13. Signaling Game: Models for Costly Information Revealing
13. Signaling Game: Models for Costly Information Revealing
本章导读 信号传递（signaling）是一种机制：拥有信息的一方可以主动以某种方式向无信息的一方披露自身信息，从而修正逆向选择模型。§13.1 MWG 劳动力市场：先证当披露无成本时，每个 SPNE 都会完全揭示信息（定理 13.1）；随后转向有成本的信号——工人选择教育/努力 \(e\)（纯信号、无生产率价值）作为信号，效用 \(U=w-c(e,\theta)\)，并施加单交叉性质 \(c_{e\theta}<0\)（高类型边际成本更低）。三类序贯均衡：分离（\(e_L^\star=0\)，高类型付出 \(e_H^\star\in[\underline e_{sep},\overline e_{sep}]\)）、混同（同一 \(e_p^\star\in[0,\overline e_{pool}]\)）、混合/杂交（低类型混合 \(0\) 与 \(e_H^\star\)）；分析各自的帕累托最优；并用 sender–receiver 博弈定义直觉准则（intuitive criterion）作为序贯均衡的精炼，结论是唯一存活的是最低成本分离均衡。§13.2 JR 保险市场：在两风险类型下重做有成本信号博弈（合约 \((B,p)\) 为二维信号），给出分离与混同均衡及其条件。所有图（图 8–16）均已转述。
13. Signaling Game: Models for Costly Information Revealing
Overview Signaling is a mechanism by which the side with information can choose to reveal it to the side without, modifying the adverse-selection model. §13.1 the MWG labor market: first, when revelation is costless, every SPNE fully reveals information (Theorem 13.1); then we turn to costly signals — a worker chooses education/effort \(e\) (pure signal, no productivity value) as a signal, with utility \(U=w-c(e,\theta)\) and the single-crossing property \(c_{e\theta}<0\) (high types have lower marginal cost). Three kinds of sequential equilibria: separating (\(e_L^\star=0\), high type exerts \(e_H^\star\in[\underline e_{sep},\overline e_{sep}]\)), pooling (a common \(e_p^\star\in[0,\overline e_{pool}]\)), and hybrid (low type mixes \(0\) and \(e_H^\star\)); we analyze the Pareto-optimality of each, then use a sender–receiver game to define the intuitive criterion as a refinement of sequential equilibrium, concluding that only the least-cost separating equilibrium survives. §13.2 the JR insurance market: redo the costly-signaling game with two risk types (the contract \((B,p)\) is a two-dimensional signal), giving separating and pooling equilibria with their conditions. All figures (8–16) are paraphrased.
信号传递作为一种机制，使拥有信息的一方能够以某种方式把信息揭示给无信息的另一方，从而修正逆向选择下的非对称信息模型。
13.1 MWG 劳动力市场问题
13.1.1 无成本信息披露的定理 / Theorem for costless information revealing
现在我们希望被掌握信息的一方（工人）尝试向企业沟通自己是谁。每个人都有激励让所有人知道自己的类型。设工人可以可信地披露其类型，博弈时序为：
- 阶段 1：自然抽取每个工人的类型 \(\theta_i\)，作为该工人的私人信息；
- 阶段 2：工人可选择是否披露类型；
- 阶段 3：企业基于阶段 2 的信息选择工资。
Signaling serves as a mechanism by which the side with information can choose to reveal that information in some way to the side without it, modifying the asymmetric-information model for adverse selection.
13.1 MWG Labor Market Problem
13.1.1 Theorem for costless information revealing
We now want the informed party (workers) to try to communicate to firms who they are. Everyone has an incentive to get everyone to know who they are. Suppose workers can credibly disclose their type. The game timing is:
- Stage 1: nature draws each worker's type \(\theta_i\), read as private information;
- Stage 2: a worker can choose whether to reveal type;
- Stage 3: firms choose wages conditional on stage 2.
定理 13.1（无成本披露下完全揭示）/ Theorem 13.1 (full revelation under costless disclosure) 在每个子博弈精炼纳什均衡（SPNE）中，信息都被完全揭示，即每个工人都向企业披露其类型。In every subgame perfect Nash equilibrium (SPNE), full information is revealed, i.e. every worker reveals his type to firms.
证明（要点）/ Proof (sketch) 设 \(\theta_1<\cdots<\theta_n\)，每个状态以一定概率出现，记为 \(\phi_1,\dots,\phi_n\)。假设有两个工人不披露，其中 \(\theta_k>\theta_m\)。\(k\) 面临什么激励？由于无人能区分池中的工人，企业只能支付池内平均 \(\frac{\theta_m+\theta_k}{2}\)，而 \(\theta_k\) 不喜欢这一点，于是会偏离去披露自己的类型。逐级向下论证，所有类型都会披露。注意，至于最后一名工人披露与否并不重要，因为可以反推出来。\(\blacksquare\)Let \(\theta_1<\cdots<\theta_n\), each state occurring with probability \(\phi_1,\dots,\phi_n\). Suppose two workers choose not to reveal, where \(\theta_k>\theta_m\). What incentive does \(k\) face? Since no one can differentiate the workers in the pool, firms can only pay the average \(\frac{\theta_m+\theta_k}{2}\), and \(\theta_k\) dislikes this, so he deviates to reveal his type. Arguing down the line, all types reveal. Note that whether the last worker reveals doesn't matter, because we can back it out. \(\blacksquare\)
模型隐含的两条性质 / Two properties the model specifies 1. 每个类型都能生成自身类型的证据（即可信的无成本沟通）；2. 单调性：人人都想被认为是"好的"（大家一致认为类型越高越好）。然而本节聚焦于信息揭示有成本的博弈。1. Each type can produce evidence of their type (i.e. credible costless communication); 2. Monotonicity: everyone wants to be thought "good" (all agree higher type is better). However, this section focuses on games where information revelation is costly.
13.1.2 有成本信号的 MWG 模型设定 / MWG model set-up for costly information revealing
当无成本沟通不可能时，工人可以选择某种东西（如本模型中的努力 \(e\)）来传递自己是高类型的信号——即便发送该信号是有成本的。考虑如下模型。
13.1.2 MWG model set-up for costly information revealing
When costless communication is not possible, a worker can choose something (e.g. effort \(e\) in this model) that signals he is a high type — even though sending that signal is costly. Consider the following model.
