设定与时序 / Set-up and timing 1. 工人效用 \(U=w-c(t,\theta)\)，其中 \(c(0,\theta)=0\)，\(c\) 关于 \(t\) 递增且凸。(a) \(t\) 是企业作为工资报价一部分所要求的任务；任务仅用于筛选，对企业没有生产价值。(b) 单交叉性质 \(c_{t\theta}<0\)（高类型边际成本更低），蕴含 \(c_\theta<0\)。(c) \(\theta_H>\theta_L>0\) 两类型，概率分别为 \(\phi\) 与 \(1-\phi\)。2. 两家企业相互竞争。 (a) 竞争需 \(\ge2\) 家企业；(b) 两家足以完成证明，故不失一般性设只有两家；(c) 记 \(j\) 为做得不更好的那家，\(\pi_j\le\tfrac12\pi_{industry}\)；(d) 均衡中无企业负利润，\(\pi_1,\pi_2\ge0\)。3. 时序：(a) 自然为工人抽取类型，工人私下得知 \(\theta\)（生产率）；(b) 企业给出包含相应任务的工资方案；(c) 工人选择最优报价接受。1. Workers' utility \(U=w-c(t,\theta)\), with \(c(0,\theta)=0\) and \(c\) increasing and convex in \(t\). (a) \(t\) is the task required by the firm as part of the wage offer; the task is only for screening and is not productive to the firm. (b) Single-crossing \(c_{t\theta}<0\) (high types have lower marginal cost), which implies \(c_\theta<0\). (c) \(\theta_H>\theta_L>0\) are the two types, with probabilities \(\phi\) and \(1-\phi\). 2. Two firms compete with each other. (a) Competition needs \(\ge2\) firms; (b) two firms are enough for the proofs, so WLOG there are only two; (c) let firm \(j\) denote the firm not doing better, \(\pi_j\le\tfrac12\pi_{industry}\); (d) in equilibrium no firm operates at negative profits, \(\pi_1,\pi_2\ge0\). 3. Timing: (a) nature chooses types for workers, who privately learn \(\theta\) (productivity); (b) firms offer wage packages including the required task associated with that wage; (c) workers choose the best offer to accept.

纯策略均衡的两种形态 / Two forms of pure-strategy equilibrium (a) 分离纯策略均衡由以下刻画：(i) 企业 \(i\)（\(i\in\{1,2\}\)）给出两份分离报价的方案 \(\psi^i=\{\psi_L^\star=(w_L^\star,t_L^\star),\,\psi_H^\star=(w_H^\star,t_H^\star)\}\)；(ii) 工人行动为 \(\{\)接受企业 1 的 \(\psi_L^\star\)，接受企业 1 的 \(\psi_H^\star\)，接受企业 2 的 \(\psi_L^\star\)，接受企业 2 的 \(\psi_H^\star\}\)。(b) 混同纯策略均衡由以下刻画：(i) 企业 \(i\) 给出唯一报价 \(\psi^i=\{\psi_p^\star=(w_p^\star,t_p^\star)\}\)；(ii) 工人行动为 \(\{\)接受企业 1 的 \(\psi_p^\star\)，接受企业 2 的 \(\psi_p^\star\}\)。(a) A separating pure-strategy equilibrium is characterized by: (i) firm \(i\) (\(i\in\{1,2\}\)) offers a package of two separate offers \(\psi^i=\{\psi_L^\star=(w_L^\star,t_L^\star),\,\psi_H^\star=(w_H^\star,t_H^\star)\}\); (ii) the workers' action is \(\{\)Accept \(\psi_L^\star\) of firm 1, Accept \(\psi_H^\star\) of firm 1, Accept \(\psi_L^\star\) of firm 2, Accept \(\psi_H^\star\) of firm 2$}\(. **(b) A pooling pure-strategy equilibrium** is characterized by: (i) firm \)i$ offers one offer \(\psi^i=\{\psi_p^\star=(w_p^\star,t_p^\star)\}\); (ii) the workers' action is \(\{\)Accept \(\psi_p^\star\) of firm 1, Accept \(\psi_p^\star\) of firm 2$}$.

