框架 / Framework 买家与拍卖人都风险中性（即都用期望货币价值直接评价效用）。共 \(n\) 个买家，均为单位需求（各只需 1 单位），且只有单一一件物品出售。\(b_i\in\mathcal{B}_i\) 为买家 \(i\) 的出价，\(\mathcal{B}_i\) 为所有可能出价之集。每个买家 \(i\) 由类型（或称信号）\(\theta_i\) 刻画，随机分布于支撑 \(\Theta_i=[\underline\theta_i,\overline\theta_i]\)，累积分布 \(F_i(\theta_i)\)、密度 \(f_i(\theta_i)\)；\(n\) 个买家的类型 \((\theta_1,\dots,\theta_n)\) 服从联合分布 \(F(\theta_1,\dots,\theta_n)\)、密度 \(f(\theta_1,\dots,\theta_n)\)。买家 \(i\) 的效用为 \(V_i(\theta_1,\dots,\theta_n)\cdot p_i-t_i\)，其中 \(V_i\) 是 \(i\) 的估值、\(p_i\) 是 \(i\) 得到物品的概率、\(t_i\) 是为得到物品的转移支付（未必等于出价）。注意 \(\theta_i\) 之间可相关、\(p_i\) 可依赖所有类型的分布、估值 \(V_i\) 可不仅依赖自身类型还依赖他人类型。假设无二级市场（不可转售）、无合谋。Both bidders and the auctioneer are risk-neutral (both evaluating utilities by expected monetary value directly). There are \(n\) bidders in total, all with unit demands (each demands only one unit), and only a single unit of the good is for sale. \(b_i\in\mathcal{B}_i\) is the bid of agent \(i\), \(\mathcal{B}_i\) the set of all possible bids. Each bidder \(i\) is characterized by a type (or signal) \(\theta_i\), randomly distributed on support \(\Theta_i=[\underline\theta_i,\overline\theta_i]\) with c.d.f. \(F_i(\theta_i)\) and p.d.f. \(f_i(\theta_i)\); the \(n\) bidders' types \((\theta_1,\dots,\theta_n)\) follow a joint distribution \(F(\theta_1,\dots,\theta_n)\) with p.d.f. \(f(\theta_1,\dots,\theta_n)\). Bidder \(i\)'s utility is \(V_i(\theta_1,\dots,\theta_n)\cdot p_i-t_i\), where \(V_i\) is \(i\)'s valuation, \(p_i\) the probability \(i\) gets the good, and \(t_i\) the transfer payment to get the good (not necessarily the bidding price). Note the \(\theta_i\)'s can be correlated, \(p_i\) can depend on the distribution of all types, and valuation \(V_i\) can depend not only on \(i\)'s own type but also on others'. Assume there is no secondary market (no resale) and no collusion.

24.2 独立私人价值 (IPV) 环境 / Independent and private values (IPV) environment 独立：所有主体的类型独立分布，\(f(\theta_1,\dots,\theta_n)=f_1(\theta_1)\times\cdots\times f_n(\theta_n)\)，\(F\) 同理。私人：每个 \(i\) 的估值只与自身类型相关，\(V_i(\theta_1,\dots,\theta_n)=\theta_i\)。（一般情况未必如此：共同价值情形如石油钻探，\(V_i(\theta_1,\dots,\theta_n)=\frac1n\sum_{i=1}^n\theta_i\)。）进一步假设所有主体在同一支撑上有相同的类型分布：\(F_i=F_j\)、\(\Theta_i=\Theta_j=[0,1]\)、\(\mathcal{B}_i=\mathcal{B}_j=\mathcal{B}\) 对 \(\forall i,j\)。Independent: the types of all agents are independently distributed, \(f(\theta_1,\dots,\theta_n)=f_1(\theta_1)\times\cdots\times f_n(\theta_n)\), similarly for \(F\). Private: each agent \(i\)'s valuation is only related to his own type, \(V_i(\theta_1,\dots,\theta_n)=\theta_i\). (In general this might not be true: in the common-values case, e.g. oil drilling, \(V_i(\theta_1,\dots,\theta_n)=\frac1n\sum_{i=1}^n\theta_i\).) We further assume all agents have the same distribution of types on the same support: \(F_i=F_j\), \(\Theta_i=\Theta_j=[0,1]\), \(\mathcal{B}_i=\mathcal{B}_j=\mathcal{B}\) for \(\forall i,j\).

