11. Brief Review of Game Theory

Jun He June 01, 2026

中文English 博弈论回顾Game Theory Review 贝叶斯纳什均衡Bayes Nash Equilibrium 完美贝叶斯均衡Perfect Bayesian Equilibrium 序贯均衡Sequential Equilibrium 一价拍卖First Price Auction

11. Brief Review of Game Theory

Note

Part III 导读 / Part III overview Part III — Price Theory III 把 Ch 10 的博弈论工具应用于信息经济学，分若干组：本章（Ch 11）回顾不完全信息博弈并统一记号；随后逆向选择组（Ch 12 Adverse Selection, 13 Signaling, 14 Cheap Talk, 15 Signal-Jamming, 16 Competitive Screening）；道德风险组（Ch 17 Hidden Action, 18 Innes 1990, 19 Holmstrom-Milgrom 1987）；机制设计组（Ch 20 Monopolistic Screening, 21 Baron-Myerson, 22 Random Mechanism, 23 Dynamic Screening）；拍卖组（Ch 24 Four Standard Auctions, 25 Revenue Equivalence, 26 All-Pay, 27 Optimal Auction）；双边贸易组（Ch 28-29）；社会有效机制设计组（Ch 30）。

Note

本章导读本章简要回顾上一部分（Ch 10）不完全信息博弈的基础，并把此前记号映射到本部分将用的记号。§11.1 静态不完全信息博弈（＝策略型不完全信息博弈）：设定、一价拍卖 FPA 例、纯/混合（实为行为）策略、贝叶斯纳什均衡 BNE。§11.2 动态不完全信息博弈（＝扩展型）：进入博弈例、弱序贯均衡（＝弱 PBE）、完美贝叶斯均衡 PBE（额外三条件）、序贯均衡（序贯理性＋一致性，强于 PBE）。

11. Brief Review of Game Theory

Note

Part III overview Part III — Price Theory III applies the game theory of Ch 10 to information economics, in several groups: this chapter (Ch 11) reviews incomplete-information games and unifies notation; then Adverse Selection (Ch 12 Adverse Selection, 13 Signaling, 14 Cheap Talk, 15 Signal-Jamming, 16 Competitive Screening); Moral Hazard (Ch 17 Hidden Action, 18 Innes 1990, 19 Holmstrom-Milgrom 1987); Mechanism Design (Ch 20 Monopolistic Screening, 21 Baron-Myerson, 22 Random Mechanism, 23 Dynamic Screening); Auctions (Ch 24 Four Standard Auctions, 25 Revenue Equivalence, 26 All-Pay, 27 Optimal Auction); Bilateral Trade (Ch 28-29); Socially Efficient Mechanism Design (Ch 30).

Note

Overview This chapter briefly reviews the basics of incomplete-information games from the previous part (Ch 10) and maps the earlier notation to the one used here. §11.1 static incomplete-information games (= strategic-form incomplete-info games): set-up, the first price auction (FPA) example, pure/mixed (really behavioral) strategy, Bayes Nash equilibrium (BNE). §11.2 dynamic incomplete-information games (= extensive form): an entry-game example, weak sequential equilibrium (= weak PBE), perfect Bayesian equilibrium (PBE) (three extra conditions), sequential equilibrium (sequential rationality + consistency, stronger than PBE).

11.1 Static games of incomplete information

此处静态不完全信息博弈等价于上部分的策略型不完全信息博弈。

11.1.1 Set-up

玩家集 $N=\{1,2,\dots,n\}$，任一玩家 $i\in N$。
类型剖面 $\theta=(\theta_1,\theta_2,\dots,\theta_n)\in\Theta$：$\theta_i$ 为玩家 $i$ 的类型，$\Theta_i$ 为其类型空间，$\Theta=\prod_{i=1}^n\Theta_i$ 为类型剖面空间。
类型结构：类型空间的分布结构（连续或离散）——$F$ 为 c.d.f.、$\phi$ 为 p.d.f.。
行动空间：每个玩家 $i\in N$ 的行动空间 $S_i$，任一可用行动 $s_i\in S_i$。
收益：每个玩家 $i\in N$ 的收益 $u_i(s_i,s_{-i};\theta)$。

