14. Cheap Talk
14. Cheap Talk
本章导读 前几章的信号传递是有成本的（教育努力或保险保额）。本章考虑信号无成本的情形——此即"废话/空谈"（cheap talk）。§14.1 设定：时序与标准 sender–receiver 博弈相同，但因消息无成本，支付与消息无关 \(U(\theta,m,a)=u(\theta,a)\)；有意义沟通的条件是均衡映射 \(\Theta\to A\) 非常数，需三件事：(1) 发送者对行动的偏好依赖于 \(\theta\)，(2) 接收者最优行动依赖于 \(\theta\)，(3) 双方偏好有共同性。§14.2 用 Alice/Bob 的两任务两类型例子展示胡言均衡与有信息均衡，并用两张反例支付矩阵说明违反条件 2 或条件 3 时沟通失效。§14.3 Crawford–Sobel (1982)：在 \(\theta\sim[0,1]\)、\(a^S(\theta)>a^R(\theta)\) 的连续模型中给出 \(n\) 步分区序贯均衡，永远存在一步胡言均衡，更高步数受偏差限制；二次偏好例子 \(a^R=\theta\)、\(a^S=\theta+b\) 求出两步均衡需 \(b<\tfrac14\)。所有表格为支付矩阵。
14. Cheap Talk
Overview Signaling in previous chapters was costly (education effort or insurance coverage). This chapter considers the case where signals are costless — this is "cheap talk". §14.1 setting: the timing is the same as the standard sender–receiver game, but since messages are costless, payoffs are independent of messages, \(U(\theta,m,a)=u(\theta,a)\); the condition for meaningful communication is that the equilibrium mapping \(\Theta\to A\) be non-constant, requiring three things: (1) the sender's preferences over actions depend on \(\theta\), (2) the receiver's optimal action depends on \(\theta\), (3) a commonality of preferences. §14.2 uses an Alice/Bob two-task two-type example to display the babbling equilibrium and an informative equilibrium, and two counter-example payoff matrices showing communication breaks down when Condition 2 or 3 is violated. §14.3 Crawford–Sobel (1982): in a continuous model with \(\theta\sim[0,1]\) and \(a^S(\theta)>a^R(\theta)\), gives the \(n\)-step partition sequential equilibrium, with a one-step babbling equilibrium always existing and higher step counts limited by the bias; the quadratic-preference example \(a^R=\theta\), \(a^S=\theta+b\) shows a two-step equilibrium needs \(b<\tfrac14\). All tables are payoff matrices.
前几节考虑了信号传递有成本的情形（如教育、或保险中不完全的保额）。现在我们考虑信号无成本的情形——信号无成本即所谓"cheap talk（废话）"。
14.1 设定 / Setting
14.1.1 时序 / Timing
时序与标准的 sender–receiver 信号博弈相同：
- 自然按某先验分布为发送者抽取类型 \(\theta_i\in\Theta\)；分布是共同知识，但 \(\theta_i\) 是发送者的私人信息。
- 发送者 \(S\) 选择消息 \(m\in M\)，\(M\) 为可行消息集；策略是关于消息的条件概率分布 \(\sigma_S(m\mid\theta)\)。
- 接收者 \(R\) 观察到 \(m\) 并形成信念 \(\mu(\theta\mid m)\)，随后选择行动 \(a\in A\)；策略是关于行动的条件概率分布 \(\sigma_R(a\mid m)\)。
由于消息无成本，支付与消息无关，故可写为下式：
The previous sections considered settings in which signaling was costly (e.g. education, or incomplete coverage in insurance). Now we consider the case where signals are costless — costless signals are "cheap talk".
14.1 Setting
14.1.1 Timing
The timing is the same as the standard sender–receiver signaling game:
- Nature chooses the type \(\theta_i\in\Theta\) for the sender according to some prior distribution; the distribution is common knowledge but \(\theta_i\) is private information to the sender.
- Sender \(S\) chooses a message \(m\in M\), where \(M\) is the set of feasible messages; a strategy is a conditional probability distribution over messages \(\sigma_S(m\mid\theta)\).
- Receiver \(R\) observes \(m\) and forms beliefs \(\mu(\theta\mid m)\), then chooses an action \(a\in A\); a strategy is a conditional probability distribution over actions \(\sigma_R(a\mid m)\).
