Political Science Quick Reference: Formulae and Theorems | 政治学公式定理速查手册

📚 Political Science Quick Reference: Formulae and Theorems | 政治学公式定理速查手册

This handbook collects the essential models, laws and quantitative formulas encountered in the CAIE IGCSE Politics course. Each entry is presented as a pair of ‘theorem’ and explanation, designed to help you quickly revise the logic of electoral systems, institutional checks and political behaviour.

本手册汇集了CAIE IGCSE政治课程中必学的核心模型、定律与量化公式（即“公式定理”）。每个条目都以“定理 + 解释”的中英配对呈现，帮助你快速梳理选举制度、机构制衡和政治行为的逻辑。

1. Duverger’s Law | 杜瓦杰定律

Duverger’s law states that single-member district plurality (SMDP) electoral systems tend to produce a two-party system, while proportional representation (PR) favours multi-party competition. The mechanism is psychological: voters avoid wasting votes on smaller parties that cannot win a plurality. SMDP ⇒ two-party system; PR ⇒ multi-party system.

杜瓦杰定律指出，单一选区相对多数决制（SMDP）倾向于产生两党制，而比例代表制（PR）则有利于多党竞争。其心理机制在于选民不愿将选票“浪费”在无法赢得相对多数的小党身上。简单多数制 ⇒ 两党制；比例代表制 ⇒ 多党制。

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2. Median Voter Theorem | 中间选民定理

In a two-candidate election with single-peaked preferences on a left–right scale, both candidates will converge towards the policy position favoured by the median voter. This theorem explains why major parties often sound similar and why extreme platforms usually lose. Optimal platform → median voter’s ideal point.

在左右光谱上偏好呈单峰分布的两党选举中，双方候选人都会向中间选民偏好的政策位置靠拢。该定理解释了为何大党在政见上往往趋同，以及极端政纲通常难以胜选。最优政纲 → 中间选民的理想点。

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3. Arrow’s Impossibility Theorem | 阿罗不可能定理

Kenneth Arrow proved that no ranked voting system can simultaneously satisfy all four reasonable fairness conditions: Unrestricted domain (U), Pareto efficiency (P), independence of irrelevant alternatives (IIA), and non-dictatorship (ND). Every real-world voting rule must sacrifice at least one of these ideals. No voting rule satisfies (U, P, IIA, ND) simultaneously.

肯尼斯·阿罗证明，任何排序投票制都无法同时满足四项合理的公平条件：无限制域（U）、帕累托效率（P）、无关选项独立性（IIA）和非独裁性（ND）。现实中每种投票规则都不得不牺牲至少其中一项理想条件。不存在同时满足 U、P、IIA、ND 的投票规则。

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4. Iron Triangle | 铁三角模型

The iron triangle describes a stable, mutually beneficial relationship among three actors in US federal policymaking: congressional committees, bureaucratic agencies and interest groups. Each provides the others with resources – funding, political support and policy implementation – making the triangle highly resistant to change. Interest Groups ⇄ Congressional Committees ⇄ Bureaucracy ⇄ Interest Groups.

铁三角模型描绘了美国联邦政策制定中三个主体之间稳定且互利的共生关系：国会委员会、官僚机构和利益集团。三者彼此提供资源——资金、政治支持与政策执行——使得该三角极难被打破。利益集团 ⇄ 国会委员会 ⇄ 官僚机构 ⇄ 利益集团。

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5. Separation of Powers & Checks and Balances | 权力分立与制衡

The doctrine of separation of powers divides government into three independent branches: legislature (L), executive (E) and judiciary (J). Checks and balances ensure that each branch can limit the others’ powers, preventing any one branch from dominating. L ↔ E ↔ J ↔ L (mutual checks). Overlap is intentionally designed: e.g., presidential veto (E over L), judicial review (J over L/E).