模型设定 / Set-up 1. 工人效用为 \(U=w-c(e,\theta)\)，其中 \(c(0,\theta)=0\)，且 \(c\) 关于 \(e\) 递增且凸。关键假设单交叉性质 \(c_{e\theta}<0\)：高类型的边际成本更低；它与凸性共同蕴含 \(c_\theta<0\)。2. \(\theta_H>\theta_L>0\)，\(r(\theta)=0\)，高类型概率 \(\phi\)、低类型概率 \(1-\phi\)。3. 时序：(a) 工人得知自己的类型 \(\theta\)（生产率）；(b) 工人选择努力水平 \(e\)（作为信号）；(c) 企业基于 \(e\) 给出工资 \(w(e)\)；(d) 工人接受或拒绝。4. 工人策略：选择努力 \(e(\theta)\ge0\)（教育水平）以赢取更高工资。此处教育仅用于信号传递，没有功能性收益（不提升生产率）。1. The worker's utility is \(U=w-c(e,\theta)\), with \(c(0,\theta)=0\) and \(c\) increasing and convex in \(e\). Key assumption — single-crossing property \(c_{e\theta}<0\): high types have lower marginal cost; together with convexity it implies \(c_\theta<0\). 2. \(\theta_H>\theta_L>0\), \(r(\theta)=0\), high type with probability \(\phi\) and low type with probability \(1-\phi\). 3. Timing: (a) workers learn their type \(\theta\) (productivity); (b) workers choose an effort level \(e\) (as a signal); (c) firms offer a wage \(w(e)\) conditional on \(e\); (d) workers accept or reject. 4. Worker's strategy: choose effort \(e(\theta)\ge0\) (the level of education) to earn a higher wage. Here education is only for signaling, with no functional benefit (it does not improve productivity).
$$ U=w-c(e,\theta),\qquad c(0,\theta)=0,\qquad c_{e\theta}<0 $$
图 8 / Figure 8（两类型无差异曲线，已转述 / indifference curves, paraphrased） 在 \(e\)–\(w\) 平面上（横轴努力 \(e\)、纵轴工资 \(w\)）：(a) 两类型 \(\theta_H,\theta_L\) 的无差异曲线都向上倾斜；(b) 向左上方移动（更少努力、更高工资）更受偏好；(c) \(c(e,\theta)\) 关于 \(e\) 的凸性使无差异曲线凸；(d) 因 \(c_{e\theta}<0\)，在每个 \(e\) 处低类型 \(\theta_L\) 的无差异曲线都比高类型 \(\theta_H\) 更陡。其中 \(U_H\equiv\{(e,w)\mid U_H=w-c(e,\theta)\}\)、\(U_L\equiv\{(e,w)\mid U_L=w-c(e,\theta)\}\) 分别为 \(\theta_H\)（效用水平 \(U_H\)）与 \(\theta_L\)（效用水平 \(U_L\)）的无差异曲线。In the \(e\)–\(w\) plane (horizontal axis effort \(e\), vertical axis wage \(w\)): (a) both types \(\theta_H,\theta_L\) have upward-sloping indifference curves; (b) moving up-left (less effort, more wage) is favorable; (c) the convexity of \(c(e,\theta)\) in \(e\) makes the indifference curves convex; (d) since \(c_{e\theta}<0\), at each \(e\) the low type \(\theta_L\)'s indifference curve is steeper than the high type \(\theta_H\)'s. Here \(U_H\equiv\{(e,w)\mid U_H=w-c(e,\theta)\}\) and \(U_L\equiv\{(e,w)\mid U_L=w-c(e,\theta)\}\) are the indifference curves for \(\theta_H\) (utility level \(U_H\)) and \(\theta_L\) (utility level \(U_L\)) respectively.
求解 SPNE，其解的形式为下式，其中 \(\mu^\star(\cdot)\) 是信念系统、\(e^\star(\cdot)\) 是一维（行为）策略、\(w^\star(e)\) 是企业向努力为 \(e\) 的工人开出的工资。下面给出三类可能的序贯均衡。
To solve the SPNE, the solution takes the form below, where \(\mu^\star(\cdot)\) is the system of beliefs, \(e^\star(\cdot)\) the one-dimensional (behavioral) strategies, and \(w^\star(e)\) the wage firms offer to a worker whose effort is \(e\). Below are the three kinds of possible sequential equilibria.
$$ \Big\{\,\underbrace{e^\star(\theta_H)}_{\equiv\, e_H^\star},\ \underbrace{e^\star(\theta_L)}_{\equiv\, e_L^\star}\,;\ w^\star(e)\,;\ \underbrace{\mu^\star(\theta_H\mid e)}_{\equiv\,\mu^\star(e)}\Big\} $$
13.1.3 分离均衡 / Separating equilibrium
按定义，分离均衡要求解 \(\{(e_H^\star,e_L^\star),w^\star(e),\mu^\star(e)\}\) 满足：\(\mu^\star(e_H^\star)=1\)、\(\mu^\star(e_L^\star)=0\)（但一般的 \(\mu^\star(\cdot)\) 未被指定）；\(w^\star(e_H^\star)=\theta_H\)、\(w^\star(e_L^\star)=\theta_L\)（但一般的 \(w^\star(\cdot)\) 未被指定）。
引理：\(e_L^\star=0\) 且 \(U_L^\star=\theta_L\)。 因为低类型已被完全从高类型中区分开，低类型的均衡工资恒为 \(\theta_L\)（无论努力多少），故付出有成本的信号对低类型不值得，于是 \(e_L^\star=0\)；又由 \(U=w-c(e,\theta)\)、\(c(0,\theta)=0\)，得 \(U_L^\star=\theta_L\)。
13.1.3 Separating equilibrium
By definition, a separating equilibrium requires the solution \(\{(e_H^\star,e_L^\star),w^\star(e),\mu^\star(e)\}\) to satisfy: \(\mu^\star(e_H^\star)=1\), \(\mu^\star(e_L^\star)=0\) (but the general \(\mu^\star(\cdot)\) is unspecified); \(w^\star(e_H^\star)=\theta_H\), \(w^\star(e_L^\star)=\theta_L\) (but the general \(w^\star(\cdot)\) is unspecified).
