Year 10 Cambridge Politics: Quick Reference Handbook of Formulas and Theorems | Year 10 剑桥政治：公式定理速查手册

📚 Year 10 Cambridge Politics: Quick Reference Handbook of Formulas and Theorems | Year 10 剑桥政治：公式定理速查手册

Although politics is not a subject of mathematical formulas, it has its own ‘laws’ and ‘theorems’ – testable propositions that explain how political systems behave. This quick reference handbook collects the essential formulas and theorems for Year 10 Cambridge Politics students, covering electoral systems, power structures, and international relations. Mastering these models will sharpen your analytical skills and help you tackle exam questions with confidence.

政治学虽然不涉及数学公式，但它有自己的“定律”和“定理”——即可检验的命题，用以解释政治制度如何运作。本速查手册汇集了Year 10剑桥政治学课程中必备的核心公式与定理，涵盖选举制度、权力结构和国际关系等领域。掌握这些模型将提升你的分析能力，让你更有信心应对考试。

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1. Duverger’s Law | 杜瓦杰定律

French sociologist Maurice Duverger observed that electoral systems shape party systems. His most famous proposition states that single-member district plurality (first-past-the-post) elections tend to produce a two-party system, while proportional representation encourages multipartyism. This is because in SMP, small parties rarely win seats, so voters concentrate on the two strongest candidates, and elites form broad coalitions before elections.

法国社会学家莫里斯·杜瓦杰观察到选举制度塑造政党制度。他最著名的命题是：单一选区相对多数决（简单多数制）倾向于产生两党制，而比例代表制则鼓励多党制。这是因为在简单多数制下，小党很难赢得席位，选民因此会将选票集中在最强的两位候选人身上，精英们也会在选前结成广泛的联盟。

SMP electoral system → Two-party dominance

PR electoral system → Multiparty system

• In the UK (SMP), Labour and Conservatives dominate, while the Liberal Democrats struggle to convert votes into seats.

  英国（简单多数制）由工党和保守党主导，自民党很难将选票转化为席位。

• In the Netherlands (PR), numerous parties gain representation according to their vote share.

  在荷兰（比例代表制），众多政党根据得票比例获得代表权。

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2. Iron Triangle | 铁三角定理

The iron triangle describes a stable, mutually beneficial relationship between a government agency, a congressional committee, and an interest group. Each corner provides something the others need: the agency gets funding and political support; the committee receives electoral support and information; the interest group gains favourable regulation and contracts. This closed network often resists change and can undermine broader public accountability.

铁三角描述了政府机构、国会委员会和利益集团之间稳定的互利关系。每个角都为其他两方提供所需：政府机构获得资金和政治支持；国会委员会获得选举支持和信息；利益集团则得到有利的监管和合同。这个封闭的网络通常抗拒变革，并可能削弱更广泛的公共问责。

Congressional Committee + Bureaucratic Agency + Interest Group = Policy Subgovernment (Iron Triangle)

For example, the Department of Defense, the House Armed Services Committee, and defence contractors form an iron triangle that steers military spending.

例如，国防部、众议院军事委员会和国防承包商构成了一个影响军事开支的铁三角。

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3. Security Dilemma | 安全困境定理

The security dilemma occurs in international relations when one state seeks to increase its security by building up arms, but this action makes other states feel less secure, prompting them to arm in response. The result can be an arms race that leaves all parties as insecure as before – or worse. This spiral illustrates why defensive actions can be misperceived as offensive threats.

安全困境出现在国际关系中：当一国试图通过扩充军备来增强自身安全时，这种举动却让其他国家感到更不安全，从而促使它们也扩军回应。结果可能导致一场军备竞赛，使所有各方和之前一样——甚至更加——不安全。这一螺旋上升说明为什么防御性行动可能被误解为进攻性威胁。

State A increases defensive arms → State B perceives threat → State B arms → A feels more threatened → Arms race

The Cold War nuclear arms race between the USA and the USSR is a classic case of the security dilemma.