Since messages are costless, payoffs are independent of messages, so we can write them as below:
$$ U^S(\theta,m,a)=u^S(\theta,a),\qquad U^R(\theta,m,a)=u^R(\theta,a) $$
14.1.2 有意义沟通的假设 / Assumptions for meaningful communication
有意义沟通的条件是：从发送者类型空间到接收者行动空间的均衡映射 \(\Theta\to A\) 非常数。这很直观——有意义的沟通至少应揭示一些关于私人类型的信息，体现为当类型实现不同时均衡行动随之改变。
14.1.2 Assumptions for meaningful communication
The condition for meaningful communication is that the equilibrium mapping \(\Theta\to A\) from the sender's type space to the receiver's action space be non-constant. This is very intuitive — meaningful communication should at least reveal some information about the privately known types, exemplified by a change in equilibrium actions when types are realized differently.
有意义沟通的三个条件 / Three conditions for meaningful communication 1. 发送者对行动 \(a\in A\) 的偏好必须依赖于 \(\theta\)。 不同类型的发送者必须对接收者的行动有不同偏好；否则不同类型的发送者都会发送同一条最有利的消息，使该消息变得没有意义。2. 接收者的最优行动必须依赖于 \(\theta\)。 否则接收者总会采取同一个最有利的行动，沟通便与他无关。3. 偏好的共同性。 发送者与接收者必须有某种共同"喜好"；这保证发送者有激励去通知接收者以某种使双方都受益的方式回应。1. The sender's preferences over actions \(a\in A\) must depend on \(\theta\). Different sender types must have different preferences over the receiver's actions; otherwise all senders of different types would send the same most-favorable message, making that message meaningless. 2. The receiver's optimal action must depend on \(\theta\). Otherwise the receiver would always take the same most-favorable action and communication is irrelevant to him. 3. Commonality of preferences. Senders and receivers must have something they "like" in common; this guarantees the sender has incentives to notify the receiver to respond in a certain way that benefits both sides.
14.2 例子 / Example
Alice 拥有一家公司、雇员为 Bob。Alice 可把两项任务之一分配给 Bob，而 Bob 是两种类型之一，这是 Bob 事前的私人信息。支付共同取决于 Bob 的类型与所分配的任务，支付矩阵如下；Bob 可以说一个词。
14.2 Example
Alice owns a company with worker Bob. Alice can assign Bob one of two tasks, and Bob is one of two types, which is ex-ante private information for Bob. The payoff jointly depends on Bob's type and the assigned task, with the payoff matrix below; Bob gets to say one word.
表 1 / Table 1：支付矩阵 / Payoff Matrix
| Nature | Bob \ Alice | Hard | Easy |
|---|---|---|---|
| \(\tfrac12\) | \(\theta_{smart}\) | $(2,2)$ | $(0,0)$ |
| \(\tfrac12\) | \(\theta_{dull}\) | $(0,0)$ | $(1,1)$ |
两类均衡 / Two sets of equilibria 1. 胡言均衡（babbling）：Bob 胡言乱语。Alice 知道 Bob 在胡言，于是不听；Bob 之所以胡言，是因为 Alice 不在乎。于是 Alice 碰运气，选 Hard——在 \((\tfrac12,\tfrac12)\) 的概率下面对 $(2,2)$ 与 $(0,0)$ 的支付。2. 有信息均衡：Bob 说 Smart 或 Dull，Alice 正确解读并把他分到正确的任务。事实上存在许多这样的均衡，因为只要 Alice 理解类型与词语之间的映射，Bob 说什么都可以。1. Babbling equilibrium: Bob is babbling. Alice knows Bob is babbling and so doesn't listen; Bob babbles because Alice doesn't care. So Alice takes her chances and chooses Hard — facing payoffs $(2,2)$ and $(0,0)$ with probabilities \((\tfrac12,\tfrac12)\). 2. Informative equilibrium: Bob says Smart or Dull and Alice correctly interprets it and assigns him to the correct task. In fact there are many such equilibria, as Bob can say anything as long as Alice understands the mapping between types and words.