于是（一般的）扩展型博弈 $\Gamma$ 定义为 $\Gamma=\{N;S_1,\dots,S_n;u_1,\dots,u_n;\Theta;F\}$。

11.1.2 Example of static incomplete information game: first price auction (FPA)

简单一价拍卖：所有竞买人同时各报一价，对自己类型有私人信息（保留价 $\theta_i$ 由自然私下告知各竞买人），拍卖人卖给出价最高者。两竞买人的简单情形：

玩家集 $N=\{1,2\}$。
类型（保留价 $\theta_i$）剖面 $\theta=(\theta_1,\theta_2)\in\Theta$，$\theta_i\in\Theta_i=\mathbb{R}_+$，$\Theta=\mathbb{R}_+^2$。
类型结构：c.d.f. 为 $F$。
行动（出价 $b_i=s_i(\theta_i)$）空间 $S_i=\mathbb{R}_+$。
收益（$i=1,2$）：

11.1 Static games of incomplete information

The static incomplete-information game here is equivalent to the strategic-form incomplete-info game of the previous part.

11.1.1 Set-up

Set of players $N=\{1,2,\dots,n\}$, any player $i\in N$.
Type profile $\theta=(\theta_1,\theta_2,\dots,\theta_n)\in\Theta$: $\theta_i$ is player $i$'s type, $\Theta_i$ its type space, $\Theta=\prod_{i=1}^n\Theta_i$ the space of type profiles.
Type structure: the distribution structure of the type space (continuous or discrete) — $F$ the c.d.f., $\phi$ the p.d.f.
Action space: each player $i\in N$ has action space $S_i$, any available action $s_i\in S_i$.
Payoff: each player $i\in N$ has payoff $u_i(s_i,s_{-i};\theta)$.

So the (generally) extensive form game $\Gamma$ is defined by $\Gamma=\{N;S_1,\dots,S_n;u_1,\dots,u_n;\Theta;F\}$.

11.1.2 Example of static incomplete information game: first price auction (FPA)

A simple first price auction: all bidders simultaneously bid a price, each with private information about its own type (reservation price $\theta_i$ revealed privately by nature), and the auctioneer sells to the highest bidder. The simple two-bidder case:

Players $N=\{1,2\}$.
Type (reservation price $\theta_i$) profile $\theta=(\theta_1,\theta_2)\in\Theta$, $\theta_i\in\Theta_i=\mathbb{R}_+$, $\Theta=\mathbb{R}_+^2$.
Type structure: c.d.f. $F$.
Action (bid $b_i=s_i(\theta_i)$) space $S_i=\mathbb{R}_+$.
Payoff ($i=1,2$):

$$u_i(s_i,s_j;\theta)=\begin{cases}\theta_i-b_i&b_i>b_j,\ i\ne j\\0&b_i

FPA 作为扩展型博弈 $\Gamma=\{\{1,2\};\mathbb{R}_+,\mathbb{R}_+;u_1,u_2;\mathbb{R}_+^2;F\}$。

11.1.3 Pure strategy and mixed strategy

纯策略 $s_i:\Theta_i=\mathbb{R}_+\to S_i=\mathbb{R}_+$。"混合策略"（实为与前文一致的行为策略）$\sigma_i:\Theta_i\times S_i\to[0,1]$，对每个纯策略按类型实现赋条件概率；对 $\theta_i\in\Theta_i$，$\sigma_i(\cdot|\theta_i)\in\Delta(S_i)$。

11.1.4 Bayesian Nash equilibrium (BNE)

BNE 定义为 $\sigma^\star=(\sigma_1^\star,\dots,\sigma_n^\star)$ 使 $\forall i\in N$、$\forall\theta\in\Theta$ 若 $\sigma_i^\star(s_i|\theta_i)>0$ 则 $\forall\hat s_i\in S_i$：

FPA as an extensive form game $\Gamma=\{\{1,2\};\mathbb{R}_+,\mathbb{R}_+;u_1,u_2;\mathbb{R}_+^2;F\}$.