冷战时期美苏之间的核军备竞赛就是安全困境的经典案例。

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4. Democratic Peace Theory | 民主和平论

Democratic peace theory asserts that liberal democracies rarely, if ever, go to war with one another. Explanations include shared democratic norms of compromise, institutional constraints (e.g., the need for public consent or parliamentary approval), and economic interdependence. The theory is not absolute but is a strong statistical pattern in modern history.

民主和平论声称，自由民主国家之间很少（即便有）相互开战。解释包括共同的民主妥协规范、制度制约（如需要公众同意或议会批准）以及经济相互依存。这一定理并非绝对，但在现代史上是一个很强的统计规律。

Democracy A × Democracy B → No armed conflict between them

While democracies have fought non-democracies, the proposition that two established democracies have never fought a full-scale war holds up under scrutiny.

虽然民主国家曾与非民主国家交战，但两个成熟的民主国家从未发生过全面战争这一命题经得起检验。

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5. Logic of Collective Action | 集体行动逻辑

Mancur Olson argued that rational, self-interested individuals will not voluntarily act to achieve their common or group interests unless the group is small or there are selective incentives. In large groups, individuals can free-ride on the efforts of others, so collective action fails without coercion or special benefits. This explains why some interests organise effectively while others remain latent.

曼瑟尔·奥尔森指出，理性的自利个体不会自愿为实现共同或集体利益而行动，除非集团规模很小或存在选择性激励。在大集团中，个人可以搭便车享受他人的努力成果，因此若无强制手段或特殊利益，集体行动便会失败。这解释了为什么某些利益能够有效组织起来，而另一些却始终潜伏。

Large group + common interest → Free riding → No collective action (without selective incentives)

A trade union offers selective benefits (legal advice, insurance) to encourage membership, overcoming the free-rider problem.

工会通过提供选择性福利（法律咨询、保险）来鼓励人们加入，从而克服搭便车问题。

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6. Median Voter Theorem | 中间选民定理

In a two-candidate election with a single left-right policy dimension, if voters have single-peaked preferences and voting is sincere, both candidates will converge to the position of the median voter. This theorem, formulated by Duncan Black and popularised by Anthony Downs, predicts centrist campaigns and explains why major parties often sound similar.

在两候选人竞选、政策维度为一维左右谱系且选民偏好单峰并诚实投票的情况下，两位候选人都会向中间选民的立场靠拢。这一定理由邓肯·布莱克提出、安东尼·唐斯推广，预测了竞选中的中间路线，并解释了为什么主要政党常常听起来很相似。

Candidate A’s platform, Candidate B’s platform → Median voter’s ideal point

In US presidential elections, candidates often shift to the centre after the primaries to capture the median voter.

在美国总统选举中，候选人在初选后通常会转向中间立场，以争取中间选民。

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7. Separation of Powers | 权力分立原则

The separation of powers divides government authority into three branches – legislative, executive, and judicial – to prevent the concentration of power and provide checks and balances. Each branch has distinct functions and can limit the others (e.g., presidential veto, judicial review, legislative oversight). This model is foundational in the US Constitution and analogous systems.

权力分立将政府权力划分为立法、行政和司法三个分支，以防止权力集中并实现制衡。每个分支都有不同的职能，并能制约其他分支（如总统否决权、司法审查、立法监督）。这一模型是美国宪法及类似制度的基础。

Legislative ←→ Executive ←→ Judicial (checks and balances)

Branch	Power	Check on others
Legislative	Makes laws	Can override veto, approve appointments, impeach
Executive	Enforces laws	Veto legislation, appoint judges
Judicial	Interprets laws	Declare laws unconstitutional

Each branch can block actions by the others, ensuring no single branch dominates.

每一个分支都能阻止其他分支的行动，确保没有任何单一分支可以独大。

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8. Federalism Formula | 联邦制公式

Federalism is a system in which sovereignty is constitutionally divided between a central government and regional units (states, provinces, cantons). Both levels have direct authority over citizens and can make laws in specified areas. The division is often enshrined in a written constitution, and disputes are resolved by a supreme court.

联邦制是一种主权在宪法上由中央政府和区域单位（州、省、州）划分的制度。两级政府都对公民拥有直接权威，并可在特定领域制定法律。这种划分通常被写入成文宪法，争议由最高法院解决。

Federal System = National Government + State Governments (shared sovereignty) → Dual polity

Examples include the United States, Germany, India, and Australia. In these systems, powers like defence are central, while education is regional.

