EC487 Advanced Microeconomics

GameTheory Micro

This course is mainly about the Game Theory stuff. This page serves as a Map of Content.

Examinable Proofs

Below are some examinable proofs you should be able to reproduce in the exam:

• Min-Max = Max-Min

Lecture 1

Definition 1.1 A finite normal-form game is a tuple $(N,A,u)$:

• $N=1,...,n$ is a finite set of players.
• Each player $j$ has a finite action set $A^j$. Let $A={\times }_{j=1}^N{A}^j$.
• Each player $j$ has a utility function $u^j:A\to \mathbb{R}$. that represents preferences over $A$. We define as $u=(u^j{)}_{j=1}^N$ .

Notice that, Mixed Strategy allows for randomization. We use the simplex notation $\Delta (A^j)$ to denote the set of all probability distributions over $A^j$:

$$S^j=\Delta (A^j)=\left\{s^j:A^j\to {\mathbb{R}}_{+}\text{s.t.}\sum _{a^j\in A^j}{s}^j(a^j)=1\right\}$$

and $S={\times }_{j=1}^N{S}^j$ . Here $S$ means the set of all possible mixed strategy profiles.

$j$ here represents a specific player.

Nash Equilibrium

Unlike in the undergrad, we give a more general definition of Nash Equilibrium here:

A strategy profile $s^{\ast }\in S$ is a Nash Equilibrium if

$$v^j(s^{\ast j},s^{\ast -j})\ge v^j(s^j,s^{\ast -j})$$

for all $s^j\in S^j$ and for all players $j\in \{1,\dots ,N\}$.

Here, $s^{-j}$ means the strategy profile of all players except player $j$.

Zero Sum Game

Lecture 2

What is $\Delta (X)$

We first need to denote that $\Delta (X)$ is the probability simplex over a finite set $X$.

$$\Delta (X)=\{p:X\to {\mathbb{R}}_{+}\text{ s.t. }\sum _{x\in X}p(x)=1\}$$

For example, $\Delta (A^j)$ is all mixed strategies for player $j$, if $A^j=\{L,R\}$, then $\Delta (A^j)=\{(p,1-p):p\in [0,1]\}$, which means all ways to randomize between left and right.

Belief

The belief is written as ${\mu }^j\in \Delta (A^{-j})$, intuitively, it means a probability distribution over what all other players might play. In Nash Equilibrium, belief about other player’s actions had to be consistent with their strategies, i.e.,

$${\mu }^j(a^{-j})=\prod _{i\ne j}{s}^i(a^i)$$

The intuition of this consistency condition is that, it requires beliefs match reality.

Again, we recall that $a^j$ is an element, it stands for what $j$ actually plays, while $s^j$ is a function, it stands for the probability distribution over all possible actions. And $s^j(a^j)$ means the probability that player $j$ plays action $a^j$. The upper letter case, like $A^j$, means the set of all possible actions.

Combining the correlated beliefs (and strategies), we could define player $j$ ‘s expected utility as follows:

$$v^j(s^j,{\mu }^j)=\sum _{a^j\in A^j}\sum _{a^{-j}\in A^{-j}}{s}^j(a^j){\mu }^j(a^{-j})u^j(a^j,a^{-j})$$

Note that, here we use ${\mu }^j$ instead of $s^{-j}$, because we want to emphasize the belief part. But in Nash Equilibrium, they are consistent, so it doesn’t matter which one you use.

This equation, to be honest, is quite abstract at the first slight, we think about it step by step:

• Step 1: We first consider if the player $j$ plays pure action $a^j$ and opponents also play pure profile $a^{-j}$, utility is very simple: $$u^j(a^j,a^{-j})$$
• Step 2: Now, if player $j$ isn’t sure what opponents will play, so he would have a belief ${\mu }^j$ about it. Then, the expected utility for player $j$ if he plays pure action $a^j$ is: $$\sum _{a^{-j}\in A^{-j}}{\mu }^j(a^{-j})u^j(a^j,a^{-j})$$
• Step 3: Now, player $j$ also randomizes his action, using his mixed strategies $s^j$, then the expected utility for player $j$ is: $$\sum _{a^j\in A^j}{s}^j(a^j)\left(\sum _{a^{-j}\in A^{-j}}{\mu }^j(a^{-j})u^j(a^j,a^{-j})\right)$$