验证 \(\bar b(\theta)\) 是对称均衡 / Verify \(\bar b(\theta)\) is a symmetric equilibrium (严格单调) 对 (24.2) 求导：\(\bar b'(\theta)=1+\dfrac{g(\theta)}{(G(\theta))^2}\int_0^\theta G(x)\,dx-\dfrac{1}{G(\theta)}G(\theta)=\dfrac{g(\theta)}{(G(\theta))^2}\int_0^\theta G(x)\,dx>0\)，故 \(\bar b\) 严格递增，亦可微。(激励相容) 主体 \(i\) 选报告 \(\hat\theta\) 解 \(\max_{\hat\theta}U(\hat\theta\mid\theta)\equiv G(\hat\theta)(\theta_i-\bar b(\hat\theta))\)。由 (24.2) 得 \(\bar b(\hat\theta)G(\hat\theta)=\hat\theta G(\hat\theta)-\int_0^{\hat\theta}G(x)\,dx\)，故 \(U(\hat\theta\mid\theta)=G(\hat\theta)(\theta-\hat\theta)+\int_0^{\hat\theta}G(x)\,dx\) (24.3)，而 \(U(\theta\mid\theta)=\int_0^\theta G(x)\,dx\) (24.4)。相减：\(\Delta U=U(\theta\mid\theta)-U(\hat\theta\mid\theta)=\int_{\hat\theta}^\theta[G(x)-G(\hat\theta)]\,dx\ge0\)（若 \(\hat\theta<\theta\) 则 \(G(x)\ge G(\hat\theta)\)；若 \(\hat\theta>\theta\) 则积分与被积函数符号同时翻转，整体仍正）。故讲真话最优，\(\bar b(\cdot)\) 为对称均衡。\(\blacksquare\)(Strict monotonicity) Differentiate (24.2): \(\bar b'(\theta)=1+\dfrac{g(\theta)}{(G(\theta))^2}\int_0^\theta G(x)\,dx-\dfrac{1}{G(\theta)}G(\theta)=\dfrac{g(\theta)}{(G(\theta))^2}\int_0^\theta G(x)\,dx>0\), so \(\bar b\) is strictly increasing and differentiable. (Incentive compatibility) Agent \(i\) chooses report \(\hat\theta\) to solve \(\max_{\hat\theta}U(\hat\theta\mid\theta)\equiv G(\hat\theta)(\theta_i-\bar b(\hat\theta))\). From (24.2), \(\bar b(\hat\theta)G(\hat\theta)=\hat\theta G(\hat\theta)-\int_0^{\hat\theta}G(x)\,dx\), so \(U(\hat\theta\mid\theta)=G(\hat\theta)(\theta-\hat\theta)+\int_0^{\hat\theta}G(x)\,dx\) (24.3), while \(U(\theta\mid\theta)=\int_0^\theta G(x)\,dx\) (24.4). Subtracting: \(\Delta U=U(\theta\mid\theta)-U(\hat\theta\mid\theta)=\int_{\hat\theta}^\theta[G(x)-G(\hat\theta)]\,dx\ge0\) (if \(\hat\theta<\theta\) then \(G(x)\ge G(\hat\theta)\); if \(\hat\theta>\theta\) both the integral and the integrand flip sign, so the whole is still positive). So truthful reporting is optimal and \(\bar b(\cdot)\) is a symmetric equilibrium. \(\blacksquare\)

注 24.1：最优出价的解读 / Remark 24.1: interpreting the optimal bid 由 (24.1)，最优出价是"在自己为最高类型的条件下、第二高类型的条件期望"（\(G(\cdot),g(\cdot)\) 是 \(n-1\) 个主体中最高者的累积分布与密度）。故 \(\bar b(\theta)=\mathbb{E}[\max_{j\ne i}\theta_j\mid\max_{j\ne i}\theta_j\le\theta_i]\)，即在自己拥有最高估值的条件下，以最便宜的方式（出第二高类型）赢得物品。From (24.1), the optimal bid is the conditional expectation of the second-highest type conditional on the agent himself having the highest type (\(G(\cdot),g(\cdot)\) being the c.d.f. and p.d.f. of the highest of \(n-1\) agents). So \(\bar b(\theta)=\mathbb{E}[\max_{j\ne i}\theta_j\mid\max_{j\ne i}\theta_j\le\theta_i]\), i.e. bid the second-highest type to win the good in the cheapest way, conditional on the bidder having the highest valuation.