若上述三个条件中任一被违反，便不会有有意义的沟通。考虑以下两种情形。
情形 1：接收者的最优行动不依赖于 \(\theta\)。 见表 2。此时 Alice 总会把 Hard 任务分给 Bob，沟通毫无作用。
If any of the three conditions above is violated, there is no meaningful communication. Consider the following two cases.
Case 1: the receiver's optimal action doesn't depend on \(\theta\). See Table 2. Here Alice would always assign the Hard question to Bob, and communication plays no role.
表 2 / Table 2：与有意义沟通不相容 / Inconsistent with Meaningful Communication
| Nature | Bob \ Alice | Hard | Easy |
|---|---|---|---|
| \(\tfrac12\) | \(\theta_{smart}\) | $(1,2)$ | $(1,0)$ |
| \(\tfrac12\) | \(\theta_{dull}\) | $(0,2)$ | $(1,1)$ |
情形 2：Bob 有激励歪曲信息，因为其偏好与 Alice 相反。 这违反偏好的共同性，见表 3。Alice 知道 Bob 想要的总是与自己的利益相悖，于是认为 Bob 的话是有意歪曲，干脆无视他的话。因此，要让有意义的信息传递在均衡中出现，发送者与接收者的偏好必须部分一致。
Case 2: Bob has an incentive to misrepresent because his preferences are opposite to Alice's. This violates the commonality of preferences; see Table 3. Alice knows that what Bob wants is always against her benefit, so she thinks Bob's talking is misrepresenting on purpose and simply ignores his words. Thus, for meaningful information transmission to arise in equilibrium, the preferences of the sender and receiver must be partially aligned.
表 3 / Table 3：与有意义沟通不相容 / Inconsistent with Meaningful Communication
| Nature | Bob \ Alice | Hard | Easy |
|---|---|---|---|
| \(\tfrac12\) | \(\theta_{smart}\) | $(0,2)$ | $(1,0)$ |
| \(\tfrac12\) | \(\theta_{dull}\) | $(2,0)$ | $(2,1)$ |
14.3 Crawford and Sobel (1982)：废话博弈 / Cheap talk
14.3.1 设定 / Set-up
14.3 Crawford and Sobel (1982): Cheap talk
14.3.1 Set-up
设定 / Set-up 1. \(\theta\sim[0,1]\)（不失一般性）。2. 消息空间 \(M=[0,1]\)。3. \(u^S(\theta,a)\) 与 \(u^R(\theta,a)\) 二次连续可微，且关于 \(a\) 严格凹并递增，\(a\in\mathbb{R}\)。4. 假设唯一最优：发送者唯一最优在 \(a^S(\theta)\)、接收者唯一最优在 \(a^R(\theta)\)；共同性部分为 \(u^S_{a\theta}>0\)、\(u^R_{a\theta}>0\)，且两效用函数都关于 \(\theta\) 递增。5. 发送者总想要比接收者更高的行动，即 \(a^S(\theta)>a^R(\theta)\)。1. \(\theta\sim[0,1]\) (WLOG). 2. Message space \(M=[0,1]\). 3. \(u^S(\theta,a)\) and \(u^R(\theta,a)\) are twice continuously differentiable, and strictly concave and increasing in \(a\), \(a\in\mathbb{R}\). 4. Assume a unique optimum: the sender has a unique optimum at \(a^S(\theta)\) and the receiver at \(a^R(\theta)\); the commonality part is \(u^S_{a\theta}>0\) and \(u^R_{a\theta}>0\), and both utility functions are increasing in \(\theta\). 5. The sender always wants a higher action than the receiver, i.e. \(a^S(\theta)>a^R(\theta)\).