图 17 / Figure 17（两类型无差异曲线，已转述 / indifference curves, paraphrased） 在 \(t\)–\(w\) 平面（横轴任务 \(t\)、纵轴工资 \(w\)）：(a) 两类型无差异曲线都向上倾斜；(b) 向左上方（更少任务、更高工资）更受偏好；(c) \(c(t,\theta)\) 关于 \(t\) 的凸性使无差异曲线凸；(d) 因 \(c_{t\theta}<0\)，在每个 \(t\) 处低类型 \(\theta_L\) 的无差异曲线比高类型 \(\theta_H\) 更陡。其中 \(U_H\equiv\{(t,w)\mid U_H=w-c(t,\theta_H)\}\)、\(U_L\equiv\{(t,w)\mid U_L=w-c(t,\theta_L)\}\)。In the \(t\)–\(w\) plane (horizontal axis task \(t\), vertical axis wage \(w\)): (a) both types' indifference curves slope up; (b) up-left (less task, more wage) is favorable; (c) the convexity of \(c(t,\theta)\) in \(t\) makes the curves convex; (d) since \(c_{t\theta}<0\), at each \(t\) the low type \(\theta_L\)'s curve is steeper than the high type \(\theta_H\)'s. Here \(U_H\equiv\{(t,w)\mid U_H=w-c(t,\theta_H)\}\) and \(U_L\equiv\{(t,w)\mid U_L=w-c(t,\theta_L)\}\).

事实 16.1（完全信息均衡）/ Fact 16.1 (perfect-information equilibrium) 在完全信息均衡中，\((w_L^\star=\theta_L,\,t_L^\star=0)\) 且 \((w_H^\star=\theta_H,\,t_H^\star=0)\)。In equilibrium with perfect information, \((w_L^\star=\theta_L,\,t_L^\star=0)\) and \((w_H^\star=\theta_H,\,t_H^\star=0)\).

证明 / Proof 第一步：证 \(w_L^\star=\theta_L\)、\(w_H^\star=\theta_H\)。 反设不然。因企业不会负利润运营，故对企业 \(i\) 至少有 \(w_L^\star<\theta_L\) 或 \(w_H^\star<\theta_H\) 之一成立。不失一般性设 \(w_H^\star<\theta_H\)，则另一家企业 \(j\ne i\) 可偏离、开出 \(w_H'=w_H^\star+\varepsilon\)（\(\varepsilon\in(0,\theta_H-w_H^\star)\)），把 \(\theta_H\) 工人从 \(i\) 抢走并赚取严格为正的利润；\(\varepsilon\) 足够小时该偏离对 \(j\) 有利，矛盾。Step 1: prove \(w_L^\star=\theta_L\) and \(w_H^\star=\theta_H\). Suppose not. Since firms won't operate at negative profits, for firm \(i\) at least one of \(w_L^\star<\theta_L\) or \(w_H^\star<\theta_H\) holds. WLOG assume \(w_H^\star<\theta_H\); then the other firm \(j\ne i\) can deviate by offering \(w_H'=w_H^\star+\varepsilon\) (\(\varepsilon\in(0,\theta_H-w_H^\star)\)), stealing the \(\theta_H\) worker from \(i\) and earning strictly positive profit. For small enough \(\varepsilon\) this deviation is profitable for \(j\), a contradiction.

第二步：证 \(t_L^\star=0\)、\(t_H^\star=0\)。 反设不然。不失一般性设企业 \(i\) 要求 \(\psi^\star=(w_H^\star=\theta_H,\,t_H^\star>0)\)。则企业 \(j\) 可偏离、开出 \(\psi'=(w_H'=\theta_H-\varepsilon,\,t_H'=t_H^\star-\delta)\)（\(\varepsilon,\delta>0\)）；\(\varepsilon\) 足够小时 \(\theta_H\) 工人更偏好 \(\psi'\) 而全部转投 \(j\)，使 \(j\) 赚取严格为正的利润，故偏离对 \(j\) 有利，矛盾。\(\blacksquare\)Step 2: show \(t_L^\star=0\) and \(t_H^\star=0\). Suppose not. WLOG assume firm \(i\) requires \(\psi^\star=(w_H^\star=\theta_H,\,t_H^\star>0)\). Then firm \(j\) can deviate by offering \(\psi'=(w_H'=\theta_H-\varepsilon,\,t_H'=t_H^\star-\delta)\) (\(\varepsilon,\delta>0\)); for small enough \(\varepsilon\), \(\theta_H\) workers prefer \(\psi'\) and all switch to \(j\), making \(j\) earn strictly positive profit, so the deviation is profitable for \(j\), a contradiction. \(\blacksquare\)