实例包括美国、德国、印度和澳大利亚。在这些制度下，国防等权力属于中央，而教育则属于地方。

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9. Electoral Threshold | 选举阈值定理

In proportional representation systems, parties often must surpass a minimum percentage of the national vote to win any seats. This electoral threshold prevents extreme fragmentation, but it can also exclude smaller parties and lead to wasted votes. Common thresholds range from 3% to 5%.

在比例代表制下，政党通常必须越过全国得票率的最低门槛才能获得任何席位。这一选举阈值可以防止极端碎片化，但也可能将小党排除在外并导致选票浪费。常见的阈值在3%至5%之间。

Vote share < Threshold (e.g., 5%) → No representation in parliament

Germany’s 5% threshold has historically kept far-right and far-left splinter groups out of the Bundestag, except when parties win three constituency seats to bypass it.

德国5%的门槛一直将极右和极左的小团体挡在联邦议院门外，除非政党赢得三个选区席位以绕过该门槛。

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10. Political Spectrum Scale | 政治光谱尺度

The left–right political spectrum simplifies ideologies into a continuum. The left generally emphasises equality, social justice, and government intervention in the economy, while the right stresses individual liberty, free markets, and tradition. Modern variations add a second axis (e.g., libertarian vs. authoritarian) to capture nuances.

左右政治光谱将意识形态简化为一条连续轴线。左翼通常强调平等、社会正义和政府干预经济，而右翼则强调个人自由、自由市场和传统。现代版本增加了第二轴（如自由意志与威权主义）以捕捉更多细微差别。

Political ideology = f(preference for equality vs. liberty)

• Left: More government ownership, progressive taxation, strong welfare state.

  左翼：更多公有制、累进税制、强大的福利国家。

• Right: Lower taxes, deregulation, traditional social values.

  右翼：低税收、放松管制、传统社会价值。

• Extremes: Far-left (communism) and far-right (fascism) often share authoritarian methods.

  极端：极左（共产主义）和极右（法西斯主义）往往具有威权主义的手段。

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11. Social Contract Theory | 社会契约论定理

Social contract theory holds that individuals consent, explicitly or implicitly, to surrender some freedoms and submit to authority in exchange for protection of their remaining rights. Philosophers like Hobbes, Locke, and Rousseau used this concept to justify the legitimacy of government. If a ruler breaks the contract, citizens may have a right to revolt.

社会契约论认为，个人明示或默示地同意放弃某些自由并服从权威，以换取对其剩余权利的保护。霍布斯、洛克和卢梭等哲学家利用这一概念为政府的合法性辩护。如果统治者违反了契约，公民有权反抗。

Consent of the governed → Legitimate authority

Locke argued that government must protect life, liberty, and property; when it fails, the contract is void and revolution is justified.

洛克主张政府必须保护生命、自由和财产；若政府未能做到，契约即告无效，革命便是正当的。

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12. Prisoner’s Dilemma in Politics | 政治中的囚徒困境定理

The prisoner’s dilemma is a game-theoretic model showing why two rational actors might not cooperate even if it appears in their best interest. In politics, it is used to explain arms races, trade wars, and failure to provide public goods. Mutual defection yields a worse outcome than mutual cooperation, but the structure of incentives leads to defection.

囚徒困境是一种博弈论模型，它表明两个理性行为体为何可能不合作，即便合作看似符合他们的最佳利益。在政治学中，它被用来解释军备竞赛、贸易战和公共物品供给失败。相互背叛导致的结果比相互合作更糟，但激励结构却驱使人们选择背叛。

	A cooperates	A defects
B cooperates	Both gain moderate reward (10,10)	A gains large, B loses (15,1)
B defects	A loses, B gains large (1,15)	Both lose (5,5)

Individual rationality → Mutual defection → Suboptimal collective outcome

In trade negotiations, two countries might both raise tariffs to protect domestic industries even though a free-trade deal would make both better off.

在贸易谈判中，两个国家可能都提高关税以保护本国产业，即便自由贸易协定能让双方获益更多。

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