By rearranging the terms, we get the equation below:

$$v^j(s^j,{\mu }^j)=\sum _{a^j\in A^j}\sum _{a^{-j}\in A^{-j}}{s}^j(a^j){\mu }^j(a^{-j})u^j(a^j,a^{-j})$$

Definition

A player $j$ action $a^j\in A^j$ is a Best Response $B{R}^j({\mu }^j)$ against belief ${\mu }^j\in \Delta (A^{-j})$ if $a^j\in {\text{argmax}}_{{\tilde{a}}^j}{v}^j(a^j,{\mu }^j)$

Rationality

If a player is rational, the weakest decision criterion for him is not to apply in any situation that are never Best Response.

Definition (Never a Best Response)

An action $a^j\in A^j$ is never a Best Response if there doesn’t exist any belief ${\mu }^j\in \Delta (A^{-j})$ such that $a^j\in B{R}^j({\mu }^j)$.

Correlated Beliefs and Uncorrelated Beliefs

This distinction is crucial for understanding the difference between Nash equilibrium and more general solution concepts.

Uncorrelated Beliefs

Uncorrelated beliefs assume opponents play independently. Player $j$ believes each opponent $i$ mixes according to some strategy $s^i$, and their joint actions are statistically independent:

${\mu }^j(a^{-j})={\prod }_{i\ne j}{s}^i(a^i)$

This means: “The probability of seeing action profile $a^{-j}$ equals the product of individual probabilities.”

Example: In a 3-player game, if Player 3 believes:

• Player 1 plays Left with probability 0.5
• Player 2 plays Up with probability 0.3

Then Player 3 must believe:

• ${\mu }^3(Left,Up)=0.5\times 0.3=0.15$
• ${\mu }^3(Left,Down)=0.5\times 0.7=0.35$
• ${\mu }^3(Right,Up)=0.5\times 0.3=0.15$
• ${\mu }^3(Right,Down)=0.5\times 0.7=0.35$

Correlated Beliefs

Correlated beliefs allow players to believe opponents’ actions are statistically dependent. Now ${\mu }^j\in △(A^{-j})$ can be any probability distribution.

Example: Player 3 could believe:

• ${\mu }^3(Left,Up)=0.6$
• ${\mu }^3(Right,Down)=0.4$
• ${\mu }^3(Left,Down)=0$
• ${\mu }^3(Right,Up)=0$

This cannot arise from independent mixing! Players 1 and 2 seem to coordinate - when one goes Left/Up, the other does too.

Dominant Strategy

Definition (Strictly Dominated)

An action $a^j$ is strictly dominated by a strategy $s^j\in \Delta (A^j)$ if

$$u^j(a^j,a^{-j})<\sum _{{\tilde{a}}^j\in A^j}{s}^j({\tilde{a}}^j)u^j({\tilde{a}}^j,a^{-j})\forall a^{-j}\in A^{-j}$$

It’s easy to come up with the idea that $a^j$ would never a BR iff it is strictly dominated.

There would be a proof in the lecture 2, redo it.

Definition (Recursive Rationalizability)

${\mu }_{t+1}^i[a^i]$

Common knowledge of rationality is “stronger” than rationality alone.

See the Problem Sets for EC487 for more applications.

Take a break, I want to summarize the notations about what we have come so far:

Notation	Meaning	Example
$A^j$	$j$ ‘s action sets	$A^j=\{U,M,D\}$
$a^j$	a specific action by player $j$	$a^j=U$

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About the exam. Half of the lecture would be about an additional idea, and give some ideas

convex of Competitive Equilibrium, and Game Theory

• There would be 3 exercises, and 6 short questions for each exercise. The third question is very heavy. The second question is a proof, and there would be a list provided.