权力分立原则将政府划分为三个独立的部门：立法（L）、行政（E）与司法（J）。制衡机制则确保每个部门都能牵制其他部门，避免单一部门独大。L ↔ E ↔ J ↔ L（相互制衡）。制度上刻意设置重叠：例如总统否决（E 制衡 L），司法审查（J 制衡 L/E）。

• Legislature makes laws but can be checked by executive veto and judicial review.

  立法机构制定法律，但受行政否决与司法审查制约。

• Executive enforces laws but can be impeached by the legislature and ruled unlawful by the courts.

  行政机构执行法律，但可被立法机构弹劾、被法院裁决违法。

• Judiciary interprets laws but appointments rely on the executive and legislature.

  司法机构解释法律，但法官任命依赖行政与立法机构。

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6. Parliamentary Sovereignty | 议会主权

Under the UK constitution, parliamentary sovereignty means that Parliament is the supreme legal authority and can make or unmake any law. No Parliament can pass laws that future Parliaments cannot change, and there is no entrenched higher law.Parliament ≡ supreme law-making body; no entrenchment; cannot bind successors.

在英国宪法下，议会主权意味着议会是最高法律权威，可制定或废除任何法律。没有任何议会能通过限制未来议会修改的法律，也不存在不可撼动的高级法。议会 ≡ 最高立法主体；法律无刚性保障；本届议会不可约束下届议会。

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7. d’Hondt Method (Seat Allocation) | 汉狄法（席位分配）

The d’Hondt formula is used in many proportional representation systems to allocate seats to parties based on their vote shares. After each seat is awarded, the winning party’s vote total is divided by (seats already won + 1). The party with the highest quotient wins the next seat. Priority quotient = Votes ÷ (s + 1), where s = seats already won.

汉狄法被许多比例代表制国家用于将选票转化为席位。每分配一个席位后，获得该席位的政党得票数除以（已获席位数 s + 1）。商数最高者赢得下一席。优先商数 = 得票数 ÷ (s + 1)，s 为己获席位数。

Party	Votes	/1	/2	/3	Seats
A	100,000	100,000 (1st)	50,000 (3rd)	33,333	2
B	80,000	80,000 (2nd)	40,000	26,667	1
C	50,000	50,000	25,000	16,667	0

在上表中，第一个席位由 A 以 100,000 的商数获得，其票数随即除以2变为50,000；第二个席位由 B 以 80,000 获得；第三个席位再次授予 A，因其商数 50,000 最高。最终 A 获2席，B 获1席，C 无席位。该公式对大党较为有利。

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8. Voter Turnout Formula | 投票率公式

Voter turnout is a measure of political participation. It is calculated as the percentage of registered voters who cast a ballot in a given election. Turnout (%) = (Number of votes cast ÷ Registered electorate) × 100%.

投票率是衡量政治参与程度的一个指标，计算方式为实际投票人数除以登记选民总数，再乘以百分百。投票率 (%) = (投票人数 ÷ 登记选民数) × 100%。

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9. Majority Thresholds | 多数决制门槛

Different electoral systems require different winning margins. An absolute majority demands more than half of all valid votes cast. Absolute majority = (Total valid votes ÷ 2) + 1. In simple plurality (FPTP), the candidate with the most votes wins, even if the total falls far short of a majority.

不同选举制度对胜选门槛有不同要求。绝对多数指获得超过总有效票数的一半再加一票。 绝对多数 = (总有效票数 ÷ 2) + 1。 而在简单相对多数制（先达者胜）中，得票最多者即可当选，哪怕远未过半。

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10. Political Spectrum Model | 政治光谱模型

The conventional left–right spectrum summarises ideological positions. In a simple unidimensional model, positions range from far-left (radical equality) through centre to far-right (traditional hierarchy). Policy preferences can be placed on a continuous scale. Ideology scale: Far Left (―10) ⟷ Centre (0) ⟷ Far Right (+10).

传统的左右光谱用以概括意识形态立场。在简单的一维模型中，立场从极左（激进平等）经中间派延续到极右（传统等级制）。政策偏好可置于一条连续坐标上。意识形态量表

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