11.1.3 Pure strategy and mixed strategy

Pure strategy $s_i:\Theta_i=\mathbb{R}_+\to S_i=\mathbb{R}_+$. "Mixed strategy" (really a behavioral strategy consistent with earlier notions) $\sigma_i:\Theta_i\times S_i\to[0,1]$, assigning a conditional probability to each pure strategy given the type realization; for $\theta_i\in\Theta_i$, $\sigma_i(\cdot|\theta_i)\in\Delta(S_i)$.

11.1.4 Bayesian Nash equilibrium (BNE)

A BNE is $\sigma^\star=(\sigma_1^\star,\dots,\sigma_n^\star)$ such that $\forall i\in N$, $\forall\theta\in\Theta$ if $\sigma_i^\star(s_i|\theta_i)>0$ then $\forall\hat s_i\in S_i$:

$$\mathbb{E}_{\theta_{-i}}\left[u_i(s_i,\sigma^\star_{-i};\theta)|\theta_i\right]\ge\mathbb{E}_{\theta_{-i}}\left[u_i(\hat s_i,\sigma^\star_{-i};\theta)|\theta_i\right]$$

其中 $\mathbb{E}_{\theta_{-i}}\left[u_i(s_i,\sigma^\star_{-i};\theta)|\theta_i\right]=\sum_{\theta\in\Theta_{-i}}\sum_{s_{-i}\in S_{-i}}u_i(s_i,\sigma^\star_{-i};\theta)\cdot\sigma^\star_{-i}(s_{-i}|\theta_{-i})\cdot\phi(\theta_{-i})$。

11.2 Dynamic games of incomplete information

此处动态不完全信息博弈等价于上部分的扩展型不完全信息博弈。

11.2.1 Motivating example

博弈树（已转述）：两玩家。进入者（玩家 1）可选 Out 离开市场（$\to(0,2)$），或进入并选 Fight($F$)/Accommodate($A$)；在位者（玩家 2）在进入者进入时选 $F/A$，但不知进入者选了 $F$ 还是 $A$（信息集 $\{2a,2b\}$）。收益：$F\to2a$、$A\to2b$；$2a$：$F\to(-3,1)$、$A\to(1,-2)$；$2b$：$F\to(-2,-1)$、$A\to(3,1)$。该动态博弈可写成策略型博弈的收益矩阵：

where $\mathbb{E}_{\theta_{-i}}\left[u_i(s_i,\sigma^\star_{-i};\theta)|\theta_i\right]=\sum_{\theta\in\Theta_{-i}}\sum_{s_{-i}\in S_{-i}}u_i(s_i,\sigma^\star_{-i};\theta)\cdot\sigma^\star_{-i}(s_{-i}|\theta_{-i})\cdot\phi(\theta_{-i})$.

11.2 Dynamic games of incomplete information

The dynamic incomplete-information game here is equivalent to the extensive form incomplete-info game of the previous part.

11.2.1 Motivating example

Game tree (paraphrased): two players. The entrant (player 1) chooses Out to leave the market ($\to(0,2)$), or enters and chooses Fight ($F$)/Accommodate ($A$); the incumbent (player 2) chooses $F/A$ when the entrant enters, but doesn't know whether the entrant chose $F$ or $A$ (info set $\{2a,2b\}$). Payoffs: $F\to2a$, $A\to2b$; $2a$: $F\to(-3,1)$, $A\to(1,-2)$; $2b$: $F\to(-2,-1)$, $A\to(3,1)$. This dynamic game can be written as a strategic-form payoff matrix:

$E\backslash I$	$F$	$A$
$Out$	$(0,2)$	$(0,2)$
$F$	$(-3,1)$	$(1,-2)$
$A$	$(-2,-1)$	$(3,1)$

显然 $(Out,F)$ 与 $(A,A)$ 是两个纯策略纳什均衡（蓝箭头）。但第一个均衡 $(Out,F)$ 不合理——进入者不应无论如何都选 Out。为排除 $(Out,F)$，需引入其他概念。

11.2.2 Weak sequential equilibrium

Clearly $(Out,F)$ and $(A,A)$ are two pure strategy Nash equilibria (blue arrows). But the first, $(Out,F)$, is unreasonable — the entrant should not always choose Out no matter what. To rule out $(Out,F)$, we need other notions.