注 16.1 / Remark 16.1 完全信息博弈的均衡 \((w_L^\star=\theta_L,t_L^\star=0),(w_H^\star=\theta_H,t_H^\star=0)\) 非常有效：既无需筛选，企业便不要求任何无意义的任务，并对两类型都给出正确工资。但当不存在完全信息时，筛选变得有意义，并伴随一项社会福利成本。The perfect-information equilibrium \((w_L^\star=\theta_L,t_L^\star=0),(w_H^\star=\theta_H,t_H^\star=0)\) is very efficient: since there is no need for screening, the firm requires no meaningless task for either type and offers correct wages to both. But when there is no perfect information, screening becomes meaningful, and there is a social welfare cost associated with screening.

结果 1 / Result 1 在任何纯策略均衡中，两家企业的期望支付都为零。In any pure-strategy equilibrium, the expected payoff of both firms is zero.

证明 / Proof 反设不然，则行业利润严格为正。以分离均衡为例，记均衡报价为 \((w_L^\star,t_L^\star)\) 与 \((w_H^\star,t_H^\star)\)（混同时取 \(w_L^\star=w_H^\star\)、\(t_L^\star=t_H^\star\)）。企业 \(j\) 可偏离、对每类型加价 \(\varepsilon>0\) 开出 \((w_L^\star+\varepsilon,t_L^\star)\) 与 \((w_H^\star+\varepsilon,t_H^\star)\)，从 \(i\) 抢走全部工人与利润；\(\varepsilon\) 足够小时偏离有利，矛盾。\(\blacksquare\)Suppose not; then industry profits are strictly positive. Take the separating equilibrium as an example, with equilibrium offers \((w_L^\star,t_L^\star)\) and \((w_H^\star,t_H^\star)\) (for pooling, \(w_L^\star=w_H^\star\), \(t_L^\star=t_H^\star\)). Firm \(j\) can deviate by adding \(\varepsilon>0\) to each type, offering \((w_L^\star+\varepsilon,t_L^\star)\) and \((w_H^\star+\varepsilon,t_H^\star)\), stealing all workers and profits from \(i\); for small enough \(\varepsilon\) the deviation is profitable, a contradiction. \(\blacksquare\)

结果 2 / Result 2 不存在混同均衡。No pooling equilibrium exists.

证明（图 18）/ Proof (Figure 18) 图 18（已转述）：设点 \(E\) 表示混同均衡，即 \((w=\mathbb{E}[\theta],\,t=t_p^\star)\)，落在 \(w=\mathbb{E}[\theta]\) 高度、两类型无差异曲线在此相切。则其右上方"蓝色阴影区"内任一点（特别是点 \(p\)）都是企业 \(j\) 的有利偏离：若企业 \(i\) 开出点 \(E\)，企业 \(j\) 改开点 \(p=(w_p',t_p')\)（更高工资、更多任务），低类型宁愿留在 \(E\)，而所有高类型转投 \(j\)；因 \(w_p'<\theta_H\)，\(j\) 赚严格正利润，与结果 1 矛盾。\(\blacksquare\)Figure 18 (paraphrased): let point \(E\) denote the pooling equilibrium, i.e. \((w=\mathbb{E}[\theta],\,t=t_p^\star)\), at height \(w=\mathbb{E}[\theta]\) where the two indifference curves are tangent. Then any point in the upper-right "blue shaded area" (in particular point \(p\)) is a profitable deviation for firm \(j\): if firm \(i\) offers point \(E\), firm \(j\) offers point \(p=(w_p',t_p')\) (higher wage, more task), the low type prefers to stay at \(E\) while all high types switch to \(j\); since \(w_p'<\theta_H\), \(j\) makes strictly positive profit, contradicting Result 1. \(\blacksquare\)