Lemma: \(e_L^\star=0\) and \(U_L^\star=\theta_L\). Since the low type is completely differentiated from the high type, the equilibrium wage for the low type is always \(\theta_L\) regardless of effort, so the costly signal is not worth it for the low type, hence \(e_L^\star=0\); and by \(U=w-c(e,\theta)\) with \(c(0,\theta)=0\), we get \(U_L^\star=\theta_L\).
分离均衡的三个条件 / Conditions for separating equilibrium 条件 1：\(w^\star(e_H^\star)=\theta_H\)，\(w^\star(e_L^\star)=\theta_L\)（企业的盈亏平衡条件）。条件 2：\(U_L^\star=\theta_L\ge\theta_H-c(e_H^\star,\theta_L)\)（保证没有低类型有激励伪装成高类型）。条件 3：\(U_H^\star\ge\theta_L\)（保证没有高类型有激励伪装成低类型）。Condition 1: \(w^\star(e_H^\star)=\theta_H\), \(w^\star(e_L^\star)=\theta_L\) (the break-even condition for firms). Condition 2: \(U_L^\star=\theta_L\ge\theta_H-c(e_H^\star,\theta_L)\) (guarantees no low type has an incentive to pretend to be high). Condition 3: \(U_H^\star\ge\theta_L\) (guarantees no high type has an incentive to pretend to be low).
例： 设 \(c(e,\theta)=e\times(k-\theta)\)。则条件 2 与条件 3 分别给出：
Example: let \(c(e,\theta)=e\times(k-\theta)\). Then Conditions 2 and 3 give, respectively:
$$ \theta_L\ge\theta_H-e_H^\star k+e_H^\star\theta_L \;\Rightarrow\; e_H^\star\ge\frac{\theta_H-\theta_L}{k-\theta_L}\equiv\underline e_{sep} $$
$$ \theta_H-e_H^\star k+e_H^\star\theta_H\ge\theta_L \;\Rightarrow\; e_H^\star\le\frac{\theta_H-\theta_L}{k-\theta_H}\equiv\overline e_{sep} $$
合并得 \(\underline e_{sep}\le e_H^\star\le\overline e_{sep}\)。一组可行的信念与工资为下面两式：
Combining, \(\underline e_{sep}\le e_H^\star\le\overline e_{sep}\). One workable belief and wage are the two expressions below:
$$
\mu^\star(e)=\begin{cases} 1 & e\ge e_H\\[2pt] 0 & e 图 9 / Figure 9（分离均衡，已转述 / Separating Equilibrium, paraphrased）
在 \(e\)–\(w\) 平面上，工资模式 \(w^\star(e)\) 是一条阶梯：\(e 按定义，混同均衡要求解满足：\(e_H^\star=e_L^\star\equiv e_p^\star\)；\(\mu^\star(e_p^\star)=\phi\)；\(w^\star(e_p^\star)=\mathbb{E}[\theta]=\phi\theta_H+(1-\phi)\theta_L\)。 By definition, a pooling equilibrium requires the solution to satisfy: \(e_H^\star=e_L^\star\equiv e_p^\star\); \(\mu^\star(e_p^\star)=\phi\); \(w^\star(e_p^\star)=\mathbb{E}[\theta]=\phi\theta_H+(1-\phi)\theta_L\). 混同均衡的三个条件 / Conditions for pooling equilibrium
条件 1：\(w^\star(e_p^\star)=\mathbb{E}[\theta]=\phi\theta_H+(1-\phi)\theta_L\)（企业盈亏平衡）。条件 2：\(U_L^\star\ge\theta_L\)（没有低类型有激励离开池）。条件 3：\(U_H^\star\ge\theta_L\)（没有高类型有激励离开池）。Condition 1: \(w^\star(e_p^\star)=\mathbb{E}[\theta]=\phi\theta_H+(1-\phi)\theta_L\) (firms break even). Condition 2: \(U_L^\star\ge\theta_L\) (no low type has an incentive to leave the pool). Condition 3: \(U_H^\star\ge\theta_L\) (no high type has an incentive to leave the pool). 例： 仍设 \(c(e,\theta)=e\times(k-\theta)\)。条件 2 要求下式，而条件 3 由条件 2 蕴含，故 \(0\le e_p^\star\le\overline e_{pool}\)： Example: still let \(c(e,\theta)=e\times(k-\theta)\). Condition 2 requires the inequality below, and Condition 3 is implied by Condition 2, so \(0\le e_p^\star\le\overline e_{pool}\): $$
\mathbb{E}[\theta]-e_p^\star k+e_p^\star\theta_L\ge\theta_L \;\Rightarrow\; e_p^\star\le\frac{\mathbb{E}[\theta]-\theta_L}{k-\theta_L}\equiv\overline e_{pool}
$$ 一组可行的信念与工资为： One workable belief and wage are: $$
\mu^\star(e)=\begin{cases} \phi & e\ge e_p^\star\\[2pt] 0 & e 图 10、图 11 / Figures 10, 11（混同均衡两版本，已转述 / two versions, paraphrased）
图 10：工资阶梯在 \(e_p^\star>0\) 处从 \(\theta_L\) 跳到 \(\mathbb{E}[\theta]\)，两类型都选同一点 \((e_p^\star,\mathbb{E}[\theta])\)，两无差异曲线在该点相切；这是付出正努力的混同。图 11（零努力混同）：\(e_p^\star=0\)，两类型都不付努力、都领 \(\mathbb{E}[\theta]\)；图中还标出了较高成本的分离阈值 \(\underline e_{sep},\overline e_{sep}\) 以资对比。Figure 10: the wage step jumps from \(\theta_L\) to \(\mathbb{E}[\theta]\) at \(e_p^\star>0\), both types pick the same point \((e_p^\star,\mathbb{E}[\theta])\), and the two indifference curves are tangent there; this is pooling with positive effort. Figure 11 (no-effort pooling): \(e_p^\star=0\), both types exert no effort and receive \(\mathbb{E}[\theta]\); the figure also marks the higher separating thresholds \(\underline e_{sep},\overline e_{sep}\) for comparison. 由图 12 可见：对高类型 \(\theta_H\)，付出 \(e_H^\star\) 努力领取 \(w_H^\star\) 是最优的；对低类型 \(\theta_L\)，要么付零努力领 \(\theta_L\)，要么付 \(e_H^\star\) 努力领 \(w_H^\star\) 同样最优——于是低类型采用混合行为策略。设低类型以 \((m,1-m)\) 的概率在 \((e_H^\star,\,0)\) 间混合。 From Figure 12: for the high type \(\theta_H\), it is optimal to exert \(e_H^\star\) effort and receive \(w_H^\star\); for the low type \(\theta_L\), it is optimal either to exert zero effort and receive \(\theta_L\), or to exert \(e_H^\star\) effort and receive \(w_H^\star\) — so the low type uses a mixing behavioral strategy. Suppose the low type mixes between \((e_H^\star,\,0)\) with probabilities \((m,1-m)\). $$
