econ-game-theory
用報酬矩陣和納許均衡分析競爭與合作的策略互動
中文翻譯
---
name: "econ-game-theory"
description: "運用基礎賽局理論的概念（納許均衡、優勢策略、囚徒困境）分析策略互動。當使用者需要為競爭決策建模、預測對手行為、設計誘因機制，或評估合作與競爭的情境時使用。就算使用者只說「我們的競爭對手會怎麼做」、「該合作還是競爭」或「怎麼設計對的誘因」也要用。"
metadata:
category: "WP-17 經濟學院"
tags: ["economics", "game-theory", "strategy"]
---
# 賽局理論基礎
## 概述
賽局理論為策略互動建模，其中每位參與者的結果取決於其他人的選擇。它提供工具來預測行為、找出穩定的結果（均衡），並設計能讓誘因一致的機制。
## 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.
```
### 分析步驟
1. **找出參與者**和他們可用的策略
2. **建立報酬矩陣**（同時賽局）或**賽局樹**（循序賽局）
3. **檢查各參與者有沒有優勢策略**
4. **找出納許均衡**：最佳回應交會的地方
5. **循序賽局**：從終點節點開始做逆向歸納
6. **評估效率**：納許均衡是否具有柏拉圖最適？如果不是，標出合作的機會
3. 最後得到的路徑就是子賽局完美均衡
## 輸出格式
```markdown
# Game Theory Analysis: {Situation}
## Game Setup
- 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) |
## Analysis
- 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?}
```
## 範例
### 正確的做法
**情境：** 兩家珍珠奶茶連鎖品牌在考慮降價
| | B 店：維持價格 | B 店：降價 |
|---|---|---|
| A 店：維持價格 | (80, 80) | (40, 100) |
| A 店：降價 | (100, 40) | (60, 60) |
- 雙方都有優勢策略：降價（100 > 80，60 > 40）
- **納許均衡：（降價, 降價）= (60, 60)**，這是個囚徒困境 ✓
- 雙方都比較想要（維持, 維持）= (80, 80)，但沒有具約束力的協議就維持不住
- **意涵**：價格戰是理性的結果。要跳出來，需要重複互動（聲譽）、合約，或是讓價格變得沒那麼重要的差異化。
### 錯誤的做法
- 「我們的競爭對手大概會合作，因為這對大家都好」→ 在一次性的囚徒困境裡，理性的參與者會背叛。合作需要重複賽局或強制執行。這違反了模型的預測。
## 常見陷阱
- **納許均衡 ≠ 最佳結果**：納許均衡是穩定的，但不一定最佳。囚徒困境的納許均衡，對雙方來說都比合作差。
- **多重均衡**：很多賽局有多個納許均衡。要用額外的準則（焦點、風險優勢、柏拉圖優勢）從中挑選。
- **估計報酬是最難的部分**：報酬都知道之後，矩陣很容易建。要估出實際的報酬需要市場研究和財務建模。
- **重複賽局讓一切都變了**：在一次性賽局裡，背叛是優勢。在重複賽局裡，以牙還牙和聲譽效應讓合作成為可能。
- **資訊很重要**：不完全資訊（你不知道對手的報酬）或不完美資訊（你看不到對手的行動）的賽局，需要用貝氏分析。
- **混合策略納許均衡是常態，不是例外**：當不存在純策略納許均衡時（例如猜硬幣），agent 常常回報「找不到均衡」，而不是去算混合策略。每個有限賽局至少有一個納許均衡，如果找不到純策略的，就去解混合機率。
## 參考資料
- 重複賽局與無名氏定理，見 `references/repeated-games.md`
- 機制設計基礎，見 `references/mechanism-design.md`英文原文
---
name: "econ-game-theory"
description: "Apply basic game theory concepts including Nash equilibrium, dominant strategies, and the Prisoner's Dilemma to analyze strategic interactions. Use this skill when the user needs to model competitive decisions, predict rival behavior, design incentive mechanisms, or evaluate cooperation vs competition scenarios — even if they say 'what will our competitor do', 'should we cooperate or compete', or 'how do we set up the right incentives'."
metadata:
category: "WP-17 經濟學院"
tags: ["economics", "game-theory", "strategy"]
---
# Game Theory Basics
## Overview
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.
## Framework
```
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
1. **Identify players** and their available strategies
2. **Build the payoff matrix** (simultaneous) or **game tree** (sequential)
3. **Check for dominant strategies** per player
4. **Find Nash Equilibrium** — where best responses intersect
5. **For sequential games**: apply backward induction from terminal nodes
6. **Evaluate efficiency** — is the NE Pareto optimal? If not, flag cooperation opportunity
3. The resulting path is the Subgame Perfect Equilibrium
## Output Format
```markdown
# Game Theory Analysis: {Situation}
## Game Setup
- 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) |
## Analysis
- 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?}
```
## Examples
### Correct Application
**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.
### Incorrect Application
- "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.
## Gotchas
- **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.
## References
- For repeated games and folk theorem, see `references/repeated-games.md`
- For mechanism design basics, see `references/mechanism-design.md`