14.3.2 序贯均衡条件 / Sequential equilibrium condition
存在多个可能的均衡。下面描述一般的 \(n\) 步均衡。在 \(n\) 步均衡中，存在 \(n\) 步分区 \(X_0,\dots,X_n\) 使得 \(\{[0,X_1],[X_1,X_2],\dots,[X_{n-1},1]\}\)，且发送者策略与信念为下面各式（发送者在同一区段内随机发送区段内的消息）：
14.3.2 Sequential equilibrium condition
There are multiple possible equilibria. The general \(n\)-step equilibrium is described below. In an \(n\)-step equilibrium, there exists an \(n\)-step partition \(X_0,\dots,X_n\) such that \(\{[0,X_1],[X_1,X_2],\dots,[X_{n-1},1]\}\), and the sender's strategy and beliefs are given below (the sender randomizes among messages within the same segment):
$$ \sigma^S(m\mid\theta)=\begin{cases} 0 & \text{if } \theta\in[X_i,X_{i+1}],\ m\notin[X_i,X_{i+1}]\\[2pt] >0 & \text{if } \theta\in[X_i,X_{i+1}],\ m\in[X_i,X_{i+1}]\end{cases} $$
$$ \sigma^S\big(m\in[X_i,X_{i+1}]\mid\theta\big)=1\quad\text{if } \theta\in[X_i,X_{i+1}] $$
$$ \mu\big(\theta\mid m\in[X_i,X_{i+1}]\big)=\begin{cases} \dfrac{p(\theta)}{p([X_i,X_{i+1}])} & \theta\in[X_i,X_{i+1}]\\[6pt] 0 & \text{otherwise}\end{cases} $$
记 \(\bar a^R\) 为最优反应，则接收者在每个区段上按后验信念最大化期望效用：
Denote \(\bar a^R\) as the best response; the receiver maximizes expected utility over each segment under the posterior belief:
$$ \bar a^R\big([X_i,X_{i+1}]\big)=\arg\max_a \int_{X_i}^{X_{i+1}} u^R(\theta,a)\,\mu\big(\theta\mid m\in[X_i,X_{i+1}]\big)\,d\theta $$
而边界类型 \(\theta=X_i\) 在相邻两个行动之间无差异：
and the boundary sender type \(\theta=X_i\) is indifferent between the adjacent actions at each \(X_i\):
$$ u^S\big(X_i,\,\bar a^R([X_{i-1},X_i])\big)=u^S\big(X_i,\,\bar a^R([X_i,X_{i+1}])\big) $$
注 14.1 / Remark 14.1 这些条件确实描述了序贯均衡之一：发送者愿意按此模式发送消息，接收者据此解读并最优反应，而该最优行动又使发送者愿意继续按此模式发送。永远存在一步均衡（\(n=1\)）：发送者在整个消息空间上随机，接收者只按整个类型空间的期望行动；故这个一步均衡就是无任何有意义信息传递的胡言均衡。在更高的非平凡均衡中，问题出在发送者想要的比接收者愿意给的更多，即 \(a^S(\theta)>a^R(\theta)\)；因此对给定的具体 \(a^S(\theta)>a^R(\theta)\)，均衡的步数存在一个上限。于是信息无法被传到完全揭示类型（\(n=\infty\)）的程度，除非 \(a^S(\theta)=a^R(\theta)\)。下面用二次例子说明。These conditions indeed describe one of the sequential equilibria: senders like to send messages in this pattern, the receiver interprets and reacts optimally, and that optimal action makes the sender like to continue sending in this pattern. There always exists a one-step equilibrium (\(n=1\)): the sender randomizes among the whole message space and the receiver just acts according to the expectation of the whole type space; so this one-step equilibrium is the babbling equilibrium with no meaningful information conveyed. In a higher non-trivial equilibrium, the problem arises because senders want more than receivers want to give, i.e. \(a^S(\theta)>a^R(\theta)\); so there is a maximum number of steps in equilibrium given each specific \(a^S(\theta)>a^R(\theta)\). Hence information cannot be sent to perfectly reveal the types (\(n=\infty\)) unless \(a^S(\theta)=a^R(\theta)\). See this in the following quadratic example.
14.3.3 二次偏好的简单废话博弈 / A simple cheap talk game with quadratic preferences
考虑如下设定，其中 \(b\) 是发送者与接收者之间的偏差：
14.3.3 A simple cheap talk game with quadratic preferences
Consider the following set-up, where \(b\) is the bias between the sender and the receiver:
$$ \theta\sim u[0,1] $$
$$ u^R(\theta,a)=-\tfrac12(a-\theta)^2 \;\Rightarrow\; a^R(\theta)=\theta $$
$$ u^S(\theta,a)=-\tfrac12(a-\theta-b)^2 \;\Rightarrow\; a^S(\theta)=\theta+b,\ \ b>0 $$
当 \(b>0\) 时发送者总偏好比接收者更高的行动。我们刻画均衡分区，并验证给定分区是否能生成博弈的预测对局。先找两步均衡，即
For \(b>0\) the sender always prefers a higher action than the receiver. We characterize the equilibrium partition and verify that a given partition generates the predicted play of the game. First we look for a two-step equilibrium, i.e.
