15. Signal-Jamming Games
15. Signal-Jamming Games
本章导读 前几章的信息揭示博弈里，掌握信息的一方先行动并设法揭示信息；本章考虑另一类——信号干扰（signal-jamming）。该名称由 Fudenberg–Tirole (1986) 在掠夺性定价的论文中提出。基本思路：自然生成关于世界状态的带噪信号，一方可利用这些噪声"干扰信号"，使另一方无法从观察中辨清、从而做出对前者有利的误判；更一般地，信号干扰模型指一方选择行动来操纵另一方的推断，且干扰方未必拥有更多信息。§15.1 Holmstrom (1982) 职业关切：风险中性经理私下知道自身类型 \(\theta=\mu_\theta+\eta\)，企业仅有先验、需在观察到第一期产出 \(X_1=\theta+a_1+\varepsilon_1\) 后用贝叶斯法则更新；经理因第二期工资 \(w_2=\mathbb{E}[X_2\mid X_1]=\mathbb{E}[\theta\mid X_1]\) 而有激励在第一期付出努力以"抬高"市场对其类型的推断，一阶条件 \(\psi'(a_1^\star)=\delta(1-\lambda)=\delta\sigma_\eta^2/(\sigma_\varepsilon^2+\sigma_\eta^2)\) 钉死最优努力；序贯均衡中无人被骗，但努力仍正。\(T=\infty\) 时类型终被完全识别、努力随之下降。无图。
15. Signal-Jamming Games
Overview In the previous information-revelation games, the informed player moves first and tries to reveal information; this chapter considers another type — signal-jamming. The name was invented by Fudenberg–Tirole (1986) in their paper on predation. The basic idea: nature generates noisy signals about the state of the world, and one side of players may take advantage of those noises ("jam these signals") so that the other side cannot tell from observations and misinterprets the signals in a way favorable to the first side; more generally, signal-jamming models involve one player choosing actions to manipulate the inferences of another, and the jamming player need not have more information. §15.1 Holmstrom (1982) Career Concerns: a risk-neutral manager privately knows his type \(\theta=\mu_\theta+\eta\), while firms have only a prior to be updated by Bayes' rule after observing the first-period output \(X_1=\theta+a_1+\varepsilon_1\); because the second-period wage is \(w_2=\mathbb{E}[X_2\mid X_1]=\mathbb{E}[\theta\mid X_1]\), the manager has an incentive to exert first-period effort to "raise" the market's inference about his type, with the first-order condition \(\psi'(a_1^\star)=\delta(1-\lambda)=\delta\sigma_\eta^2/(\sigma_\varepsilon^2+\sigma_\eta^2)\) pinning down the optimal effort; in the sequential equilibrium no one is fooled, yet effort is still positive. With \(T=\infty\) the type is eventually fully identified and effort declines. No figures.
在前几节的信息揭示博弈中，掌握信息的一方先行动并设法把信息揭示给无信息的一方。本节考虑另一类信息揭示博弈——信号干扰博弈。这一名称由 Fudenberg 与 Tirole (1986) 在关于掠夺行为的论文中提出。
基本思路是：自然生成关于世界状态的带噪信号，一方玩家可能利用这些噪声（"干扰这些信号"），使另一方无法从观察中辨清，并以对第一方有利的方式误读信号。更一般地，信号干扰模型涉及一方选择行动来操纵另一方的推断；注意，与此前的信号模型不同，干扰信号的一方未必拥有更多信息。考虑下面的例子：经理有激励改变自己的行动以操纵市场结果，从而影响企业对其类型的推断，以最大化自身支付。
15.1 Holmstrom (1982)：职业关切 / Career concerns
一位风险中性的经理拥有关于自身类型的私人信息。他从企业获得的报酬是其类型与努力的函数。经理希望付出高努力以说服市场相信他的类型 \(\theta\) 较高，从而赢得更高工资。
15.1.1 设定 / Set-up
In the previous sections on information-revelation games, the informed player moves first and tries to reveal the information to the uninformed player. This section considers another type of information-revelation game — signal-jamming games. The name was invented by Fudenberg and Tirole (1986) in their paper about predation.
The basic idea is that noisy signals about the state of the world are generated by nature, and one side of players might take advantage of those noises ("jam these signals") so that the other side cannot tell from observations and misinterprets the signals in a way favorable to the first side. More generally, signal-jamming models involve one player choosing actions to manipulate the inferences of another; note that, unlike the earlier signaling models, the player who jams the signal need not have more information. Consider the following example: managers have an incentive to change their actions to manipulate the market outcome and thus affect the firm's inference about their type, so as to maximize their payoff.
15.1 Holmstrom (1982): Career concerns
A risk-neutral manager has private information about his type. His compensation from the firm is a function of his type and effort. The manager wants to exert high effort to convince the market that his type \(\theta\) is high, in order to garner a higher wage.