\mu^\star(e_H^\star)=\frac{\mathbb{P}(\text{high type})}{\mathbb{P}(\text{high type})+\mathbb{P}(\text{low type, chooses }e_H^\star)}=\frac{\phi}{\phi+(1-\phi)m}
$$ 图 12 / Figure 12（杂交均衡，已转述 / Hybrid Equilibrium, paraphrased）
工资阶梯在 \(e_H^\star\) 处从 \(\theta_L\) 跳到 \(w_H^\star\)（\(w_H^\star\) 介于 \(\theta_L\) 与 \(\theta_H\) 之间）。高类型确定地选 \((e_H^\star,w_H^\star)\)；低类型在 \((0,\theta_L)\) 与 \((e_H^\star,w_H^\star)\) 之间混合，其无差异曲线 \(U_L^\star\) 同时通过这两点（无差异），刚好与高类型的 \(U_H^\star\) 在阶梯处相切。The wage step jumps from \(\theta_L\) to \(w_H^\star\) at \(e_H^\star\) (with \(w_H^\star\) between \(\theta_L\) and \(\theta_H\)). The high type chooses \((e_H^\star,w_H^\star)\) for sure; the low type mixes between \((0,\theta_L)\) and \((e_H^\star,w_H^\star)\), its indifference curve \(U_L^\star\) passing through both points (indifferent), just tangent to the high type's \(U_H^\star\) at the step. 一组可行的信念与序贯理性工资为： One workable belief and the sequentially rational wage are: $$
\mu^\star(e)=\begin{cases} 0 & e 其中 \(w_H^\star\) 由信念加权给出： where \(w_H^\star\) is the belief-weighted average: $$
w_H^\star=\mu^\star(e_H^\star)\,\theta_H+\big(1-\mu^\star(e_H^\star)\big)\theta_L=\frac{\phi\theta_H}{\phi+(1-\phi)m}+\frac{(1-\phi)m\,\theta_L}{\phi+(1-\phi)m}
$$ 最后只剩 \(m\) 待定：需令低类型 \(\theta_L\) 在"选 \(0\)"与"选 \(e_H^\star\)"之间无差异，从而序贯理性地严格混合，即下式（其中花括号上方分别标注两种选择的支付）；对每个给定的 \(c(e_H^\star,\theta_L)\)，式 (13.1) 都钉死一个 \(m\)（并连带钉死 \(w_H^\star\) 与 \(\mu^\star(\cdot)\)），使低类型严格混合，从而杂交均衡严格成立： Only \(m\) remains to be specified: we need the low type \(\theta_L\) to be indifferent between "playing \(0\)" and "playing \(e_H^\star\)" so that strict mixing is sequentially rational, i.e. the equation below (the braces mark the payoffs of the two choices); for every specified \(c(e_H^\star,\theta_L)\), equation (13.1) pins down a specific \(m\) (and pins down \(w_H^\star\) and \(\mu^\star(\cdot)\)) such that the low type is sequentially rational for strictly mixing, so the hybrid equilibrium is strictly hybrid: $$
\frac{\phi\theta_H}{\phi+(1-\phi)m}+\frac{(1-\phi)m\,\theta_L}{\phi+(1-\phi)m}-c(e_H^\star,\theta_L)=\theta_L \tag{13.1}
$$ 注 13.1、注 13.2 / Remarks 13.1, 13.2
注 13.1：若企业知道两类型的无差异曲线，便可从四种均衡中直接挑一个；但若无法获得关于工人偏好的完全信息，企业能做的只是设定工资模式，并预期四种均衡之一会发生，而事前并不知道是哪一种。但序贯地看，企业会更新信念、调整工资模式；序贯均衡正是用来刻画那些无需序贯调整的策略组合。注 13.2：此外，企业并不真正在意是哪个均衡，因为在所有均衡中企业利润都为零。Remark 13.1: if a firm knows the indifference curves of both types, it can pick one equilibrium from the four; but if it is impossible to have complete information about workers' preferences, all firms can do is set wage patterns and expect one of the four to happen without knowing which ex ante. But sequentially, firms update beliefs and adjust the wage pattern; the sequential equilibrium describes exactly the strategy profiles that require no sequential adjustment. Remark 13.2: furthermore, firms don't really care which equilibrium it is, since firms always make zero profit in all equilibria. 分离均衡： 最低成本分离是 \(e_H^\star=\underline e_{sep}\)，满足下面第一式；最高成本分离是 \(e_H^\star=\overline e_{sep}\)，满足第二式。最低成本分离帕累托占优所有其他分离均衡，因为企业与低类型都无差异，而高类型严格更好。 Separating equilibria: the least-cost separating is \(e_H^\star=\underline e_{sep}\), satisfying the first equation below; the highest-cost separating is \(e_H^\star=\overline e_{sep}\), satisfying the second. The least-cost separating Pareto dominates all other separating equilibria, since firms and the low type are indifferent while the high type is strictly better off. $$
\theta_H-c(\underline e_{sep},\theta_L)=\theta_L\qquad\text{and}\qquad \theta_H-c(\overline e_{sep},\theta_H)=\theta_L
$$ 混同均衡： 最低成本混同是 \(e_p^\star=\underline e_{pool}=0\)；最高成本混同是 \(e_p^\star=\overline e_{pool}\)，满足 \(\mathbb{E}[\theta]-c(\overline e_{pool},\theta_L)=\theta_L\)。显然有下面的排序；最低成本混同帕累托占优所有其他混同均衡（企业无差异，而高、低类型都严格更好）。 Pooling equilibria: the least-cost pooling is \(e_p^\star=\underline e_{pool}=0\); the highest-cost pooling is \(e_p^\star=\overline e_{pool}\), satisfying \(\mathbb{E}[\theta]-c(\overline e_{pool},\theta_L)=\theta_L\). Clearly we have the ordering below; the least-cost pooling Pareto dominates all other pooling equilibria (firms indifferent, both high and low types strictly better off). $$