第二高类型的密度 / Density of the second-highest type \(n(n-1)f(\theta)F(\theta)^{n-2}(1-F(\theta))\) 确实是第二高类型的密度：对某主体 \(i\) 为最高、\(j\) 以类型 \(\theta\) 为第二高，需 \(i\) 高于 \(j\)（概率 \(1-F(\theta)\)）、其余 \(n-2\) 个低于 \(j\)（概率 \(F(\theta)^{n-2}\)）、\(j\) 类型为 \(\theta\) 的密度 \(f(\theta)\)，联合 \(f(\theta)F(\theta)^{n-2}(1-F(\theta))\)；\(i\) 有 \(n-1\) 个候选且互斥，并集 \((n-1)f(\theta)F(\theta)^{n-2}(1-F(\theta))\)；\(j\) 有 \(n\) 个候选，并集 \(n(n-1)f(\theta)F(\theta)^{n-2}(1-F(\theta))\)。\(n(n-1)f(\theta)F(\theta)^{n-2}(1-F(\theta))\) is indeed the density of the second-highest type: for some agent \(i\) to be highest and \(j\) with type \(\theta\) to be second-highest, we need \(i\) higher than \(j\) (prob \(1-F(\theta)\)), all other \(n-2\) lower than \(j\) (prob \(F(\theta)^{n-2}\)), and the density of \(j\) having type \(\theta\) is \(f(\theta)\), joint \(f(\theta)F(\theta)^{n-2}(1-F(\theta))\); there are \(n-1\) mutually exclusive candidates for \(i\), union \((n-1)f(\theta)F(\theta)^{n-2}(1-F(\theta))\); and \(n\) candidates for \(j\), union \(n(n-1)f(\theta)F(\theta)^{n-2}(1-F(\theta))\).

断言 24.1 / Claim 24.1 对玩家 \(i\) 而言，出价 \(b_i=\theta_i\) 是（弱）占优策略。It is a (weakly) dominant strategy for player \(i\) to bid \(b_i=\theta_i\).

证明 / Proof 定义 \(b_j^\star\equiv\max_{j\ne i}b_j\)。情形 1：\(\theta_i>b_j^\star\)。 出 \(b_i=\theta_i\) 则胜，支付 \(\theta_i-b_j^\star>0\)。(1) 出 \(b_i>\theta_i\) 仍胜，支付仍 \(\theta_i-b_j^\star>0\)，无激励偏离；(2) 出 \(b_i\le b_j^\star\) 则败、支付 0，劣于 \(\theta_i-b_j^\star\)；(3) 出 \(b_j^\star情形 2：\(\theta_i (1) 出 \(b_i\ge\theta_i\)：要么败（支付 0），要么以严格负支付胜（\(b_i>b_j^\star\)）；都不优于出 \(\theta_i\) 得 0；(2) 出 \(b_i<\theta_i\) 败、支付 0。故 \(b_i=\theta_i\) 最优。情形 3：\(\theta_i=b_j^\star\)。 无论谁胜，\(i\) 的支付总为 0。(1) 出 \(b_i>\theta_i\) 胜得严格负支付，更差；(2) 出 \(b_i<\theta_i\) 败得 0，不更优。故 \(b_i=\theta_i\) 最优。综上三种情形，断言成立。\(\blacksquare\)Define \(b_j^\star\equiv\max_{j\ne i}b_j\). Case 1: \(\theta_i>b_j^\star\). Bidding \(b_i=\theta_i\) wins, payoff \(\theta_i-b_j^\star>0\). (1) Bidding \(b_i>\theta_i\) still wins, payoff still \(\theta_i-b_j^\star>0\), no incentive to deviate; (2) bidding \(b_i\le b_j^\star\) loses, payoff \(0<\theta_i-b_j^\star\); (3) bidding \(b_j^\starCase 2: \(\theta_i (1) Bidding \(b_i\ge\theta_i\): either loses (payoff 0) or wins with strictly negative payoff (\(b_i>b_j^\star\)); neither better than bidding \(\theta_i\) and getting 0; (2) bidding \(b_i<\theta_i\) loses, payoff 0. So \(b_i=\theta_i\) is optimal. Case 3: \(\theta_i=b_j^\star\). Whoever wins, \(i\)'s payoff is always 0. (1) Bidding \(b_i>\theta_i\) wins with strictly negative payoff, worse; (2) bidding \(b_i<\theta_i\) loses, payoff 0, not better. So \(b_i=\theta_i\) is optimal. Based on the three cases, the claim holds. \(\blacksquare\)

