有随机均衡结果 \(\phi^\star(q,t\mid\theta)\) 的机制 \(\Gamma\)，都存在直接机制 \(\hat\Gamma\)，使买家最优策略为讲真话，且原均衡结果被实现 \(\{\hat\phi(q,t\mid\theta)=\phi^\star(q,t\mid\theta)\}_{\theta\in\Theta}\)。证明类似：令 \(\hat\phi(q,t\mid\hat\theta)=\phi^\star(q,t\mid\hat\theta)\)，由 \(\mathbb{E}_{\hat\theta}[u(q,\theta)-t\mid\theta]\ge\mathbb{E}_{\hat\theta}[u(q,\theta)-t\mid\hat\theta]\) 得讲真话。For any mechanism \(\Gamma\) with random equilibrium outcome \(\phi^\star(q,t\mid\theta)\), there exists a direct mechanism \(\hat\Gamma\) such that the buyer's optimal strategy is telling the truth, and the original equilibrium outcome is implemented \(\{\hat\phi(q,t\mid\theta)=\phi^\star(q,t\mid\theta)\}_{\theta\in\Theta}\). The proof is similar: let \(\hat\phi(q,t\mid\hat\theta)=\phi^\star(q,t\mid\hat\theta)\), and from \(\mathbb{E}_{\hat\theta}[u(q,\theta)-t\mid\theta]\ge\mathbb{E}_{\hat\theta}[u(q,\theta)-t\mid\hat\theta]\) truth-telling follows.

注 20.5、注 20.6 / Remarks 20.5, 20.6 注 20.5：显示原理的直觉是卖家可跳过通往末端分布的所有步骤，直接给出正确对应于类型的末端分布，并靠显示偏好的逻辑让买家揭示真实类型；关键是卖家必须知道原分布中类型与结果的正确对应。注 20.6：由显示原理，任何复杂机制（含其均衡）都等价（在均衡结果意义上）于一个完全揭示的直接机制。故以下讨论只聚焦于完全揭示的直接机制。Remark 20.5: the intuition of the revelation principle is that the seller can skip all the steps leading to the final distribution and directly offer the correctly type-dependent final distribution, letting the buyer reveal the true type by the logic of revealed preference; the key is that the seller must know the correct correspondence between type and outcome in the original distribution. Remark 20.6: by the revelation principle, any complicated mechanism (with its equilibrium) is equivalent (in equilibrium outcome) to a direct mechanism consistent with full revelation. So in the following discussion we focus only on direct mechanisms featuring full revelation.

定义 20.3、20.4 与注 20.7–20.9 / Definitions 20.3, 20.4 and Remarks 20.7–20.9 定义 20.3（激励相容）：定义报告真实类型的支付 \(U(\theta)\equiv u(q(\theta),\theta)-t(\theta)\)。机制 \(\{q(\theta),t(\theta)\}_{\theta\in\Theta}\) 激励相容当且仅当 \(U(\theta)\ge u(q(\hat\theta),\theta)-t(\hat\theta)\) 对 \(\forall\theta,\hat\theta\)。注 20.7：此 IC 条件恰是 (20.1) 中讲真话的 IC。定义 20.4（可实施性）：配置 \(\{q(\theta),U(\theta)\}_{\theta\in\Theta}\) 可实施，若对 \(t(\theta)=u(q(\theta),\theta)-U(\theta)\)，机制 \(\{q(\theta),t(\theta)\}_{\theta\in\Theta}\) 激励相容。注 20.8：可实施性即"无人会撒谎，故分布 \(\{q(\theta),U(\theta)\}\) 可按报告类型正确执行"。注 20.9：可实施性与 IC 等价，都关乎完全揭示；但 IC 针对机制 \(\{q(\theta),t(\theta)\}\)，可实施性针对配置 \(\{q(\theta),U(\theta)\}\)。Definition 20.3 (Incentive compatibility): define the payoff of reporting the true type by \(U(\theta)\equiv u(q(\theta),\theta)-t(\theta)\). A mechanism \(\{q(\theta),t(\theta)\}_{\theta\in\Theta}\) is incentive compatible iff \(U(\theta)\ge u(q(\hat\theta),\theta)-t(\hat\theta)\) for \(\forall\theta,\hat\theta\). Remark 20.7: this IC condition is exactly the truth-telling IC in (20.1). Definition 20.4 (Implementability): an allocation \(\{q(\theta),U(\theta)\}_{\theta\in\Theta}\) is implementable if for \(t(\theta)=u(q(\theta),\theta)-U(\theta)\), the mechanism \(\{q(\theta),t(\theta)\}_{\theta\in\Theta}\) is incentive compatible. Remark 20.8: implementability simply says no one will have an incentive to lie, so the distribution \(\{q(\theta),U(\theta)\}\) can be correctly carried out based on reported types. Remark 20.9: implementability and IC are equivalent, both focusing on full revelation; but IC is defined for a mechanism \(\{q(\theta),t(\theta)\}\), whereas implementability is for an allocation \(\{q(\theta),U(\theta)\}\).