0=\underline e_{pool}<\overline e_{pool}<\underline e_{sep}<\overline e_{sep}
$$ 分离与混同之间 / Between separating and pooling
当 \(\phi\) 非常大（接近 \(1\)）时，最低成本混同帕累托占优最低成本分离：因为 \(\phi\approx1\) 时 \(\mathbb{E}[\theta]\approx\theta_H\)，从最低成本分离转到最低成本混同时高类型工资损失很小；而 \(e_H^\star\) 未必接近零，故努力的下降一般很大，高类型严格更好；低类型在最低成本混同下也领到严格更高的工资，也更好；企业无差异。当 \(\phi\) 不够接近 \(1\) 时，两者之间没有确定的帕累托占优。When \(\phi\) is very large (almost \(1\)), the least-cost pooling Pareto dominates the least-cost separating: because \(\mathbb{E}[\theta]\approx\theta_H\) when \(\phi\approx1\), the high type's wage loss from least-cost separating to least-cost pooling is small; while \(e_H^\star\) is not necessarily close to zero, so the reduction in effort is generally big and the high type is strictly better off; the low type also receives a strictly higher wage under least-cost pooling and is better off; firms are indifferent. When \(\phi\) is not sufficiently close to \(1\), there is no certain Pareto dominance between the two. 直觉准则的引子来自图 10（混同序贯均衡）。见图 13 转述。 The motivating example of the intuitive criterion comes from Figure 10 (the pooling sequential equilibrium). See the paraphrase of Figure 13. 图 13 / Figure 13（偏离路径精炼，已转述 / Off-path refinement, paraphrased）
对高类型而言，点 \(n\) 位于比混同序贯均衡点 \(p\) 更好的无差异曲线上。设高类型选努力 \(\hat e\) 而非 \(e_p^\star\)，且 \(\hat e\) 对低类型严格比 \(e_p^\star\) 更差。那么企业观察到此信号，便有信心相信选 \(\hat e\) 的是高类型；均衡中企业必须给他开 \(\theta_H\)，否则别的企业可开 \(w=\theta_H-\varepsilon\) 赚正利润，直到 \(w=\theta_H\)。每个高类型都这样做，于是变成分离均衡。故当我们无法阻止高类型这样做时，混同（序贯）均衡无法存活。For the high type, point \(n\) lies on a better indifference curve than the pooling sequential equilibrium at point \(p\). Suppose a high type chooses effort \(\hat e\) instead of \(e_p^\star\), where \(\hat e\) is strictly worse than \(e_p^\star\) for low types. Then a firm observes this signal and is confident the one who chooses \(\hat e\) is a high type; in equilibrium the firm must offer him \(\theta_H\), else another firm can offer \(w=\theta_H-\varepsilon\) to earn positive profit until \(w=\theta_H\). Every high type does the same, so it becomes a separating equilibrium. Hence the pooling (sequential) equilibrium fails to survive when we cannot prevent high types from doing this. 为何还需要更强的精炼 / Why a stronger refinement is needed
然而，混同均衡仍是一个序贯均衡：因为高类型的上述"干扰"需要企业与之"合作"，所以它实际上是双边偏离而非单边偏离。序贯均衡并不排除双边偏离，故这种干扰发生在序贯均衡路径之外。因此我们需要一个比序贯均衡更强的条件来精炼均衡集——序贯均衡有许多不同层次的精炼，这里介绍其中之一：直觉准则。However, the pooling equilibrium is still a sequential equilibrium: because such interference by high types needs the firms to "cooperate" with them, it is actually a bilateral deviation, not a unilateral one. Sequential equilibrium does not preclude bilateral deviation, so such interference happens off the sequential-equilibrium path. Hence we need a condition stronger than sequential equilibrium to refine the equilibrium set — there are many levels of refinement; here we present one, the intuitive criterion. 为正式引入直觉准则，考虑 sender–receiver（发送者–接收者）博弈，时序为： 效用：\(U^S(\theta,m,a)\) 是类型 \(\theta\) 发送 \(m\)、面对回应 \(a\) 时发送者的效用；\(U^\star(\theta)\) 是序贯均衡下发送者的效用；\(U^R(\theta,m,a)\) 是面对类型 \(\theta\) 的发送者、收到 \(m\)、以 \(a\) 回应时接收者的效用。定义 \(\hat\Theta(m)\subseteq\Theta\) 为下式： To formally introduce the intuitive criterion, consider the sender–receiver game, with timing: Utilities: \(U^S(\theta,m,a)\) is the sender's utility of having type \(\theta\), sending \(m\), and facing response \(a\); \(U^\star(\theta)\) is the sender's utility in the sequential equilibrium; \(U^R(\theta,m,a)\) is the receiver's utility facing a type-\(\theta\) sender, receiving \(m\), and responding with \(a\). Define \(\hat\Theta(m)\subseteq\Theta\) by the expression below: $$
\theta\in\hat\Theta(m) \iff U^\star(\theta)>\max_{a\in BR(\Theta,m)}U^S(\theta,m,a)
$$ $$
BR(\mu,m)=\arg\max_{a\in A}\sum_\theta U^R(\theta,m,a)\cdot\mu(\theta\mid m),\qquad BR(S,m)=\bigcup_{\mu\in\Delta(S)}BR(\mu,m)
$$ 直觉解释 / Intuitive explanation
\(BR(\mu,m)\) 是给定接收者信念 \(\mu\) 与收到消息 \(m\) 时的最优反应集（接收者诸最优行动）。\(BR(S,m)\) 是对类型子空间 \(S\subseteq\Theta\) 上所有可能信念取 \(BR(\mu,m)\) 的并集——即对 \(S\) 上任意信念的所有最优行动；\(BR(\Theta,m)\) 则是对任意信念的所有最优行动。\(U^\star(\theta)>\max_{a\in BR(\Theta,m)}U^S(\theta,m,a)\) 表示：当发送者为类型 \(\theta\) 时，靠发送消息 \(m\) 无法达到序贯均衡效用 \(U^\star(\theta)\)。因此 \(\hat\Theta(m)\) 是在序贯均衡中绝不会发送 \(m\) 的类型集合。\(BR(\mu,m)\) is the best-response set given the receiver's belief \(\mu\) and message \(m\) (the receiver's optimal actions). \(BR(S,m)\) is the union of \(BR(\mu,m)\) over all possible beliefs on the type subspace \(S\subseteq\Theta\) — i.e. all optimal actions for any belief over \(S\); \(BR(\Theta,m)\) is all optimal actions for any belief. \(U^\star(\theta)>\max_{a\in BR(\Theta,m)}U^S(\theta,m,a)\) means the sequential-equilibrium utility \(U^\star(\theta)\) is not achievable by sending \(m\) when the sender's type is \(\theta\). So \(\hat\Theta(m)\) is the set of types that will never send \(m\) in the sequential equilibrium. 定义 13.1（直觉准则）/ Definition 13.1 (Intuitive criterion)