15.1.1 Set-up
设定与时序 / Set-up and timing 经理的类型为 \(\theta\in\Theta\)，其中 \(\theta=\mu_\theta+\eta\)、\(\eta\sim\mathcal{N}(0,\sigma_\eta^2)\)。即经理类型 \(\theta\) 是经理自己的私人信息，对企业而言只是一个先验分布，需在观察到第一期市场产出后更新。只有两期 \(T=1,T=2\)。时序：\(T=1\) 经理被雇佣，选择不可观测的行动 \(a_1\)、成本 \(\psi(a_1)\)；市场观察到产出 \(X_1=\theta+a_1+\varepsilon_1\)，\(\varepsilon_1\sim\mathcal{N}(0,\sigma_\varepsilon^2)\)。\(T=2\) 市场给出工资 \(w_2=\mathbb{E}[X_2\mid X_1]\)。关键在于：\(\mathbb{E}[X_2\mid X_1]=\mathbb{E}[\theta+a_2+\varepsilon_2\mid X_1]\)，而经理将在第二期后退休，故第二期无激励付出努力 \(a_2^\star=0\)，且 \(\mathbb{E}[\varepsilon_2]=0\)，于是 \(\mathbb{E}[X_2\mid X_1]=\mathbb{E}[\theta\mid X_1]\)。The manager is of type \(\theta\in\Theta\), where \(\theta=\mu_\theta+\eta\) with \(\eta\sim\mathcal{N}(0,\sigma_\eta^2)\). That is, the manager's type \(\theta\) is privately known to the manager himself, but is a prior distribution to the firms that needs to be updated after observing the first-period market output. There are only two periods \(T=1,T=2\). Timing: at \(T=1\) the manager is employed and chooses an unobserved action \(a_1\) at cost \(\psi(a_1)\); the market observes output \(X_1=\theta+a_1+\varepsilon_1\) with \(\varepsilon_1\sim\mathcal{N}(0,\sigma_\varepsilon^2)\). At \(T=2\) the market offers the wage \(w_2=\mathbb{E}[X_2\mid X_1]\). The key feature: \(\mathbb{E}[X_2\mid X_1]=\mathbb{E}[\theta+a_2+\varepsilon_2\mid X_1]\), and because the manager retires after the second period he has no incentive to exert effort in period two, so \(a_2^\star=0\), and \(\mathbb{E}[\varepsilon_2]=0\), hence \(\mathbb{E}[X_2\mid X_1]=\mathbb{E}[\theta\mid X_1]\).
设 \(a_1^\star\) 是经理在第一期想付出的最优努力，则有下式（\(\theta=\mu_\theta+\eta\)）：
Let \(a_1^\star\) be the optimal effort the manager wants to make in period 1; then we have the following (\(\theta=\mu_\theta+\eta\)):
$$ X_1=\theta+a_1^\star+\varepsilon_1,\qquad \theta=(X_1-a_1^\star)-\varepsilon_1,\qquad \theta=\mu_\theta+\eta $$
于是经理类型的条件期望由下式给出，其中 \(\lambda=\dfrac{\sigma_\varepsilon^2}{\sigma_\varepsilon^2+\sigma_\eta^2}\)。该结果来自贝叶斯法则（证明见 Econometrics Notes）：
So the conditional expectation of the manager's type is given below, where \(\lambda=\dfrac{\sigma_\varepsilon^2}{\sigma_\varepsilon^2+\sigma_\eta^2}\). This result comes from Bayes' rule (see the proof in the Econometrics Notes):
$$ \mathbb{E}[\theta\mid X_1]=\lambda\mu_\theta+(1-\lambda)(X_1-a_1^\star) $$
15.1.2 均衡 / Equilibrium
考虑经理在第一期的决策。设效用贴现因子 \(\delta<1\)，则总净贴现支付为下式（已展开标出努力的成本与收益）：
15.1.2 Equilibrium
Consider the manager's decision in period 1. Let the utility discounting factor be \(\delta<1\); then the total net discounted payoff is given below (expanded to mark the cost and benefit of effort):
$$ \begin{aligned} \text{Total net discounted payoff} &= -\psi(a_1)+\delta\big(\lambda\mu_\theta+(1-\lambda)(X_1-a_1^\star)\big)\\[2pt] &= \underbrace{-\psi(a_1)}_{\text{cost of effort}}+\delta(\lambda\mu_\theta)+\delta(1-\lambda)\Big(\theta+\underbrace{a_1}_{\text{benefit of effort}}+\varepsilon_1-a_1^\star\Big) \end{aligned} $$
由一阶条件给出最优性条件，从而钉死第一期的最优努力 \(a_1^\star\)：
The first-order condition gives the optimality condition, which pins down the optimal effort \(a_1^\star\) in period 1:
$$ \psi'(a_1^\star)=\delta(1-\lambda)=\frac{\delta\sigma_\eta^2}{\sigma_\varepsilon^2+\sigma_\eta^2} $$
序贯均衡中无人被骗 / No one is fooled in the sequential equilibrium 在序贯均衡中，没有人会被欺骗——企业也能做这同样的计算。但工人（经理）仍会选择某一努力水平，使提高未来工资的边际收益等于努力的边际成本。In the sequential equilibrium, no one is fooled — the firm can also do this same calculation. But the worker (manager) will still choose an effort level such that the marginal benefit of increasing future wages equals the marginal cost of effort.
15.1.3 推广 / Generalizations 在 \(T=\infty\) 的情形下，市场与经理会随时间完全学到主体的类型。因此在早期阶段经理会付出努力，但随着其类型被逐渐识别清楚，他会付出更低的努力——因为此时他将按其被精确识别的类型获得报酬。In the \(T=\infty\) case, the market and the manager will perfectly learn the agent's type over time. Therefore, in early periods the manager will exert effort, but as his type is resolved he will exert lower effort, as he is compensated according to his exactly identified type.
参考文献 / References
- Holmstrom, B. (1982). Managerial Incentive Problems: A Dynamic Perspective.（职业关切模型）
- Fudenberg, D., & Tirole, J. (1986). A "Signal-Jamming" Theory of Predation.（信号干扰名称的出处）
References
- Holmstrom, B. (1982). Managerial Incentive Problems: A Dynamic Perspective. (the career-concerns model)
- Fudenberg, D., & Tirole, J. (1986). A "Signal-Jamming" Theory of Predation. (origin of the signal-jamming name)