A-Level CAIE Politics: Formula and Theorem Quick Revision Guide | A-Level CAIE 政治：公式定理速查手册

📚 A-Level CAIE Politics: Formula and Theorem Quick Revision Guide | A-Level CAIE 政治：公式定理速查手册

This quick revision guide collects the ‘formulas’ and ‘theorems’ frequently tested in CAIE A-Level Politics. They are not mathematical equations but concise analytical devices that help you define, compare and evaluate key concepts across the syllabus. Use them to turn detailed case studies into sharp exam answers.

本速查手册汇集了 CAIE A-Level 政治中常考的 ‘公式’ 与 ‘定理’。它们不是数学方程，而是简洁的分析工具，帮助你定义、比较和评价大纲中的核心概念。用它们把详细案例转化为清晰的考试答案。

1. Power, Authority and Legitimacy | 权力、权威与合法性

In CAIE Politics, power is the capacity to influence or coerce others, while authority is power that is accepted as rightful. Legitimacy is the belief that a ruler or institution has the right to govern. Weber identified three ideal types of authority: traditional, charismatic and legal-rational.

在 CAIE 政治中，权力是影响或强制他人的能力，而权威是被接受为正当的权力。合法性是指人们相信统治者或机构有统治权利的信念。韦伯提出了三种理想型权威：传统型、魅力型和法理型。

Authority = Power + Legitimacy

Traditional authority rests on established customs, charismatic authority on personal qualities, and legal-rational authority on constitutional rules and office. These categories explain obedience, revolution and state stability in both the UK and the USA.

传统型权威建立在既有习俗之上，魅力型权威依靠个人品质，法理型权威则基于宪法规则和职位。这些类型有助于解释英国和美国的服从、革命与国家稳定。

2. Sovereignty and the State | 主权与国家

The modern state is defined by four features: territory, population, government and sovereignty. Sovereignty has two dimensions: internal supremacy over all other associations and external recognition by other states. In the UK, legal sovereignty lies with Parliament.

现代国家由四个特征界定：领土、人口、政府和主权。主权有两个维度：对内高于一切其他团体，对外获得其他国家承认。在英国，法律主权属于议会。

State = Territory + Population + Government + Sovereignty

Dicey’s doctrine of parliamentary sovereignty states that Parliament can make or unmake any law, and no person or body has the right to override or set aside legislation. Political sovereignty, however, may be said to lie with the electorate at general elections.

戴雪的议会主权学说认为，议会可以制定或废除任何法律，任何个人或机构都无权推翻或搁置立法。然而，政治主权可以说在大选中属于选民。

3. Democracy: Direct and Representative | 民主：直接民主与代议制民主

Democracy combines popular control over public decision-making with political equality among citizens. Direct democracy involves citizens voting on policy questions themselves, as in referendums; representative democracy involves choosing representatives through regular, free and fair elections.

民主将人民对公共决策的控制与公民之间的政治平等结合起来。直接民主由公民自己对政策问题投票，如全民公投；代议制民主通过定期、自由和公正的选举选择代表。

Democracy = Participation + Competition + Civil Liberties + Accountability

Liberal democracy adds the rule of law, protection of minority rights and limited government. Direct democracy can increase legitimacy but may be slow and impractical on a large scale; representative democracy allows effective governance but creates a principal-agent problem between voters and politicians.

自由民主制还增加了法治、保护少数权利和有限政府。直接民主可以提高合法性，但可能在大范围内缓慢且不切实际；代议制民主能够有效治理，但会在选民与政治家之间产生委托-代理问题。

4. Electoral Systems and Key Formulas | 选举制度与核心公式

Electoral systems are procedures for converting votes into seats. The three main families are majoritarian, proportional and mixed. Key formulas include seat share, the effective number of parties and the Gallagher Index of disproportionality.

选举制度是将选票转化为席位的程序。三大类型是多数制、比例代表制和混合制。关键公式包括席位份额、有效政党数和加拉格尔非比例性指数。

Seat share (%) = (Votes for party ÷ Total valid votes) × 100

Effective number of parties (ENP) = 1 ÷ (p₁² + p₂² + p₃² + …)

Gallagher Index = √( ½ Σ (vᵢ − sᵢ)² )

In the Gallagher Index, vᵢ is the vote share and sᵢ is the seat share of each party. A result close to 0 means high proportionality; a high score means the system distorts the popular vote. FPTP usually produces a high Gallagher Index, while list PR produces a low one.

在加拉格尔指数中，vᵢ 是每个政党的得票率，sᵢ 是其席位份额。结果越接近 0 表示比例性越高；得分越高表示制度扭曲了普选票。简单多数制通常产生较高的加拉格尔指数，而名单比例代表制则产生较低的指数。