称一个序贯均衡满足直觉准则，当且仅当不存在 \(\tilde\theta\in\Theta\) 使得下式成立。We say a sequential equilibrium satisfies the intuitive criterion if and only if there is no \(\tilde\theta\in\Theta\) such that the inequality below holds. $$
U^\star(\tilde\theta)<\min_{a\in BR\left(\Theta\setminus\hat\Theta(m),\ m\right)}U^S(\tilde\theta,m,a)
$$ 直觉解释与结论 / Intuitive explanation and conclusion
\(\Theta\setminus\hat\Theta(m)\) 是所有可能在序贯均衡中发送 \(m\) 的类型之集合；\(BR(\Theta\setminus\hat\Theta(m),m)\) 是发送者发送 \(m\) 时接收者所有可能的反应。直觉准则是说：不存在一种类型的玩家——无论接收者对他持何种信念，他都总愿意偏离。换言之，即便接收者对偏离序贯均衡路径持某种"疯狂"的信念，也不会有任何类型的发送者总愿意偏离。\(\Theta\setminus\hat\Theta(m)\) is the set of all types that could possibly send \(m\) in the sequential equilibrium; \(BR(\Theta\setminus\hat\Theta(m),m)\) is all possible receiver responses when the sender sends \(m\). The intuitive criterion says there does not exist a type of player who would always want to deviate whatever the receiver's beliefs over him. In other words, even if the receiver has some "crazy" beliefs off the sequential-equilibrium path, there still won't be any type of sender who would always deviate. 回到图 13。当发送者（工人）发送 \(m\)：\(e=\hat e\) 时，\(\Theta\setminus\hat\Theta(m)=\{\theta_H\}\)，故企业信念必为 \(\mu(\hat e)=1\)，从而 \(BR(\Theta\setminus\hat\Theta(m),m)=\{\theta_H\}\) 为单点集，且 \(U^\star(\theta_H)<\min_{a\in BR(\Theta\setminus\hat\Theta(m),m)}U^S(\theta_H,m,a)\)，其中右端是 \(\theta_H\) 在点 \(n\) 的效用、\(U^\star(\theta_H)\) 是 \(\theta_H\) 在点 \(p\) 的效用。于是直觉准则排除所有混同均衡。 由完全相同的逻辑，直觉准则也排除除最低成本分离之外的所有分离均衡（因为在非最低成本分离中，工人总可发送某个 \(m:\hat e\in[\underline e_{sep},\,e_H^\star)\)）。因此，唯一存活于直觉准则的序贯均衡是最低成本分离均衡；可见直觉准则是序贯均衡的一种精炼。 Back to Figure 13. When the sender (worker) sends \(m\): \(e=\hat e\), we have \(\Theta\setminus\hat\Theta(m)=\{\theta_H\}\), so the firm's belief must be \(\mu(\hat e)=1\), hence \(BR(\Theta\setminus\hat\Theta(m),m)=\{\theta_H\}\) is a singleton, and \(U^\star(\theta_H)<\min_{a\in BR(\Theta\setminus\hat\Theta(m),m)}U^S(\theta_H,m,a)\), where the right side is \(\theta_H\)'s utility at point \(n\) and \(U^\star(\theta_H)\) is \(\theta_H\)'s utility at point \(p\). So the intuitive criterion rules out all pooling equilibria. By exactly the same logic, the intuitive criterion also rules out all separating equilibria except the least-cost one (since in a non-least-cost separating equilibrium the worker could always send some \(m:\hat e\in[\underline e_{sep},\,e_H^\star)\)). Hence the only sequential equilibrium surviving the intuitive criterion is the least-cost separating equilibrium; the intuitive criterion is a refinement of sequential equilibrium. 注 13.3 / Remark 13.3
在 13.1.6 中我们论证最低成本混同帕累托占优最低成本分离；此处又论证唯一存活的是最低成本分离。看似矛盾，其实不然。最低成本混同对两类型仍更有利，但高类型有严格激励去做得更好：他选稍多一点努力，并知道企业会识别并"合作"地把 \(\theta_L\) 工资让给低类型；这样就变成分离，低类型不会跟随。努力被一直推高到低类型再无激励跟随的程度，即最低成本分离——对高类型反而严格更差。那为什么高类型一开始会发起这种偏离？设想：除一名高类型外所有人都预见到这点而不发起偏离，那唯一一个高类型仍有激励偏离（无需把工资改变太多）；但这单次偏离一旦扣动扳机，其他高类型就会跟随。这类似于囚徒困境，以及不稳定寡头在无报复时的合谋瓦解。In §13.1.6 we argued the least-cost pooling Pareto dominates the least-cost separating; here we argued the only survivor is the least-cost separating. It seems a contradiction but isn't. The least-cost pooling is still more favorable to both types, but the high type has a strict incentive to do even better: he picks slightly more effort, knowing firms will recognize it and "cooperate" by offering \(\theta_L\) to the low type; this becomes separating and the low type won't follow. Effort is pushed up to the level where the low type has no incentive to follow — the least-cost separating, which is strictly worse off for the high type. Why then do high types start this deviation? Suppose all but one high type anticipate this and don't start the deviation; that single high type still has an incentive to deviate (without changing the wage too much); but this single deviation pulls the trigger and the other high types follow. This is similar to the prisoner's dilemma and to unstable oligarchy collusion without retaliation. 针对有成本信号博弈，重新考察 JR 保险市场。 Let's revisit the JR insurance market problem for the costly-signaling game. 设定 / Set-up