引理 20.1（必要性）/ Lemma 20.1 (Necessity) 若直接显示机制 \(\{q(\theta),t(\theta)\}_{\theta\in\Theta}\) 激励相容，则 (1) \(q(\cdot)\) 非降；(2) \(U(\theta)=U(\underline\theta)+\int_{\underline\theta}^\theta u_\theta(q(s),s)\,ds\)。（注 20.10：(2) 的等价条件 (2)′：\(t(\theta)=u(q(\theta),\theta)-U(\underline\theta)-\int_{\underline\theta}^\theta u_\theta(q(s),s)\,ds\)，来自恒等式 \(U(\theta)\equiv u(q(\theta),\theta)-t(\theta)\)。）If a direct revelation mechanism \(\{q(\theta),t(\theta)\}_{\theta\in\Theta}\) is incentive compatible, then (1) \(q(\cdot)\) is non-decreasing; (2) \(U(\theta)=U(\underline\theta)+\int_{\underline\theta}^\theta u_\theta(q(s),s)\,ds\). (Remark 20.10: equivalent condition (2)′: \(t(\theta)=u(q(\theta),\theta)-U(\underline\theta)-\int_{\underline\theta}^\theta u_\theta(q(s),s)\,ds\), from the identity \(U(\theta)\equiv u(q(\theta),\theta)-t(\theta)\).)

证明（必要性）/ Proof (necessity) IC 蕴含 \(U(\theta)\ge u(q(\hat\theta),\theta)-t(\hat\theta)\)。因 \(U(\hat\theta)=u(q(\hat\theta),\hat\theta)-t(\hat\theta)\)，得 \(U(\theta)-U(\hat\theta)\ge u(q(\hat\theta),\theta)-u(q(\hat\theta),\hat\theta)\) (20.2)。由 \(\theta,\hat\theta\) 对称，(20.2) 也给出 \(U(\theta)-U(\hat\theta)\le u(q(\theta),\theta)-u(q(\theta),\hat\theta)\) (20.3)。合并：IC implies \(U(\theta)\ge u(q(\hat\theta),\theta)-t(\hat\theta)\). Since \(U(\hat\theta)=u(q(\hat\theta),\hat\theta)-t(\hat\theta)\), we get \(U(\theta)-U(\hat\theta)\ge u(q(\hat\theta),\theta)-u(q(\hat\theta),\hat\theta)\) (20.2). By symmetry between \(\theta,\hat\theta\), (20.2) also gives \(U(\theta)-U(\hat\theta)\le u(q(\theta),\theta)-u(q(\theta),\hat\theta)\) (20.3). Combining:

$$u(q(\theta),\theta)-u(q(\hat\theta),\theta)\ge U(\theta)-U(\hat\theta)\ge u(q(\hat\theta),\theta)-u(q(\hat\theta),\hat\theta) \tag{20.4}$$

$$\Rightarrow u(q(\theta),\theta)-u(q(\hat\theta),\theta)\ge u(q(\theta),\hat\theta)-u(q(\hat\theta),\hat\theta) \tag{20.5}$$