Review

Possible Test Point

Lecture 8

Lecture 9

• How to draw Edgeworth Box

Question For Christopher

• For the proof of L8: $x\in \arg \max$ utilitarian welfare $⟹$ $x$ Pareto efficient, does it refer to Proposition 8.1 in the lecture notes? do we have to learn the $<=$ direction as well?
• where is L3: first-mover advantage in supermodular games / disadvantage in ZSG

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Problem Sets 归纳

您好！我很高兴能为您提供复习协助。EC487《高级微观经济学》课程涵盖了博弈论和竞争性均衡两大核心主题。根据您上传的往年试卷、Problem Sets（PS）和讲义大纲，我已经完成了题目的归类和知识点定位。

本学期的课程大纲（来自您的讲义）分为两个主要部分：

• 第一部分：博弈论（Lecture 1-5）： 涵盖纳什均衡、零和博弈、纳什议价、相关均衡、可理性化性、超模和势博弈、扩展式博弈（子博弈完美均衡、逆向归纳）、前向归纳、斯塔尔和鲁宾斯坦议价、承诺和重复博弈。
• 第二部分：竞争性均衡（Lecture 6-10）： 涵盖消费者理论、竞争性均衡的（存在性、唯一性）、福利定理、丛林博弈、竞争性均衡的基础和去中心化匹配。

以下是基于您提供的材料的复习分析和归类表格。

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复习重点和建议

基于您提供的材料和复习目标，以下是您应重点关注的方面：

1. 重点关注和重合度分析（需求 2、5、6）

您提供的 Problem Sets (PS 1-9) 覆盖了本学期所有的 Lecture Slides (L1-L10) 内容，因此它们是您复习的 最高优先级，代表了教授对每个主题深度和广度的要求。

重点程度	类别	建议和说明（基于源文件）
最高	Problem Sets (PS 1-9)	PS难度通常较高，因为有更充足的时间完成。它们涵盖了核心证明和应用，例如：零和博弈的 Minmax 定理；Nash 议价解的特性；可理性化性和严格劣势策略的迭代消除；势博弈；Cournot 和 Stackelberg 的比较；动态不一致和世代自我博弈；家庭中的竞争性均衡模型；Edgeworth Box 的存在性和唯一性；丛林博弈的制度比较。这些题目是复习的基石。
高	核心证明题	很多试卷中重复出现了理论基石的证明题，例如：零和博弈中 Minmax=Maxmin 的证明；独立性公理（Independence Axiom）；风险厌恶与效用函数凹性等价；Nash 议价解的表示定理；福利定理。
中高	Auction Theory 和 Moral Hazard/Signaling	尽管这些主题在 Lecture 大纲的标题中没有独立列出（多被包含在 L5 或应用中），但它们在所有近期的 Summer Exam (ST) 和部分 Winter Exam (LT) 中占据了很大的比重。这表明它们是课程应用的重点。特别是 拍卖机制的对称贝叶斯纳什均衡 (BNE) 求解和 Moral Hazard/Signaling/Cheap Talk 模型。
中	年代久远的题目	2018年以前的题目仍可用于练习核心概念，但可能包含与当前大纲重合度较低的主题。应优先处理与 PS/Lecture 紧密相关的部分，例如，2018年以前涉及 Nash/SPE/CE 基础概念的题目。

2. 新冠疫情时期试卷难度（需求 4）

新冠疫情时期的试卷（主要指 2020年夏季和2021年冬季/夏季）允许学生在 24小时 内完成，因此这些试卷的难度可能更高，要求更深入的分析和计算。

• 2020年夏季考试 (ST 2020)：涉及复杂拍卖机制（Q1）、带有严格限制的 Moral Hazard（Q2）和风险理论的完整证明（Q3），要求深入的理论推导。
• 2021年冬季考试 (LT 2021)：包含一个复杂的 Auction + Stackelberg 扩展式博弈（Q2），以及需要详细计算的无限重复博弈中的 TIOLI 议价（Q3）和 Coase/Externality 模型（Q4）。

建议： 将这些 24小时考试的题目作为对您理解能力的终极测试。如果能解决这些问题，您对课程内容的掌握就非常扎实。

3. Problem Sets 与考试重合度（需求 3）

有几个 Problem Set 的练习直接出现在了往年的考试中，这表明这些知识点是必考的：

• PS 1 Q3/Q4 (Zero-sum games with a star player)： 几乎完整地出现在 LT 2022 Exercise 2 中。
• PS 9 Q3/EC487LT_2023 Q2 (Walrasian bargaining)： 几乎完整地重复出现，涉及 Walrasian 均衡、Nash 议价解和交替出价博弈。
• PS 3 Q4/EC487LT_2022 Q2 (Bank Run Model/Potential Games)： 银行挤兑模型被要求讨论其博弈论基础并证明其是势博弈。