两种风险类型（发生损失 \(L\) 的概率）：\(0<\pi_L<\pi_H<1\)；两类型收入均为 \(y\)。（客观）低风险类型概率 \(\phi\)、高风险类型概率 \(1-\phi\)。保额 \(B\) 为损失 \(L\) 发生时保险公司支付的金额。时序：1. 自然抽取类型并向每位客户私下揭示 \(\pi\)；2. 客户选择 \((B,p)\) 并向保险公司提案，\(p\) 为保额 \(B\) 对应保单的保费；3. 保险公司接受（\(A\)）或拒绝（\(R\)）该提案。类型 \(i\in\{L,H\}\) 的效用见下式，其中 \(U(\cdot)\) 为两类型共享的递增凹效用函数。Two risk types (probability of suffering loss \(L\)): \(0<\pi_L<\pi_H<1\); both types have income \(y\). The (objective) probability of the low type is \(\phi\) and of the high type is \(1-\phi\). Coverage \(B\) is the amount paid by the insurance company when the loss \(L\) takes place. Timing: 1. Nature chooses the type and privately reveals \(\pi\) to each customer; 2. The customer chooses \((B,p)\) and proposes it to the insurance firm, where \(p\) is the premium for the policy with coverage \(B\); 3. Firms either accept (\(A\)) or reject (\(R\)) the proposal. The utility of type \(i\in\{L,H\}\) is given below, where \(U(\cdot)\) is an increasing concave utility function shared by both types. $$
U_i(B,p)=\pi_i\cdot U(y-p-L+B)+(1-\pi_i)\cdot U(y-p)
$$ 图 14 / Figure 14（两类型无差异曲线，已转述 / indifference curves, paraphrased）
在 \(B\)–\(p\) 平面（横轴保额 \(B\)、纵轴保费 \(p\)）：(1) 两类型无差异曲线都向上倾斜；(2) 向右下方（更低保费、更高保额）更受偏好；(3) \(U(\cdot)\) 的凹性使无差异曲线凹；(4) 因 \(\pi_H>\pi_L\) 且 \(U\) 凹，在每个 \(B\) 处高风险 \(\pi_H\) 的无差异曲线比低风险 \(\pi_L\) 更陡；(5) 精算公平保单边界：标注"High Risks"的直线斜率为 \(\pi_H\)，标注"Low Risks"的直线斜率为 \(\pi_L\)；边界上方区域对企业有利、下方区域对客户有利；两类型混合的精算公平保单落在两条直线之间。In the \(B\)–\(p\) plane (horizontal axis coverage \(B\), vertical axis premium \(p\)): (1) both types' indifference curves slope up; (2) lower-right (less premium, more coverage) is favorable; (3) the concavity of \(U(\cdot)\) makes the indifference curves concave; (4) since \(\pi_H>\pi_L\) and \(U\) is concave, at each \(B\) the high-risk \(\pi_H\) indifference curve is steeper than the low-risk \(\pi_L\); (5) actuarially fair policy boundary: the line marked "High Risks" has slope \(\pi_H\), the line marked "Low Risks" has slope \(\pi_L\); the region above the boundary is favorable to firms, below to customers; the actuarially fair policy for the mixture lies between the two lines. 解的形式为 \(\{(\psi_H^\star,\psi_L^\star),\,\sigma^\star(a\mid\psi),\,\mu^\star(\pi_L\mid\psi)\}\)，其中 \(\psi_i^\star=(B_i^\star,p_i^\star)\) 是类型 \(i\in\{H,L\}\) 的均衡保单提案，\(\mu^\star(\cdot)\) 是信念系统，\(\sigma^\star(a\mid\psi)\) 是企业的反应（在 \(a=A\) 接受、\(a=R\) 拒绝之间混合）。注意此处客户发送的消息是二维的。 按定义，高风险类型被完全区分出来，故 \(p_H^\star=\pi_H\cdot L\)（企业按精算公平价收费）、\(B_H^\star=L\)（主体风险厌恶，故偏好对可能的损失全额投保的精算公平保单），即 \(\psi_H^\star=(L,\,\pi_H L)\)。 The solution takes the form \(\{(\psi_H^\star,\psi_L^\star),\,\sigma^\star(a\mid\psi),\,\mu^\star(\pi_L\mid\psi)\}\), where \(\psi_i^\star=(B_i^\star,p_i^\star)\) is the equilibrium policy proposal of type \(i\in\{H,L\}\), \(\mu^\star(\cdot)\) the system of beliefs, and \(\sigma^\star(a\mid\psi)\) the firms' response (mixing between \(a=A\) accept and \(a=R\) reject). Note that the message sent by customers here is two-dimensional. By definition, the high-risk type is completely differentiated out, so \(p_H^\star=\pi_H\cdot L\) (firms charge the actuarially fair price) and \(B_H^\star=L\) (the agent is risk-averse, so he loves actuarially fair full insurance against his possible loss), i.e. \(\psi_H^\star=(L,\,\pi_H L)\). 分离均衡的四个条件 / Conditions for separating equilibrium
条件 1：\(B_H^\star=L\) 且 \(p_H^\star=\pi_H\cdot L\)，即 \(\psi_H^\star=(L,\pi_H L)\)（由定义直接给出，保证企业不在高类型上亏损）。条件 2：\(p_L^\star\ge\pi_L\cdot B_L^\star\)（保证企业不在低类型上亏损）。条件 3：\(U_H^\star\ge U_H(B_L^\star,p_L^\star)\)（没有高类型有激励伪装成低类型）。条件 4：\(U_L^\star\ge\max_B U_L(B,\pi_H\cdot B)\)（没有低类型有激励伪装成高类型）。这四个条件给出企业愿意接受任何对应于该区域内点的保单提案、从而支持分离均衡的最大区域。在 MWG 劳动力市场中消息一维，故企业反应在一维消息上；这里 JR 保险模型中消息二维，故反应在一个二维区域上。Condition 1: \(B_H^\star=L\) and \(p_H^\star=\pi_H\cdot L\), i.e. \(\psi_H^\star=(L,\pi_H L)\) (direct from definition; firms don't lose on the high type). Condition 2: \(p_L^\star\ge\pi_L\cdot B_L^\star\) (firms don't lose on the low type). Condition 3: \(U_H^\star\ge U_H(B_L^\star,p_L^\star)\) (no high type has an incentive to deviate to the low type). Condition 4: \(U_L^\star\ge\max_B U_L(B,\pi_H\cdot B)\) (no low type has an incentive to deviate to the high type). These four conditions give the largest region where firms accept any policy proposal corresponding to points in that region, supporting separating equilibrium. In the MWG labor market the message is one-dimensional, so the firms' response is over one message; here in the JR insurance model the message is two-dimensional, so the response is over a two-dimensional region. 一组可行的信念与反应为下面两式： One workable belief and response are the two expressions below: $$
