econ-game-theory

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Game Theory Basics

Game Theory基础

Overview

概述

Game theory models strategic interactions where each player's outcome depends on others' choices. It provides tools to predict behavior, identify stable outcomes (equilibria), and design mechanisms that align incentives.

Game Theory用于对策略互动进行建模，在这类互动中，每个参与者的结果取决于其他参与者的选择。它提供了预测行为、识别稳定结果（均衡状态）以及设计激励对齐机制的工具。

Framework

框架

IRON LAW: Define Players, Strategies, and Payoffs BEFORE Analyzing

Every game requires three elements explicitly defined:
1. Players — who are the decision-makers?
2. Strategies — what choices does each player have?
3. Payoffs — what does each player get for each combination of choices?

Analyzing a "game" without a payoff matrix is just storytelling.

IRON LAW: Define Players, Strategies, and Payoffs BEFORE Analyzing

Every game requires three elements explicitly defined:
1. Players — who are the decision-makers?
2. Strategies — what choices does each player have?
3. Payoffs — what does each player get for each combination of choices?

Analyzing a "game" without a payoff matrix is just storytelling.

Analysis Steps

分析步骤

Identify players and their available strategies
Build the payoff matrix (simultaneous) or game tree (sequential)
Check for dominant strategies per player
Find Nash Equilibrium — where best responses intersect
For sequential games: apply backward induction from terminal nodes
Evaluate efficiency — is the NE Pareto optimal? If not, flag cooperation opportunity
The resulting path is the Subgame Perfect Equilibrium

识别参与者及其可用策略
构建收益矩阵（同时博弈）或博弈树（序贯博弈）
检查每个参与者的占优策略（dominant strategies）
找到Nash equilibrium——即最优反应的交集点
对于序贯博弈：从终端节点反向推导
评估效率——该Nash equilibrium是否帕累托最优？若否，标记合作机会
最终推导路径即为子博弈完美均衡（Subgame Perfect Equilibrium）

Output Format

输出格式

markdown

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markdown

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Game Theory Analysis: {Situation}

Game Setup

Players: {list}
Strategies: Player 1: {A, B}, Player 2: {X, Y}
Type: Simultaneous / Sequential

Players: {list}
Strategies: Player 1: {A, B}, Player 2: {X, Y}
Type: Simultaneous / Sequential

Payoff Matrix (simultaneous) or Game Tree (sequential)

	Player 2: X	Player 2: Y
Player 1: A	(a1, a2)	(b1, b2)
Player 1: B	(c1, c2)	(d1, d2)

	Player 2: X	Player 2: Y
Player 1: A	(a1, a2)	(b1, b2)
Player 1: B	(c1, c2)	(d1, d2)

Analysis

Dominant strategies: {if any}
Nash Equilibrium: {strategy combination, payoffs}
Pareto optimal? {yes/no — if no, explain the cooperation opportunity}

Dominant strategies: {if any}
Nash Equilibrium: {strategy combination, payoffs}
Pareto optimal? {yes/no — if no, explain the cooperation opportunity}

Strategic Implications

{What should each player do? What mechanism could improve outcomes?}

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{What should each player do? What mechanism could improve outcomes?}

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Examples

示例

Correct Application

正确应用场景

Scenario: Two bubble tea chains considering price cut

	Chain B: Hold Price	Chain B: Cut Price
Chain A: Hold Price	(80, 80)	(40, 100)
Chain A: Cut Price	(100, 40)	(60, 60)

Both have dominant strategy: Cut Price (100 > 80, 60 > 40)
Nash Equilibrium: (Cut, Cut) = (60, 60) — a Prisoner's Dilemma ✓
Both would prefer (Hold, Hold) = (80, 80) but can't sustain it without a binding agreement
Implication: Price wars are the rational outcome. To escape, need repeated interaction (reputation), contracts, or differentiation that makes price less relevant.

场景： 两家奶茶连锁店考虑降价

	Chain B: Hold Price	Chain B: Cut Price
Chain A: Hold Price	(80, 80)	(40, 100)
Chain A: Cut Price	(100, 40)	(60, 60)

双方均拥有占优策略（dominant strategy）：降价（100 > 80，60 > 40）
Nash Equilibrium: (Cut, Cut) = (60, 60) —— 符合Prisoner's Dilemma ✓
双方更倾向于（Hold, Hold）=（80,80）的结果，但若无具有约束力的协议则无法维持该状态
启示：价格战是理性决策的结果。要打破该困境，需要重复互动（建立声誉）、签订合约或进行差异化经营，降低价格的重要性。

Incorrect Application

错误应用场景

"Our competitor will probably cooperate because it's better for everyone" → In a one-shot Prisoner's Dilemma, rational players defect. Cooperation requires repeated games or enforcement. Violates the model's prediction.

“我们的竞争对手可能会选择合作，因为这对所有人都有利” → 在单次Prisoner's Dilemma中，理性参与者会选择背叛。合作需要重复博弈或强制执行机制。该说法违背了模型的预测。

Gotchas

注意事项

Nash Equilibrium ≠ best outcome: NE is stable, not optimal. The Prisoner's Dilemma NE is worse for both players than cooperation.
Multiple equilibria: Many games have multiple NE. Additional criteria (focal points, risk dominance, Pareto dominance) help select among them.
Payoff estimation is the hard part: The matrix is easy once payoffs are known. Estimating realistic payoffs requires market research and financial modeling.
Repeated games change everything: In one-shot games, defection dominates. In repeated games, tit-for-tat and reputation effects enable cooperation.
Information matters: Games with incomplete information (you don't know opponent's payoffs) or imperfect information (you don't see opponent's moves) require Bayesian analysis.
Mixed-strategy NE is the default, not the exception: When no pure-strategy NE exists (e.g., matching pennies), agents often report "no equilibrium found" instead of computing the mixed strategy. Every finite game has at least one NE — if you can't find a pure one, solve for the mixing probabilities.

Nash equilibrium ≠ 最优结果：NE是稳定状态，但并非最优。Prisoner's Dilemma中的NE对双方而言都比合作结果更差。
多重均衡：许多博弈存在多个NE。额外的判断标准（焦点、风险占优、帕累托占优）有助于从中选择合适的均衡。
收益估算最具挑战性：一旦确定收益，构建矩阵就会变得简单。估算真实收益需要市场调研和财务建模。
重复博弈会彻底改变结果：在单次博弈中，背叛是占优策略。在重复博弈中，“以牙还牙”策略和声誉效应能够促成合作。
信息至关重要：存在不完全信息（不了解对手的收益）或不完美信息（看不到对手的行动）的博弈需要贝叶斯分析。
混合策略NE是常态而非例外：当不存在纯策略NE时（如猜硬币游戏），人们常报告“未找到均衡”，而非计算混合策略。每个有限博弈至少存在一个NE——如果找不到纯策略NE，就求解混合概率。

References

参考资料

For repeated games and folk theorem, see
```
references/repeated-games.md
```
For mechanism design basics, see
```
references/mechanism-design.md
```

关于重复博弈和民间定理，请参阅
```
references/repeated-games.md
```
关于机制设计基础，请参阅
```
references/mechanism-design.md
```