因 \(u_{q\theta},u_\theta>0\)，对 \(\theta>\hat\theta\)，(20.5) 蕴含 \(q(\theta)\ge q(\hat\theta)\)，故 \(q(\cdot)\) 非降。再令 \(\theta>\hat\theta\) 且 \(\hat\theta\to\theta\)，若 \(u\) 在 \(\theta\) 连续，(20.4) 两端除以 \((\theta-\hat\theta)\) 取极限得 \(u_\theta(q(\theta),\theta)\ge U'(\theta)\ge u_\theta(q(\theta),\theta)\)，即 \(U'(\theta)=u_\theta(q(\theta),\theta)\) (20.6)，从而 \(U(\theta)=U(\underline\theta)+\int_{\underline\theta}^\theta u_\theta(q(s),s)\,ds\) (20.7)。\(\blacksquare\)Since \(u_{q\theta},u_\theta>0\), for \(\theta>\hat\theta\), (20.5) implies \(q(\theta)\ge q(\hat\theta)\), so \(q(\cdot)\) is non-decreasing. Now let \(\theta>\hat\theta\) and \(\hat\theta\to\theta\); if \(u\) is continuous at \(\theta\), dividing (20.4) by \((\theta-\hat\theta)\) and taking the limit gives \(u_\theta(q(\theta),\theta)\ge U'(\theta)\ge u_\theta(q(\theta),\theta)\), i.e. \(U'(\theta)=u_\theta(q(\theta),\theta)\) (20.6), hence \(U(\theta)=U(\underline\theta)+\int_{\underline\theta}^\theta u_\theta(q(s),s)\,ds\) (20.7). \(\blacksquare\)

注 20.11–20.13 / Remarks 20.11–20.13 注 20.11：引理说任何 IC（完全揭示）直接机制必有可实施配置 \(\{q(\theta),U(\theta)\}\)，使 \(q(\cdot)\) 非降且 \(U(\theta)\) 是 \(u_\theta(q(\theta),\theta)\) 的积分。注 20.12（一阶结果）：定义真实类型 \(\theta\)、报告 \(\hat\theta\) 的支付 \(U(\hat\theta\mid\theta)\equiv u(q(\hat\theta),\theta)-t(\hat\theta)\)，IC 给出 \(U(\theta)\ge\max_{\hat\theta}U(\hat\theta\mid\theta)\) (20.8)；记 \(U_1,U_2\) 为关于 \(\hat\theta,\theta\) 的偏导，则 \(U_1(\theta\mid\theta)=0\)，且 \(U'(\theta)=U_1(\theta\mid\theta)+U_2(\theta\mid\theta)=U_2(\theta\mid\theta)=u_\theta(q(\theta),\theta)\)，即 (20.7)。注 20.13（二阶结果）：由 \(U_1(\theta\mid\theta)=0\) 全微分得 \(U_{11}+U_{12}=0\)；局部二阶条件 \(U_{11}\le0\Rightarrow U_{12}\ge0\)，而 \(U_{12}(\theta\mid\theta)=u_{q\theta}(q(\theta),\theta)q'(\theta)\ge0\)，因 \(u_{q\theta}\ge0\) 故 \(q'(\theta)\ge0\)，\(q(\cdot)\) 非降。Remark 20.11: the lemma says any IC (full-revelation) direct mechanism must have an implementable allocation \(\{q(\theta),U(\theta)\}\) such that \(q(\cdot)\) is non-decreasing and \(U(\theta)\) is an integral of \(u_\theta(q(\theta),\theta)\). Remark 20.12 (first-order result): define the payoff of reporting \(\hat\theta\) at true type \(\theta\) by \(U(\hat\theta\mid\theta)\equiv u(q(\hat\theta),\theta)-t(\hat\theta)\); IC gives \(U(\theta)\ge\max_{\hat\theta}U(\hat\theta\mid\theta)\) (20.8); denoting \(U_1,U_2\) the partials w.r.t. \(\hat\theta,\theta\), we have \(U_1(\theta\mid\theta)=0\) and \(U'(\theta)=U_1(\theta\mid\theta)+U_2(\theta\mid\theta)=U_2(\theta\mid\theta)=u_\theta(q(\theta),\theta)\), i.e. (20.7). Remark 20.13 (second-order result): from \(U_1(\theta\mid\theta)=0\), the total derivative gives \(U_{11}+U_{12}=0\); the local second-order condition \(U_{11}\le0\Rightarrow U_{12}\ge0\), and \(U_{12}(\theta\mid\theta)=u_{q\theta}(q(\theta),\theta)q'(\theta)\ge0\), so since \(u_{q\theta}\ge0\) we get \(q'(\theta)\ge0\), \(q(\cdot)\) non-decreasing.