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题目归类与知识点定位（Markdown 表格）

以下表格按核心知识点对 Problem Sets 和历年考试题目进行归类。

知识点类别	核心概念 (Lecture 编号)	Problem Set (PS) 题号	往年试卷及题号	重点关注原因/深度要求
博弈论基础 (L1/L2)	Nash 均衡、零和博弈、Minimax/Maxmin、Affine 变换、相关均衡、劣势策略、可理性化性	PS 1 Q1-4, PS 2 Q1-5, PS 3 Q1, Q5	LT 2022 E2-E3, LT 2023 Q1.4	极高重合度 (PS 1-2 基本概念，PS 2 涉及 Traveller’s Dilemma，PS 1 Q3 几乎是 LT 2022 E2 原题)，要求证明 Minmax 性质。
扩展式博弈/序贯博弈 (L3/L4)	SPE、逆向归纳、Cournot vs Stackelberg、Centipede Game、前向归纳、Hold-up 问题、承诺	PS 3 Q2, PS 4 Q1-2	LT 2019 Q4 (Hold-up), LT 2021 Q2a (Auction+Stackelberg), LT 2022 Q1.4, LT 2023 Q1.1 (Forward Induction)	核心应用 (Hold-up 和 Stackelberg 是经典应用)；PS 4 深入探讨 Cournot/Stackelberg 产出比较和减增差异性。
重复博弈/Folk 定理 (L5)	Grim Trigger 策略、个体理性（Individual Rationality）、可行性（Feasibility）、Folk Theorem	PS 5 Q5-6 (Secret Contracting), PS 6 Q1 (Political Manifestos)	LT 2020 Q4, LT 2022 Q1.5-6	关键理论 (Folk Theorem 是 L5 重点)，要求证明或给出 Grim Trigger 策略的应用场景和失败案例。
议价理论 (L1/L4/L9)	Nash 议价解（NBS）、效用乘积最大化、比较静力学、交替出价（Rubinstein/Stahl）、Walrasian 议价、交替关税	PS 1 Q3, PS 4 Q3, PS 9 Q3	LT 2018 Q4 (Random Proposer), LT 2023 Q2 (Walrasian Bargaining)	高重合度 (PS 9 Q3 与 LT 2023 Q2 几乎完全一致)，考察 Nash 议价解与 Walrasian 均衡和动态议价的关系。
消费者理论 (L6)	效用函数性质、局部非饱和性、凸性、Marshallian/Hicksian 需求、Slutsky 方程、劳动供给模型、风险厌恶	PS 6 Q2, PS 7 Q1	LT 2018 Q1, LT 2019 Q1, LT 2021 Q1, ST 2020 Q3, ST 2022 Q3, ST 2024 Q1	基础核心 (L6 占了很大比重)。风险厌恶的定义、等价性（凹性）和 CARA 性质是必考的证明题。
竞争性均衡 (L7)	存在性、唯一性、Gross Substitutes、Edgeworth Box、CES/Cobb-Douglas 需求	PS 7 Q2, PS 7 Q3	LT 2018 Q2, Q3, LT 2020 Q2, LT 2022 E4	核心应用 (需要掌握 Cobb-Douglas 等效用函数的 Marshallian 需求和均衡价格求解)；需要理解均衡存在失败的条件（例如：非凹性效用或边界问题）。
福利定理/效率 (L8)	第一/第二福利定理、Pareto 效率、Taxation 影响、Coase 定理、丛林博弈	PS 8 Q3 (Jungle)	LT 2020 Q1, LT 2021 Q4 (Externality/Coase), LT 2023 Q1.6, ST 2024 Q1	理论要点 (福利定理的假设和结论必须清晰)；丛林博弈作为 L8 独特应用，在 PS 8 中被详细考察。
信息经济学应用	拍卖（BNE/收益）、道德风险（Moral Hazard）、信号（Signaling/Cheap Talk）、机制设计	PS 5 Q1-2	ST 2018 Q1, Q4, ST 2020 Q1, Q2, ST 2024 Q2-3, SP 2025 Q1-4	高频考点 (占 Summer Exam 50%)。包括 Spence 信号模型、Beer-Quiche 信号博弈、Chatterjee & Samuelson 双边拍卖等。

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