结果 3 / Result 3 在任何分离均衡中，工人获得等于自身生产率的工资，即 \(w_H^\star=\theta_H\)、\(w_L^\star=\theta_L\)。此结果强于结果 1：它要求企业在两类型上分别零利润，从而排除"一类型赚、另一类型亏、净利润为零"（结果 1 未排除此情形）。In any separating equilibrium, a worker gets a wage equal to their own productivity, i.e. \(w_H^\star=\theta_H\), \(w_L^\star=\theta_L\). This is stronger than Result 1: it requires firms to earn zero profit over each type separately, ruling out the case where firms earn positive profit on one type and lose on the other with net profit zero (not ruled out by Result 1).

证明（图 19）/ Proof (Figure 19) 第一步：设 \(w_L^\star<\theta_L\)，则低类型有利可图。企业 \(j\) 可偏离开出 \(w_L'=w_L^\star+\varepsilon\)（\(\varepsilon\in(0,\theta_L-w_L^\star)\)）。若高类型转投 \(j\)，\(j\) 赚严格正利润；若高类型还跑来 \(j\) 伪装成低类型，\(j\) 赚得更多。故偏离有利，矛盾，得 \(w_L^\star\ge\theta_L\)。第二步：由结果 1 净利润为零，故 \(w_L^\star\le\theta_L\) 蕴含 \(w_H^\star\ge\theta_H\)。设 \(w_H^\star<\theta_H\)，则图 19（已转述）中"蓝色阴影区"内任一点 \(p\) 是有利偏离（同结果 2 证法），矛盾。故 \(w_H^\star=\theta_H\)、\(w_L^\star=\theta_L\)。\(\blacksquare\)Step 1: suppose \(w_L^\star<\theta_L\), then the low type is profitable. Firm \(j\) can deviate by offering \(w_L'=w_L^\star+\varepsilon\) (\(\varepsilon\in(0,\theta_L-w_L^\star)\)). If high types switch to \(j\), \(j\) makes strictly positive profit; if high types also show up at \(j\) pretending to be low, \(j\) makes even more. So the deviation is profitable, a contradiction, giving \(w_L^\star\ge\theta_L\). Step 2: by Result 1 net profit is zero, so \(w_L^\star\le\theta_L\) implies \(w_H^\star\ge\theta_H\). Suppose \(w_H^\star<\theta_H\); then in Figure 19 (paraphrased) any point \(p\) in the "blue shaded area" is a profitable deviation (same argument as Result 2), a contradiction. So \(w_H^\star=\theta_H\), \(w_L^\star=\theta_L\). \(\blacksquare\)

结果 4 / Result 4 在任何分离均衡中，\(t_L^\star=0\)。In any separating equilibrium, \(t_L^\star=0\).

证明（图 20）/ Proof (Figure 20) 反设 \(t_L^\star>0\)，则图 20（已转述）中"蓝色阴影区"内任一点 \(p\)（更高工资、更少任务）是企业 \(j\) 的有利偏离，矛盾。故 \(t_L^\star=0\)。\(\blacksquare\)Suppose not, i.e. \(t_L^\star>0\); then in Figure 20 (paraphrased) any point \(p\) in the "blue shaded area" (higher wage, less task) is a profitable deviation for firm \(j\), a contradiction. So \(t_L^\star=0\). \(\blacksquare\)

结果 5 / Result 5 在任何分离均衡中，\(t_H^\star=\hat t\)。In any separating equilibrium, \(t_H^\star=\hat t\).

证明（图 21）/ Proof (Figure 21) 反设 \(t_H^\star>\hat t\)，则图 21（已转述）中"蓝色阴影区"内任一点 \(p\)（在不诱使低类型模仿的前提下，给高类型更少任务）是企业 \(j\) 的有利偏离，矛盾。故 \(t_H^\star=\hat t\)。\(\blacksquare\)Suppose not, i.e. \(t_H^\star>\hat t\); then in Figure 21 (paraphrased) any point \(p\) in the "blue shaded area" (less task for the high type while still not tempting the low type to mimic) is a profitable deviation for firm \(j\), a contradiction. So \(t_H^\star=\hat t\). \(\blacksquare\)