引理 20.2（充分性）/ Lemma 20.2 (Sufficiency) 直接显示机制 \(\{q(\theta),t(\theta)\}_{\theta\in\Theta}\) 激励相容、配置 \(\{q(\theta),U(\theta)\}_{\theta\in\Theta}\) 可实施，若 (1) \(q(\cdot)\) 非降；(2) \(U(\theta)=U(\underline\theta)+\int_{\underline\theta}^\theta u_\theta(q(s),s)\,ds\) 对 \(\forall\theta\)。A direct revelation mechanism \(\{q(\theta),t(\theta)\}_{\theta\in\Theta}\) is incentive compatible and \(\{q(\theta),U(\theta)\}_{\theta\in\Theta}\) implementable if (1) \(q(\cdot)\) is non-decreasing; (2) \(U(\theta)=U(\underline\theta)+\int_{\underline\theta}^\theta u_\theta(q(s),s)\,ds\) for \(\forall\theta\).

证明（充分性）/ Proof (sufficiency) 欲证对任意 \(\theta\)，\(U(\theta)\ge u(q(\hat\theta),\theta)-t(\hat\theta)=U(\hat\theta)+u(q(\hat\theta),\theta)-u(q(\hat\theta),\hat\theta)\)，即 \(U(\theta)-U(\hat\theta)\ge u(q(\hat\theta),\theta)-u(q(\hat\theta),\hat\theta)\)。由条件 (2)，IC 等价于 \(\int_{\hat\theta}^\theta u_\theta(q(s),s)\,ds\ge\int_{\hat\theta}^\theta u_\theta(q(\hat\theta),s)\,ds\)。因 \(q(\cdot)\) 非降，对 \(\theta>\hat\theta\) 与 \(\forall s\in[\hat\theta,\theta]\) 有 \(q(s)\ge q(\hat\theta)\)；又 \(u_{\theta q}>0\)，故 \(u_\theta(q(s),s)\ge u_\theta(q(\hat\theta),s)\)，从而上式成立，等价于 IC。\(\blacksquare\)We want to show for any \(\theta\), \(U(\theta)\ge u(q(\hat\theta),\theta)-t(\hat\theta)=U(\hat\theta)+u(q(\hat\theta),\theta)-u(q(\hat\theta),\hat\theta)\), i.e. \(U(\theta)-U(\hat\theta)\ge u(q(\hat\theta),\theta)-u(q(\hat\theta),\hat\theta)\). By condition (2), IC is equivalent to \(\int_{\hat\theta}^\theta u_\theta(q(s),s)\,ds\ge\int_{\hat\theta}^\theta u_\theta(q(\hat\theta),s)\,ds\). Since \(q(\cdot)\) is non-decreasing, for \(\theta>\hat\theta\) and \(\forall s\in[\hat\theta,\theta]\) we have \(q(s)\ge q(\hat\theta)\); and \(u_{\theta q}>0\), so \(u_\theta(q(s),s)\ge u_\theta(q(\hat\theta),s)\), giving the inequality, equivalent to IC. \(\blacksquare\)

注 20.14 / Remark 20.14 称 \(\tilde u(q,\theta)\) 为虚拟效用，是因为卖家在对 \(\tilde u(q,\theta)\) 最大化。若把 \(\tilde u(q,\theta)\) 看作买家的效用函数，则风险中性卖家就是在最大化总产出，看起来像在最大化社会剩余的社会规划者。We call \(\tilde u(q,\theta)\) virtual utility because the seller is maximizing over \(\tilde u(q,\theta)\). If we regard \(\tilde u(q,\theta)\) as the buyer's utility function, the risk-neutral seller is maximizing total output, looking like a social planner maximizing social surplus.