\mu^\star(\psi)=\begin{cases} 1 & \psi=\psi_L^\star\\[2pt] 0 & \psi\ne\psi_L^\star\end{cases}
\qquad
\sigma^\star(A\mid\psi)=\begin{cases} 1 & \psi=\psi_L^\star\ \text{or}\ \psi=\psi_H^\star\ \text{or}\ p\ge\pi_H B\\[2pt] 0 & \text{otherwise}\end{cases}
$$ 图 15 / Figure 15（分离均衡，已转述 / Separating Equilibrium, paraphrased）
在 \(B\)–\(p\) 平面，\(\sigma^\star(a\mid\psi)=A\) 当且仅当落在"黄线"区域（即斜率 \(\pi_H\) 的高风险精算公平线及其上方可接受区）。高类型选 \((L,\pi_H L)\)，低类型选某点 \(p\)，于是得到分离均衡。注意当低类型选点 \(p\) 时得到的是最低成本分离均衡，它帕累托占优所有其他分离均衡，也是唯一存活于（信息）直觉准则的均衡。In the \(B\)–\(p\) plane, \(\sigma^\star(a\mid\psi)=A\) if and only if the point lies on the "yellow line" region (the high-risk actuarially-fair line of slope \(\pi_H\) and the acceptable region above it). The high type chooses \((L,\pi_H L)\), the low type chooses some point \(p\), giving a separating equilibrium. Note that when the low type chooses point \(p\) we get the least-cost separating equilibrium, which Pareto dominates all other separating equilibria and is also the only one surviving the (information) intuitive criterion. 按定义，两类型都偏好被完全混合；\(p^\star\ge\mathbb{E}[\pi]\cdot B\)，因为企业在均衡中不亏钱，而由混同定义其期望风险恰为 \(\mathbb{E}[\pi]\)。 By definition, both types prefer to be completely pooled; \(p^\star\ge\mathbb{E}[\pi]\cdot B\) because the firm won't lose money in equilibrium, and by the definition of pooling the expected risk is exactly \(\mathbb{E}[\pi]\). 混同均衡的三个条件 / Conditions for pooling equilibrium
条件 1：\(p^\star\ge\mathbb{E}[\pi]\cdot B\)（序贯理性下企业不亏钱）。条件 2：\(U_H^\star\ge U_H(L,\pi_H\cdot L)\equiv\underline U_H\)（没有高类型有激励偏离去被识别为高类型、进而玩分离均衡）。条件 3：\(U_L^\star\ge\max_B U_L(B,\pi_H\cdot B)\equiv\underline U_L\)（没有低类型有激励偏离去提案一份不会被拒的保单、进而被识别为低类型）。这三个条件给出企业愿意接受提案 \(\psi_{pool}^\star\)、从而支持混同均衡的最大区域。Condition 1: \(p^\star\ge\mathbb{E}[\pi]\cdot B\) (firms don't lose money under sequential rationality). Condition 2: \(U_H^\star\ge U_H(L,\pi_H\cdot L)\equiv\underline U_H\) (no high type has an incentive to deviate to be identified as high and then play the separating equilibrium). Condition 3: \(U_L^\star\ge\max_B U_L(B,\pi_H\cdot B)\equiv\underline U_L\) (no low type has an incentive to deviate to propose a non-rejectable policy and then be identified as low). These three conditions give the largest region where firms accept the proposal \(\psi_{pool}^\star\), supporting pooling equilibrium. 一组可行的信念与反应为下面两式： One workable belief and response are the two expressions below: $$
\mu^\star(\psi)=\begin{cases} \phi & \psi=\psi_{pool}^\star\\[2pt] 0 & \psi\ne\psi_{pool}^\star\end{cases}
\qquad
\sigma^\star(A\mid\psi)=\begin{cases} 1 & \psi=\psi_{pool}^\star\ \text{or}\ p\ge\pi_H B\\[2pt] 0 & \text{otherwise}\end{cases}
$$ 图 16 / Figure 16（混同均衡，已转述 / Pooling Equilibrium, paraphrased）
在 \(B\)–\(p\) 平面，混同保单 \(\psi_{pool}^\star\) 落在斜率介于 \(\pi_L\) 与 \(\pi_H\) 之间的"混合精算公平线" \(\mathbb{E}[\pi]\) 附近、且位于高风险线 \(\pi_H L\) 标记处；两类型无差异曲线 \(U_H,U_L\) 都通过该混同点，可接受区为 \(p\ge\pi_H B\) 或 \(\psi=\psi_{pool}^\star\)。本设定下同样可定义杂交均衡，其性质与 13.1.5（MWG 劳动力市场）的杂交均衡相同。In the \(B\)–\(p\) plane, the pooling policy \(\psi_{pool}^\star\) lies near the "mixed actuarially-fair line" \(\mathbb{E}[\pi]\) (slope between \(\pi_L\) and \(\pi_H\)) and at the high-risk mark \(\pi_H L\); both types' indifference curves \(U_H,U_L\) pass through this pooling point, with acceptable region \(p\ge\pi_H B\) or \(\psi=\psi_{pool}^\star\). A hybrid equilibrium can also be defined in this set-up, with the same flavor as the hybrid equilibrium in §13.1.5 (the MWG labor market).13.1.4 混同均衡 / Pooling equilibrium
13.1.4 Pooling equilibrium
13.1.5 混合（杂交）均衡 / Hybrid equilibrium
13.1.5 Hybrid equilibrium
13.1.6 各均衡的帕累托最优性 / Pareto optimality of the equilibria
13.1.6 Pareto optimality of the equilibria
13.1.7 序贯均衡的精炼：直觉准则 / Refinement: the intuitive criterion
13.1.7 Refinement of sequential equilibrium: the intuitive criterion
13.2 JR 保险市场问题
13.2.1 设定 / Set-up
13.2 JR Insurance Market Problem
13.2.1 Set-up
13.2.2 分离均衡 / Separating equilibrium
13.2.2 Separating equilibrium
13.2.3 混同均衡 / Pooling equilibrium
13.2.3 Pooling equilibrium
参考文献 / References
References