$$ m\in\begin{cases} [0,X_1] & \text{when } \theta\in[0,X_1)\\[2pt] [X_1,1] & \text{when } \theta\in[X_1,1]\end{cases} $$
在均衡中，后验信念必为下面两式：
In equilibrium, the posterior beliefs must be the two expressions below:
$$ \mu\big(\theta\mid m\in[0,X_1]\big)=\begin{cases} \dfrac{1}{X_1} & \theta\in[0,X_1)\\[6pt] 0 & \theta\in[X_1,1]\end{cases} $$
$$ \mu\big(\theta\mid m\in[X_1,1]\big)=\begin{cases} \dfrac{1}{1-X_1} & \theta\in[X_1,1]\\[6pt] 0 & \theta\in[0,X_1)\end{cases} $$
与这些信念相一致，接收者最优选择为：
Consistent with these beliefs, the receiver optimally chooses:
$$ \bar a^R([0,X_1])=\frac{X_1}{2},\qquad \bar a^R([X_1,1])=\frac{X_1+1}{2} $$
接下来选 \(X_1\) 使发送者在 \(\theta=X_1\) 处对两条消息无差异，于是有：
Now we choose \(X_1\) such that the sender is indifferent between the two messages at \(\theta=X_1\), giving:
$$ u^S\big(X_1,\bar a^R([0,X_1])\big)=u^S\big(X_1,\bar a^R([X_1,1])\big) $$
$$ -\tfrac12\left(\frac{X_1}{2}-X_1-b\right)^2=-\tfrac12\left(\frac{X_1+1}{2}-X_1-b\right)^2 $$
$$ \Rightarrow\ X_1=\frac12-2b \;\Rightarrow\; \frac12-2b>0 \;\Rightarrow\; b<\frac14 $$
结论 / Conclusion 对任意 \(b\) 值都有一步（胡言）均衡；但要存在两步均衡，需 \(b<\tfrac14\)。即有意义的沟通要求双方偏好之差不能太大。For any value of \(b\) there is a one-step (babbling) equilibrium; but for a two-step equilibrium we need \(b<\tfrac14\). That is, meaningful communication requires that the difference in preferences not be very big.
14.3.4 废话模型的应用 / Applications of cheap talk models 1. Dessein (2002) 提出组织内沟通的框架：接收者是企业所有者、发送者是经理；接收者可选择是否授权。视偏好共同性（即偏差 \(b\) 的大小），授权程度更高或更低。2. Farrell–Gibbons (1989) 在 Chatterjee–Samuelson (CS) 讨价还价博弈前考虑一轮废话；废话可以起作用，某些买卖双方更偏好"先进行一轮有信息的两步废话、再玩 CS 博弈"，而非直接玩 CS 博弈。1. Dessein (2002) presents a framework for communication in organizations: the receiver is the owner of a firm and the sender is the manager; the receiver can choose to delegate or not. Depending on the commonality of preferences (the magnitude of \(b\)), this results in greater or less delegation. 2. Farrell–Gibbons (1989) considers a round of cheap talk before the Chatterjee–Samuelson (CS) bargaining game; cheap talk can matter, and some types of buyers and sellers prefer an informative two-step round of cheap talk followed by the CS game to simply playing the CS game.
参考文献 / References
- Crawford, V. P., & Sobel, J. (1982). Strategic Information Transmission.（废话博弈的开创模型）
- Dessein, W. (2002). Authority and Communication in Organizations.
- Farrell, J., & Gibbons, R. (1989). Cheap Talk Can Matter in Bargaining.
References
- Crawford, V. P., & Sobel, J. (1982). Strategic Information Transmission. (the foundational cheap-talk model)
- Dessein, W. (2002). Authority and Communication in Organizations.
- Farrell, J., & Gibbons, R. (1989). Cheap Talk Can Matter in Bargaining.