20.4.5 五步法总结 / Summary for the 5 steps 步骤 1：用充要条件刻画 IC/可实施机制：(a) \(q(\cdot)\) 非降；(b) \(U(\theta)=U(\underline\theta)+\int_{\underline\theta}^\theta u_\theta(q(s),s)\,ds\)。步骤 2：用该条件化简 \(\mathbb{E}_\theta[U(\theta)]\) 得 (20.11)。步骤 3：代入卖家目标 (20.16) 并（在正则条件下）对 \(q(\cdot)\) 逐 \(\theta\) 点态取 f.o.c.：\(\max_{\{q(\cdot)\}}\mathbb{E}_\theta[u(q(\theta),\theta)-c(q(\theta))-U(\underline\theta)-u_\theta(q(\theta),\theta)\frac{1-F(\theta)}{f(\theta)}]\)。步骤 4：由 \(q(\cdot)\) 恢复 \(U(\theta)=U(\underline\theta)[=0\text{ by IR}]+\int_{\underline\theta}^\theta u_\theta(q(s),s)\,ds\)、\(t(\theta)=u(q(\theta),\theta)-U(\theta)\)。步骤 5：找实施直接机制 \(\{q(\theta),U(\theta)\}\) 的 \(P(q)\)（征税原理，脚注 20.4：与税无关，名称源于其与税法一样寻找数量与价格的对应菜单）——它是寻找直接机制的逆。\(P(q)\) 为买 \(q\) 的总价，\(P(q)=\{t(\theta)\text{ if }q=q(\theta)\text{ for any }\theta;\ \infty\text{ otherwise}\}\)；因 \(q(\cdot)\) 未必严增、\(q^{-1}\) 未必良定义，定义 \(\overline\theta(q)=\sup_\theta\{\theta:q(\theta)=q\}\)，则 \(P(q)=t(\overline\theta(q))\) 即所求价格菜单。Step 1: characterize the IC/implementable mechanism with the necessary-sufficient conditions: (a) \(q(\cdot)\) non-decreasing; (b) \(U(\theta)=U(\underline\theta)+\int_{\underline\theta}^\theta u_\theta(q(s),s)\,ds\). Step 2: simplify \(\mathbb{E}_\theta[U(\theta)]\) using the condition to obtain (20.11). Step 3: incorporate into the seller's objective (20.16) and (under regularity) take the f.o.c. to maximize over \(q(\cdot)\) point-wise at each \(\theta\): \(\max_{\{q(\cdot)\}}\mathbb{E}_\theta[u(q(\theta),\theta)-c(q(\theta))-U(\underline\theta)-u_\theta(q(\theta),\theta)\frac{1-F(\theta)}{f(\theta)}]\). Step 4: recover \(U(\theta)=U(\underline\theta)[=0\text{ by IR}]+\int_{\underline\theta}^\theta u_\theta(q(s),s)\,ds\) and \(t(\theta)=u(q(\theta),\theta)-U(\theta)\) from \(q(\cdot)\). Step 5: find \(P(q)\) implementing the direct mechanism \(\{q(\theta),U(\theta)\}\) (the taxation principle, footnote 20.4: nothing to do with taxes; the name comes from the similarity to a tax code finding a menu of correspondence between quantity and price) — the reverse of finding a direct mechanism. \(P(q)\) is the total price paid for \(q\), \(P(q)=\{t(\theta)\text{ if }q=q(\theta)\text{ for any }\theta;\ \infty\text{ otherwise}\}\); since \(q(\cdot)\) is not necessarily strictly increasing and \(q^{-1}\) may not be well-defined, define \(\overline\theta(q)=\sup_\theta\{\theta:q(\theta)=q\}\), then \(P(q)=t(\overline\theta(q))\) is the desired price menu.