例 24.1（二价拍卖的非讲真话均衡）/ Example 24.1 (Non-truth-telling equilibrium for SPA) IPV 环境，\(n\) 个买家类型同分布 \(F_i=F_j\) 于 \([\underline\theta,\overline\theta]\)。一个均衡是：买家 1 总出 \(\overline\theta\)，所有其他买家出 0。检验无人偏离：对买家 1，给定他人出 0，他总付 0 并得物品，无法改进；对买家 \(2,\dots,n\)，给定买家 1 总出 \(\overline\theta\)，他们出价低于 \(\overline\theta\) 则不会赢，与出 0 无异；若出价等于或高于 \(\overline\theta\)，其最高支付只有当类型为 \(\overline\theta\) 时为 0、否则为负，弱劣于出 0 得 0。故买家 \(2,\dots,n\) 不偏离 0。因此该结果确为均衡。注意此均衡下拍卖人期望收入为 0——这并不违反（特例的）收入等价定理 25.1，因为此均衡下物品并不总是分给最高类型买家、最低类型买家也不总有零支付。In an IPV environment with \(n\) bidders whose types follow symmetric distributions \(F_i=F_j\) on \([\underline\theta,\overline\theta]\), an equilibrium is that bidder 1 always bids \(\overline\theta\) and all other bidders bid 0. Check no one deviates: for bidder 1, given others bid 0, he always pays 0 and gets the good, cannot improve; for bidders \(2,\dots,n\), given bidder 1 always bids \(\overline\theta\), they won't win if they bid lower than \(\overline\theta\), which makes no difference from bidding 0; if they bid equal to or higher than \(\overline\theta\), their highest payoff is 0 only when their type is \(\overline\theta\) and negative otherwise, weakly worse than bidding 0 and getting 0. So bidders \(2,\dots,n\) won't deviate from bidding 0. Therefore this outcome is indeed an equilibrium. Notice that in this equilibrium the expected revenue to the auctioneer is 0 — which does not contradict the (special-case) Revenue Equivalence Theorem 25.1, because under this equilibrium the good is not always awarded to the highest-type bidder and the lowest-type bidder does not always have zero payoff.

英式 \(\equiv\) 二价、荷式 \(\equiv\) 一价 / English \(\equiv\) second-price, Dutch \(\equiv\) first-price 英式（升价）：拍卖人从低价起，价格上升直到只剩一人愿出价，最后留下者赢、付自己最终出价。相比二价拍卖，这是涉及多轮的动态过程；但在 IPV 下，结果是最高类型者赢、付第二高出价（即他成为唯一出价者的临界点）加上一分钱（可忽略），以最大化支付；最优策略是一直加价直到价格超过自身真实估值 \(\theta\)（每人最终出价＝真实估值 \(\theta\)），与二价拍卖结果完全相同。荷式（钟）：拍卖人从高价起，价格下降直到第一人愿出价，第一出价者赢、付自己初始出价。相比一价拍卖，这是关于"何时出价"的动态猜测过程；但在 IPV 下，结果是最高类型者赢、付"他理性预期另一买家将加入"的期望第二高出价加一分钱；最优策略是一直等到价格降到"自身为最高条件下第二高的期望值"才出价（最终出价＝条件期望的第二高 \(\theta\)），与一价拍卖结果完全相同。故英式 \(\equiv\) 二价、荷式 \(\equiv\) 一价；四种拍卖给拍卖人相同期望收入，由下一章的收入等价定理推广。English (ascending): the auctioneer starts low, the price rises until only one person is willing to bid, the last remaining wins and pays his own final bid. Compared with the second-price auction, this is a dynamic process of several rounds; but under IPV, the result is that the highest type wins and pays the second-highest bid (the cutoff point where he becomes alone in bidding) plus a penny (negligible) to maximize his payoff; the optimal strategy is always to keep bidding up until the price goes over his true valuation \(\theta\) (everyone's last and ending bid = true valuation \(\theta\)), exactly the same result as the second-price auction. Dutch (clock): the auctioneer starts high, the price drops until the first person is willing to bid, the first bidder wins and pays his own initial bid. Compared with the first-price auction, this is a dynamic process of guessing when to bid; but under IPV, the result is that the highest type wins and pays the expected second-highest bid (the cutoff where he rationally expects another bidder to join) plus a penny; the optimal strategy is to wait until the price goes down to the expected value of the second-highest conditional on the agent having the highest, with the ending bid = the conditionally expected second-highest \(\theta\), exactly the same result as the first-price auction. So English \(\equiv\) second-price, Dutch \(\equiv\) first-price; the four auctions yield the same expected revenue to the auctioneer, generalized by the Revenue Equivalence Theorem of the next chapter.