结果 6（前五结果的总结）/ Result 6 (summary of the previous five) 若存在纯策略均衡，则：它必为分离均衡；\(w_H^\star=\theta_H\)、\(w_L^\star=\theta_L\)；\(t_H^\star=\hat t\)、\(t_L^\star=0\)，其中 \(\hat t\) 满足 \(\theta_L-c(0,\theta_L)=\theta_H-c(\hat t,\theta_L)\)。这意味着唯一可能的纯策略均衡是最低成本分离均衡。If a pure-strategy equilibrium exists, then: it must be a separating equilibrium; \(w_H^\star=\theta_H\), \(w_L^\star=\theta_L\); \(t_H^\star=\hat t\), \(t_L^\star=0\), where \(\hat t\) satisfies \(\theta_L-c(0,\theta_L)=\theta_H-c(\hat t,\theta_L)\). This means the only possible pure-strategy equilibrium is the least-cost separating one.

设定与时序 / Set-up and timing 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 \(y\)。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。设有两家企业相互竞争。时序：1. 自然抽取类型并向每位客户私下揭示 \(\pi\)；2. 保险公司向客户提供合约；3. 客户选择接受最优合约。类型 \(i\in\{L,H\}\) 的效用见下式，\(U(\cdot)\) 为两类型共享的递增凹函数。Two risk types (probability of suffering loss \(L\)): \(0<\pi_L<\pi_H<1\); both types have income \(y\). The low type has probability \(\phi\) and the high type \(1-\phi\). Coverage \(B\) is the amount paid by the insurance company when the loss \(L\) takes place. Suppose there are two firms competing. Timing: 1. Nature chooses the type and privately reveals \(\pi\) to each customer; 2. insurance firms offer contracts to customers; 3. customers choose to accept the best contract. The utility of type \(i\in\{L,H\}\) is below, where \(U(\cdot)\) is an increasing concave function shared by both types.

图 24 / Figure 24（两类型无差异曲线，已转述 / indifference curves, paraphrased） 在 \(B\)–\(p\) 平面：(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) 精算公平边界："High Risks" 线斜率 \(\pi_H\)、"Low Risks" 线斜率 \(\pi_L\)，上方区域对企业有利、下方对客户有利，混合类型的精算公平保单落在两线之间。In the \(B\)–\(p\) plane: (1) both types' indifference curves slope up; (2) lower-right (less premium, more coverage) is favorable; (3) the concavity of \(U(\cdot)\) makes the curves concave; (4) since \(\pi_H>\pi_L\) and \(U\) is concave, at each \(B\) the high-risk \(\pi_H\) curve is steeper than the low-risk \(\pi_L\); (5) actuarially fair boundary: the "High Risks" line has slope \(\pi_H\), the "Low Risks" line has slope \(\pi_L\), above is favorable to firms and below to customers, with the mixture's fair policy between the two lines.

结果 1 / Result 1 在任何纯策略均衡中，两家企业的期望支付都为零。In any pure-strategy equilibrium, the expected payoff of both firms is zero.