检验结果与注 20.15 / Check the result and Remark 20.15 检验：价格菜单 (20.19) 让买家选最大化卖家利润的 \(q\)，即 \(q=2\theta-3\)。买家问题 \(\max_q u(q,\theta)-P(q)\Leftrightarrow\max_q\theta q-\tfrac12 q^2-\tfrac32 q+\tfrac14 q^2\)，f.o.c. \(\theta-q-\tfrac32+\tfrac12 q=0\Rightarrow q=2\theta-3\)，与目标一致。注 20.15：数量折扣即 \(P'(q)>0\) 且 \(P''(q)<0\)。由买家 f.o.c. \(u_q(q,\theta)=P'(q)\)；价格折扣等价于 \(P'(q)=u_q(q,\theta)>0\) 与 \(P''(q)=u_{qq}(q,\theta)-u_{q\theta}(q,\theta)\dfrac{\Lambda_{qq}(q,\theta)}{\Lambda_{q\theta}(q,\theta)}<0\)。Check: the price menu (20.19) makes the buyer choose the \(q\) that maximizes the seller's profit, \(q=2\theta-3\). The buyer's problem \(\max_q u(q,\theta)-P(q)\Leftrightarrow\max_q\theta q-\tfrac12 q^2-\tfrac32 q+\tfrac14 q^2\), f.o.c. \(\theta-q-\tfrac32+\tfrac12 q=0\Rightarrow q=2\theta-3\), matching the target. Remark 20.15: a quantity discount means \(P'(q)>0\) and \(P''(q)<0\). From the buyer's f.o.c. \(u_q(q,\theta)=P'(q)\); a price discount is equivalent to \(P'(q)=u_q(q,\theta)>0\) and \(P''(q)=u_{qq}(q,\theta)-u_{q\theta}(q,\theta)\dfrac{\Lambda_{qq}(q,\theta)}{\Lambda_{q\theta}(q,\theta)}<0\).

20.5.3 简化：IR1 + IC2 ⟹ IR2 / Simplification: IR1 + IC2 ⟹ IR2 因 \(\theta_2>\theta_1\)、\(q_1\) 非负，故 \(\theta_2 q_1-\tfrac12 q_1^2-t_1>\theta_1 q_1-\tfrac12 q_1^2-t_1\)；代入 IC2：\(\theta_2 q_2-\tfrac12 q_2^2-t_2\ge\theta_2 q_1-\tfrac12 q_1^2-t_1>\theta_1 q_1-\tfrac12 q_1^2-t_1\)；由 IR1 \(\theta_1 q_1-\tfrac12 q_1^2-t_1\ge0\)，故 \(\theta_2 q_2-\tfrac12 q_2^2-t_2>0\)（IR2）。于是可剔除冗余的 IR2，简化问题为 \(\max\phi(t_2-cq_2)+(1-\phi)(t_1-cq_1)\) s.t. IR1、IC2、IC1。Since \(\theta_2>\theta_1\) and \(q_1\) non-negative, \(\theta_2 q_1-\tfrac12 q_1^2-t_1>\theta_1 q_1-\tfrac12 q_1^2-t_1\); plug into IC2: \(\theta_2 q_2-\tfrac12 q_2^2-t_2\ge\theta_2 q_1-\tfrac12 q_1^2-t_1>\theta_1 q_1-\tfrac12 q_1^2-t_1\); by IR1 \(\theta_1 q_1-\tfrac12 q_1^2-t_1\ge0\), so \(\theta_2 q_2-\tfrac12 q_2^2-t_2>0\) (IR2). So we eliminate the redundant IR2 and simplify the problem to \(\max\phi(t_2-cq_2)+(1-\phi)(t_1-cq_1)\) s.t. IR1, IC2, IC1.