11.2.2 Weak sequential equilibrium

Important

Definition 11.1（Weak sequential equilibrium）对信念系统 $\mu$ 与行为策略剖面 $\sigma$，评估 $(\mu,\sigma)$ 是弱序贯均衡，当且仅当：1. $(\mu,\sigma)$ 满足序贯理性（$\sigma$ 关于 $\mu$ 序贯理性）；2. 对任一信息集 $h$ 使到达概率 $\mathbb{P}(h|\sigma)>0$，$\mu$ 满足贝叶斯法则，即 $\forall x\in h$，$\mu(x)=\dfrac{\mathbb{P}(x|\sigma)}{\mathbb{P}(h|\sigma)}$。弱序贯均衡等价于弱完美贝叶斯均衡（弱 PBE）。然而它不足以排除动机例中的 $(Out,F)$。For a system of beliefs $\mu$ and behavioral strategy profile $\sigma$, the assessment $(\mu,\sigma)$ is a weak sequential equilibrium iff: 1. $(\mu,\sigma)$ is sequentially rational ($\sigma$ sequentially rational w.r.t. $\mu$); 2. for any info set $h$ with reaching probability $\mathbb{P}(h|\sigma)>0$, $\mu$ satisfies Bayes' rule, i.e. $\forall x\in h$, $\mu(x)=\dfrac{\mathbb{P}(x|\sigma)}{\mathbb{P}(h|\sigma)}$. A weak sequential equilibrium is equivalent to a weak perfect Bayesian equilibrium (weak PBE). However, it is not enough to rule out $(Out,F)$ in the motivating example.

11.2.3 Perfect Bayesian equilibrium (PBE)

要使评估为 PBE，需在弱 PBE／弱序贯均衡之上加三个条件。PBE 满足：

$(\mu,\sigma)$ 序贯理性（$\sigma$ 关于 $\mu$ 序贯理性）；
对任一 $\mathbb{P}(h|\sigma)>0$ 的信息集 $h$，$\mu$ 满足贝叶斯法则 $\mu(x)=\dfrac{\mathbb{P}(x|\sigma)}{\mathbb{P}(h|\sigma)}$；
独立后验（类型独立分布）；
不要泄露你不知道的（若玩家的行动不基于额外信息，则该行动不改变其他玩家可得的信息或其信念）；
离均衡行动的共同信念（所有玩家对 0 概率被达的信息集有相同的信念系统）。

11.2.4 Sequential equilibrium

如上部分所述，评估 $(\mu,\sigma)$ 是序贯均衡当且仅当满足：1. 序贯理性；2. 一致性。序贯均衡略强于 PBE，即序贯均衡蕴含 PBE；但在某些条件下二者等价。

11.2.3 Perfect Bayesian equilibrium (PBE)

For an assessment to be a PBE, add three conditions on top of weak PBE / weak sequential equilibrium. A PBE satisfies:

$(\mu,\sigma)$ sequentially rational ($\sigma$ sequentially rational w.r.t. $\mu$);
for any info set $h$ with $\mathbb{P}(h|\sigma)>0$, $\mu$ satisfies Bayes' rule $\mu(x)=\dfrac{\mathbb{P}(x|\sigma)}{\mathbb{P}(h|\sigma)}$;
independent posteriors (types independently distributed);
don't signal what you don't know (if a player's action is not based on additional information, it won't change the information available to other players or their beliefs);
common beliefs over out-of-equilibrium actions (all players have the same system of beliefs over information sets of zero probability of being reached).

11.2.4 Sequential equilibrium

As introduced in the previous part, an assessment $(\mu,\sigma)$ is a sequential equilibrium iff it satisfies: 1. sequential rationality; 2. consistency. A sequential equilibrium is slightly stronger than PBE, i.e. it implies PBE; but under some conditions the two are equivalent.

\(E\backslash I\)	\(F\)	\(A\)
\(Out\)	$(0,2)$	$(0,2)$
\(F\)	$(-3,1)$	$(1,-2)$
\(A\)	$(-2,-1)$	$(3,1)$