证明 / Proof 反设不然，则行业利润严格为正。第一步（混同情形）：记均衡报价为 \((B_p^\star,p_p^\star)\)，企业 \(j\) 可偏离、开出 \((B_p^\star+\varepsilon,p_p^\star)\)（\(\varepsilon>0\)）从 \(i\) 抢走全部客户与利润，\(\varepsilon\) 足够小时有利，矛盾。第二步（分离情形）：记均衡报价为 \((B_L^\star,p_L^\star)\) 与 \((B_H^\star,p_H^\star)\)。由分离定义，\(U_L(B_L^\star,p_L^\star)\ge U_L(B_H^\star,p_H^\star)\) 且 \(U_H(B_H^\star,p_H^\star)\ge U_H(B_L^\star,p_L^\star)\)，且至少一式严格（否则见第一步）。不失一般性设 \(U_L(B_L^\star,p_L^\star)>U_L(B_H^\star,p_H^\star)\)。则取足够小的 \(\varepsilon,\delta>0\)（\(\varepsilon\) 更小），企业 \(j\) 偏离开出 \((B_L^\star+\varepsilon,p_L^\star)\) 与 \((B_H^\star+\delta,p_H^\star)\)，使 \(U_L(B_L^\star+\varepsilon,p_L^\star)\ge U_L(B_H^\star+\delta,p_H^\star)\)、\(U_H(B_H^\star+\delta,p_H^\star)\ge U_H(B_L^\star+\varepsilon,p_L^\star)\)，从 \(i\) 抢走全部客户与利润。该有利偏离使原报价不成其为均衡。\(\blacksquare\)Suppose not; then industry profits are strictly positive. Step 1 (pooling case): denote the equilibrium offer \((B_p^\star,p_p^\star)\); firm \(j\) can deviate by offering \((B_p^\star+\varepsilon,p_p^\star)\) (\(\varepsilon>0\)), stealing all customers and profits from \(i\), profitable for small \(\varepsilon\), a contradiction. Step 2 (separating case): denote the equilibrium offers \((B_L^\star,p_L^\star)\) and \((B_H^\star,p_H^\star)\). By the definition of separating, \(U_L(B_L^\star,p_L^\star)\ge U_L(B_H^\star,p_H^\star)\) and \(U_H(B_H^\star,p_H^\star)\ge U_H(B_L^\star,p_L^\star)\), with at least one strict (else see Step 1). WLOG \(U_L(B_L^\star,p_L^\star)>U_L(B_H^\star,p_H^\star)\). Then for small enough \(\varepsilon,\delta>0\) (\(\varepsilon\) even smaller), firm \(j\) deviates to \((B_L^\star+\varepsilon,p_L^\star)\) and \((B_H^\star+\delta,p_H^\star)\), making \(U_L(B_L^\star+\varepsilon,p_L^\star)\ge U_L(B_H^\star+\delta,p_H^\star)\) and \(U_H(B_H^\star+\delta,p_H^\star)\ge U_H(B_L^\star+\varepsilon,p_L^\star)\), stealing all customers and profits from \(i\). This profitable deviation makes the previous offers not an equilibrium. \(\blacksquare\)

结果 2 / Result 2 不存在混同均衡。No pooling equilibrium exists.

证明（图 25）/ Proof (Figure 25) 图 25（已转述）：设点 \(E\) 表示混同均衡（落在混合精算公平线 \(\mathbb{E}[\pi]\) 与价格 \(p^\star\)、保额 \(B^\star\) 处，两类型无差异曲线在此相切）。则总存在一点 \(p\) 是企业 \(j\) 的有利偏离（位于低风险无差异曲线 \(U_L\) 与高风险线之间的"蓝色阴影区"，吸引低风险者而排斥高风险者），使 \(E\) 不成其为均衡。\(\blacksquare\)Figure 25 (paraphrased): let point \(E\) denote the pooling equilibrium (at the mixed actuarially-fair line \(\mathbb{E}[\pi]\), price \(p^\star\), coverage \(B^\star\), where the two indifference curves are tangent). Then there always exists a point \(p\) that is a profitable deviation for firm \(j\) (in the "blue shaded area" between the low-risk curve \(U_L\) and the high-risk line, attracting low risks while repelling high risks), making \(E\) not an equilibrium. \(\blacksquare\)

结果 3 / Result 3 在任何分离均衡中，\(U_H^\star\ge U_H(L,\pi_H L)\)。In any separating equilibrium, \(U_H^\star\ge U_H(L,\pi_H L)\).

证明 / Proof 反设不然，则分离均衡中高风险主体获得的效用低于按精算公平价全额投保。企业 \(j\) 可偏离、开出 \((L,(\pi_H+\varepsilon)L)\)（\(\varepsilon>0\)）从 \(i\) 抢走全部高风险客户；\(\varepsilon\) 足够小时对高风险有吸引力，偏离有利，矛盾。\(\blacksquare\)Suppose not; then in the separating equilibrium the high-risk agents get utility less than from full insurance at the actuarially fair rate. Firm \(j\) can deviate by offering \((L,(\pi_H+\varepsilon)L)\) (\(\varepsilon>0\)), stealing all high-risk customers from \(i\); for small enough \(\varepsilon\) it is attractive to high risks, so the deviation is profitable, a contradiction. \(\blacksquare\)

结果 4 / Result 4 在任何分离均衡中，\(p_L^\star=\pi_L B_L^\star\)。In any separating equilibrium, \(p_L^\star=\pi_L B_L^\star\).