IC2 与 IR1 在松弛问题中均绑定（证明）/ IC2 and IR1 both binding (proof) IC2 绑定：反设 \(\theta_2 q_2-\tfrac12 q_2^2-t_2>\theta_2 q_1-\tfrac12 q_1^2-t_1\)，则有 \(\varepsilon>0\) 使 \(\tilde t_2=t_2+\varepsilon\) 仍满足 IC2；\(t_2\) 只在 IC2 中出现，IR1 不受影响，两约束在 \(\tilde t_2\) 下都成立；卖家在 \(\tilde t_2\) 下赚更多，故 \(t_2\) 非解，任何解须 IC2 绑定。IR1 绑定：反设 \(\theta_1 q_1-\tfrac12 q_1^2-t_1>0\)，则有 \(\varepsilon>0\) 使 \(\tilde t_1=t_1+\varepsilon\) 满足 IR1；IC2 中更高的 \(t_1\) 使 IC2 更易成立，两约束在 \(\tilde t_1\) 下都成立；卖家在 \(\tilde t_1\) 下赚更多，故 IR1 须绑定。\(\blacksquare\)IC2 binding: suppose not, \(\theta_2 q_2-\tfrac12 q_2^2-t_2>\theta_2 q_1-\tfrac12 q_1^2-t_1\); then there is \(\varepsilon>0\) with \(\tilde t_2=t_2+\varepsilon\) still satisfying IC2; \(t_2\) only appears in IC2, so IR1 is unaffected and both constraints hold with \(\tilde t_2\); the seller makes more with \(\tilde t_2\), so \(t_2\) cannot be the solution and any solution has IC2 binding. IR1 binding: suppose not, \(\theta_1 q_1-\tfrac12 q_1^2-t_1>0\); then there is \(\varepsilon>0\) with \(\tilde t_1=t_1+\varepsilon\) satisfying IR1; a higher \(t_1\) in IC2 makes IC2 hold more easily, so both constraints hold with \(\tilde t_1\); the seller makes more with \(\tilde t_1\), so IR1 must bind. \(\blacksquare\)

20.5.5 检验原问题与注 20.16 / Check the original problem and Remark 20.16 把结果代入 IC1（用 (20.20)、(20.21)）：\(0\ge(\theta_2-\theta_1)(q_1-q_2)=(\theta_2-\theta_1)\left(-\dfrac{\phi}{1-\phi}(\theta_2-\theta_1)-(\theta_2-\theta_1)\right)\)，因 \(\theta_2>\theta_1\) 该式成立，IC1 满足。故松弛问题的结果确为原问题的解。注 20.16：定义类型 \(i\) 的第一最优产量 \(q_i^{FB}\in\arg\max_q u(q,\theta_i)-c(q)\)，此处 f.o.c. 得 \(q_i^{FB}=\theta_i-c\)。高类型 \(\theta_2\) 得第一最优产量（\(q_2=\theta_2-c=q_2^{FB}\)）；低类型 \(\theta_1\) 得低于第一最优的产量（\(q_1Plug the result into IC1 (using (20.20), (20.21)): \(0\ge(\theta_2-\theta_1)(q_1-q_2)=(\theta_2-\theta_1)\left(-\dfrac{\phi}{1-\phi}(\theta_2-\theta_1)-(\theta_2-\theta_1)\right)\), which holds since \(\theta_2>\theta_1\), so IC1 is satisfied. Thus the relaxed problem's result is indeed the solution to the original. Remark 20.16: define type \(i\)'s first-best quantity \(q_i^{FB}\in\arg\max_q u(q,\theta_i)-c(q)\), here the f.o.c. gives \(q_i^{FB}=\theta_i-c\). The high type \(\theta_2\) gets the first-best quantity (\(q_2=\theta_2-c=q_2^{FB}\)); the low type \(\theta_1\) gets less than first best (\(q_1

定义 20.5、20.6 与引理 20.3 / Definitions 20.5, 20.6 and Lemma 20.3 定义 20.5（向下局部激励约束 DLIC）：每个类型 \(i\) 无激励伪装成紧邻其下的类型：\(u(q_i,\theta_i)-t_i\ge u(q_{i-1},\theta_i)-t_{i-1}\)。定义 20.6（向上局部激励约束 ULIC）：每个类型 \(i\) 无激励伪装成紧邻其上的类型：\(u(q_i,\theta_i)-t_i\ge u(q_{i+1},\theta_i)-t_{i+1}\)。引理 20.3：在 \(u_{q\theta}>0\) 的假设下，\(\{DLIC\}_{i=2}^N+\{ULIC\}_{i=1}^{N-1}\) 等价于 \(N(N-1)\) 个 IC 约束。Definition 20.5 (Downward local incentive constraint, DLIC): each type \(i\) has no incentive to pretend to be the type right below: \(u(q_i,\theta_i)-t_i\ge u(q_{i-1},\theta_i)-t_{i-1}\). Definition 20.6 (Upward local incentive constraint, ULIC): each type \(i\) has no incentive to pretend to be the type right above: \(u(q_i,\theta_i)-t_i\ge u(q_{i+1},\theta_i)-t_{i+1}\). Lemma 20.3: under the assumption \(u_{q\theta}>0\), \(\{DLIC\}_{i=2}^N+\{ULIC\}_{i=1}^{N-1}\) is equivalent to the \(N(N-1)\) IC constraints.

