📚 IGCSE CAIE Politics: Formula and Theorem Quick Reference Handbook | IGCSE CAIE 政治：公式定理速查手册
This quick reference handbook compiles the essential formulas, theorems, and conceptual models necessary for mastering IGCSE CAIE Politics. It translates core political principles into accessible formulas and frameworks, enabling students to analyse electoral systems, governance structures, and democratic indicators with precision.
本速查手册汇集了掌握 IGCSE CAIE 政治所需的基本公式、定理和概念模型。它将核心政治原理转化为易于理解的公式和框架，使学生能够精准分析选举制度、治理结构和民主指标。
1. Duverger's Law | 杜瓦杰定律
Duverger' to produce a two-party system. The law is often expressed as: SMDP → Two-Party Dominance.
杜瓦杰定律指出，单一选区相对多数决（SMDP）选举制度倾向于产生两党制。该定律常表示为：SMDP → 两党主导。
The mechanism behind this is the mechanical effect (only one winner per district) and the psychological effect (voters avoid wasting votes on unlikely winners).
其背后的机制是机械效应（每个选区仅一个获胜者）和心理效应（选民不愿将选票投给无望获胜的候选人）。
Duverger's Law: Plurality rule + Single-member districts → Two-party system
杜瓦杰定律：相对多数规则 + 单一选区 → 两党制
- English: It explains why countries like the United States and the United Kingdom have long been dominated by two major parties.
- 中文：它解释了为何美国、英国等国家长期由两大政党主导。
- English: The converse, Duverger's hypothesis, suggests that proportional representation (PR) favours multi-party systems.
- 中文：其逆命题，杜瓦杰假设，提出比例代表制（PR）有利于多党制。
2. Proportional Representation Seat Allocation Formula | 比例代表制席位分配公式
To allocate seats under proportional representation, the D'Hondt method uses divisors: 1, 2, 3, … for each party's total votes. The party with the highest resulting quotient wins the next seat.
在比例代表制下分配席位，D'Hondt 法使用除数 1、2、3……对每个政党的总得票数进行除法计算，商最高的政党赢得次一席位。
Quotient = Total votes of party ÷ (Seats already allocated to that party + 1)
商数 = 政党总得票数 ÷ (该党已获席位 + 1)
This process repeats until all seats are filled. An alternative is the Sainte-Laguë method using divisors 1, 3, 5, ….
此过程重复直至所有席位分配完毕。另一种方法是 Sainte-Laguë 法，使用除数 1、3、5……
| Party 政党 | Votes 得票 | ÷1 | ÷2 | ÷3 |
|---|---|---|---|---|
| A | 4500 | 4500 (1st) | 2250 (3rd) | 1500 (5th) |
| B | 3500 | 3500 (2nd) | 1750 (4th) | 1167 |
ets second (3500), Party A gets third (2250), Party B gets fourth (1750). Seats: A=2, B=2.
示例：在一个有 4 个席位的选区，政党 A 获第一席（4500），政党 B 获第二席（3500），政党 A 获第三席（2250），政党 B 获第四席（1750）。席位：A=2, B=2。
3. Effective Number of Parties | 有效政党数
The effective number of parties, developed by Laakso and Taagepera, measures party system fragmentation. It counts parties weighted by their size, providing a more realistic picture than the raw number of parties.
由 Laakso 和 Taagepera 提出的有效政党数，衡量政党体系的碎片化程度。它根据各党规模加权计算，比单纯的政党数量更能反映真实状况。
E = 1 ÷ (p₁² + p₂² + p₃² + …)
有效政党数 E = 1 ÷ (p₁² + p₂² + p₃² + …)
decimal vote shares (or seat shares) of each party. For example, if three parties have 50%, 30%, and 20%, E = 1 ÷ (0.5² + 0.3² + 0.2²) = 1 ÷ (0.25+0.09+0.04) = 1 ÷ 0.38 ≈ 2.63 effective parties.
其中 p₁、p₂、p₃ 为各政党的得票率（或席位率）的小数形式。例如，三个政党得票率分别为 50%、30% 和 20%，则 E = 1 ÷ (0.5² + 0.3² + 0.2²) = 1 ÷ 0.38 ≈ 2.63 个有效政党。
- English: E ≈ 2 in a two-party system; E > 3 indicates a fragmented multi-party system.
- 中文：两党制下 E ≈ 2；E > 3 表明碎片化的多党制。
- English: This index helps compare electoral outcomes and the impact of electoral rules.
- 中文：该指数有助于比较选举结果和选举规则的影响。
4. Democracy Index Formula | 民主指数公式
The Democracy Index, published by the Economist Intelligence Unit, rates countries on a 0–10 scale across five categories: electoral process and pluralism, functioning of government, political participation, political culture, and civil liberties.
民主指数由经济学人智库发布，基于五个类别对各国进行 0–10 分评估：选举进程与多元化、政府运作、政治参与、政治文化和公民自由。
Democracy Index = (Score₁ + Score₂ + Score₃ + Score₄ + Score₅) ÷ 5
民主指数 = (得分₁ + 得分₂ + 得分₃ + 得分₄ + 得分₅) ÷ 5
Each category is scored based on 60 indicators or expert assessment, and the simple average yields the overall index. Regimes are classified as full democracies (8–10), flawed democracies (6–7.9), hybrid regimes (4–5.
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