证明 / Proof 反设 \(p_L^\star>\pi_L B_L^\star\)，则企业 \(j\) 可偏离、开出 \((B_L^\star,p_L^\star-\varepsilon)\)（\(\varepsilon\in(0,p_L^\star-\pi_L B_L^\star)\)）从 \(i\) 抢走全部低风险客户；对低风险有吸引力，偏离有利，矛盾。\(\blacksquare\)Suppose not, i.e. \(p_L^\star>\pi_L B_L^\star\); then firm \(j\) can deviate by offering \((B_L^\star,p_L^\star-\varepsilon)\) (\(\varepsilon\in(0,p_L^\star-\pi_L B_L^\star)\)), stealing all low-risk customers from \(i\); it is attractive to low risks, so the deviation is profitable, a contradiction. \(\blacksquare\)

结果 5 / Result 5 在任何分离均衡中，\(B_H^\star=L\) 且 \(p_H^\star=\pi_H L\)。In any separating equilibrium, \(B_H^\star=L\) and \(p_H^\star=\pi_H L\).

证明 / Proof 第一步：证 \(B_H^\star=L\)。 反设 \(B_H^\star0\)）从 \(i\) 抢走全部高风险客户。因主体风险厌恶，在同等费率下更高的（但仍不足额的）保额更受偏好，对高风险有吸引力，偏离有利，矛盾。第二步：证 \(p_H^\star=\pi_H L\)， 由结果 1 与结果 4 直接得出。\(\blacksquare\)Step 1: prove \(B_H^\star=L\). Suppose not, i.e. \(B_H^\star0\)), stealing all high-risk customers from \(i\). Since agents are risk-averse, higher (but still less than full) coverage at the same rate is more favorable, attractive to high risks, so the deviation is profitable, a contradiction. Step 2: prove \(p_H^\star=\pi_H L\), which follows directly from Results 1 and 4. \(\blacksquare\)

结果 6 / Result 6 在任何分离均衡中，\(B_L^\star=\hat B\)。In any separating equilibrium, \(B_L^\star=\hat B\).

证明（图 26）/ Proof (Figure 26) 反设 \(B_L^\star<\hat B\)，则图 26（已转述）中"蓝色阴影区"内任一点 \(p\)（在不诱使高风险模仿的前提下，给低风险更高保额）是企业 \(j\) 的有利偏离，矛盾。故 \(B_L^\star=\hat B\)。\(\blacksquare\)Suppose not, i.e. \(B_L^\star<\hat B\); then in Figure 26 (paraphrased) any point \(p\) in the "blue shaded area" (higher coverage for low risks while still not tempting high risks to mimic) is a profitable deviation for firm \(j\), a contradiction. So \(B_L^\star=\hat B\). \(\blacksquare\)

六个结果的总结 / Summary of the six results 若存在纯策略均衡，则：它必为分离均衡；\(p_H^\star=\pi_H L\)、\(p_L^\star=\pi_L\hat B\)，其中 \(\hat B\) 满足 \(U_H(L,\pi_H L)=U_H(\hat B,\pi_L\hat B)\)；\(B_H^\star=L\)、\(B_L^\star=\hat B\)。这意味着唯一可能的纯策略均衡是最大保额分离均衡。这里"最大保额"指在两类型被完全分离的条件下两者都享有最高保额（由结果 5、6 刻画），可解读为另一种形式的"最低成本"——其中成本是保额缩减带来的福利损失。If a pure-strategy equilibrium exists, then: it must be a separating equilibrium; \(p_H^\star=\pi_H L\), \(p_L^\star=\pi_L\hat B\), where \(\hat B\) satisfies \(U_H(L,\pi_H L)=U_H(\hat B,\pi_L\hat B)\); \(B_H^\star=L\), \(B_L^\star=\hat B\). This means the only possible pure-strategy equilibrium is the most-coverage separating one. Here "most-coverage" means both types enjoy the most coverage conditional on being perfectly separated (characterized by Results 5 and 6), and can be interpreted as another form of "least-cost" where the cost is the welfare loss from reduced insurance coverage.