引理 20.3 证明 / Proof of Lemma 20.3 先证 ULIC \(u(q_i,\theta_i)-t_i\ge u(q_{i+1},\theta_i)-t_{i+1}\) 蕴含 \(u(q_i,\theta_i)-t_i\ge u(q_{i+k},\theta_i)-t_{i+k}\)（\(k\ge1\)），只需证 \(u(q_{i+1},\theta_i)-t_{i+1}\ge u(q_{i+k},\theta_i)-t_{i+k}\) (20.22)。利用相邻绑定约束（\(u(q_{i+1},\theta_{i+1})-t_{i+1}=0\) 型恒等式）整理 (20.22)，并用 \(u_\theta>0\) 与 \(q(\cdot)\) 非降（\(q_{i+k}\ge q_{i+1}\)）：First show ULIC \(u(q_i,\theta_i)-t_i\ge u(q_{i+1},\theta_i)-t_{i+1}\) implies \(u(q_i,\theta_i)-t_i\ge u(q_{i+k},\theta_i)-t_{i+k}\) (\(k\ge1\)), which only requires showing \(u(q_{i+1},\theta_i)-t_{i+1}\ge u(q_{i+k},\theta_i)-t_{i+k}\) (20.22). Using the adjacent binding constraints (identities of the \(u(q_{i+1},\theta_{i+1})-t_{i+1}=0\) type) to rearrange (20.22), and using \(u_\theta>0\) and \(q(\cdot)\) non-decreasing (\(q_{i+k}\ge q_{i+1}\)):

$$ > \begin{aligned} > &u(q_{i+1},\theta_i)-t_{i+1}-u(q_{i+k},\theta_i)+t_{i+k}\\ > &=u(q_{i+1},\theta_i)-u(q_{i+1},\theta_{i+1})-u(q_{i+k},\theta_i)+u(q_{i+k},\theta_{i+k})\\ > &>u(q_{i+1},\theta_i)-u(q_{i+1},\theta_{i+1})-u(q_{i+k},\theta_i)+u(q_{i+k},\theta_{i+1})\\ > &=[u(q_{i+k},\theta_{i+1})-u(q_{i+k},\theta_i)]-[u(q_{i+1},\theta_{i+1})-u(q_{i+1},\theta_i)]\ge0 > \end{aligned} > $$

最后一步由 \(u_{q\theta}>0\) 成立，故 (20.22) 成立。同理（用 DLIC、\(q(\cdot)\) 非降 \(q_{i-k}\le q_{i-1}\)、\(u_\theta>0\)）可证 DLIC \(u(q_i,\theta_i)-t_i\ge u(q_{i-1},\theta_i)-t_{i-1}\) 蕴含 \(u(q_i,\theta_i)-t_i\ge u(q_{i-k},\theta_i)-t_{i-k}\)（\(k\ge1\)），即 (20.23) \(u(q_{i-1},\theta_i)-t_{i-1}\ge u(q_{i-k},\theta_i)-t_{i-k}\) 成立。综合两向，所有 \(N(N-1)\) 个 IC 都由局部约束推出。\(\blacksquare\)The last step holds by \(u_{q\theta}>0\), so (20.22) holds. Similarly (using DLIC, \(q(\cdot)\) non-decreasing \(q_{i-k}\le q_{i-1}\), \(u_\theta>0\)) one shows DLIC \(u(q_i,\theta_i)-t_i\ge u(q_{i-1},\theta_i)-t_{i-1}\) implies \(u(q_i,\theta_i)-t_i\ge u(q_{i-k},\theta_i)-t_{i-k}\) (\(k\ge1\)), i.e. (20.23) \(u(q_{i-1},\theta_i)-t_{i-1}\ge u(q_{i-k},\theta_i)-t_{i-k}\) holds. Combining both directions, all \(N(N-1)\) IC constraints follow from the local constraints. \(\blacksquare\)