📚 Year 11 Cambridge Politics: Formulas and Theorems Quick Reference Handbook | 剑桥十一年级政治：公式定理速查手册
Political science, like mathematics, has its own set of 'formulas' and 'theorems' that help us understand how governments, elections, and international relations operate. This quick reference handbook collects key models, rules of thumb, and formal relationships that Cambridge IGCSE Politics students need to recall under timed conditions. From electoral quotas to the principles of separation of powers, each entry is presented as a concise statement followed by its logical meaning.
政治学与数学一样，也有自己的“公式”和“定理”，帮助我们理解政府、选举和国际关系的运作方式。这份速查手册汇集了剑桥 IGCSE 政治学生需要在限时条件下回忆的关键模型、经验法则和形式化关系。从选举配额到权力分立原则，每个条目都以简明陈述呈现，并附上其逻辑含义。
1. Droop Quota Formula | Droop 配额公式
The Droop quota is used in Single Transferable Vote (STV) elections to determine the minimum number of votes a candidate needs to be elected. It ensures that no more candidates can reach the quota than there are seats to fill.
Droop 配额用于单一可转移投票（STV）选举，以确定候选人当选所需的最低票数。它确保达到配额的候选人不会多于应选议席。
Q = ( V ÷ ( S + 1 ) ) + 1
Here, Q is the quota, V is the total number of valid votes cast, and S is the number of seats to be filled. The formula is intentionally designed to be one vote more than the minimum needed to fill all seats absolutely.
此处 Q 为配额，V 为有效票总数，S 为待填补的议席数。该公式被特意设计为比完全填满所有议席所需的最低票数多一票。
otes is guaranteed election and any surplus is transferred according to voter preferences.
如果一个选区有 1000 名选民投下有效票，共有 4 个议席，则 Droop 配额为 (1000 ÷ (4+1)) + 1 = 200 + 1 = 201 票。任何获得 201 票的候选人都保证当选，剩余选票则按选民的偏好转移。
2. Hare Quota Formula | 赫尔配额公式
The Hare quota is the simplest quota used in largest remainder proportional representation and some STV systems. It represents the exact number of votes per seat if all votes were equally distributed.
赫尔配额是用于最大余数比例代表制和某些 STV 系统的最简单配额。它代表了如果所有选票平均分配的话每个议席的精确票数。
Q = V ÷ S
V is the total number of valid votes and S is the number of seats. Unlike the Droop quota, the Hare quota does not add an extra vote, which can make it harder to fill all seats without remainders.
V 是有效票总数，S 是议席数。与 Droop 配额不同，Hare 配额不额外加票，这使得可能不容易在没有余数的情况下填满全部议席。
a higher threshold that often requires subsequent rounds of surplus transfers or the use of the largest remainder method to allocate remaining seats.
在同样的 1000 票 4 席例子中，Hare 配额为 1000 ÷ 4 = 250 票，门槛更高，常常需要后续的剩余票转移轮次或使用最大余数法来分配剩下的议席。
3. Proportional Seat Allocation Formula | 比例代表制席位分配公式
In pure proportional representation, each party receives a number of seats directly proportional to its share of the total vote. This formula translates percentage support into legislative seats.
在纯粹的比例代表制中，每个政党获得的席位数与其得票占总票数的份额直接成正比。这个公式将得票百分比转化为立法席位。
SP = ( VP ÷ Vtotal ) × Stotal
SP is the number of seats for party P, VP is the votes won by that party, Vtotal is the total valid votes, and Stotal is the total number of seats to be allocated.
SP 为政党 P 的席位数，VP 为该政党赢得的票数，Vtotal 为有效票总数，Stotal 为待分配的总议席数。
For example, if a party gains 30% of votes in a 100-seat legislature elected by pure PR, it would theoretically obtain 30 seats. Real-world systems may use rounding methods or thresholds, but the core formula remains proportional.
例如，如果某政党在一个采用纯粹比例代表制的 100 席立法机构中获得 30% 的选票，理论上它将获得 30 个席位。现实中的制度可能采用取整方法或设置门槛，但核心公式仍是成比例的。
4. Winner-Takes-All Theorem | 赢者通吃定理
The winner-takes-all theorem describes Single-Member Plurality (SMP) systems, where the candidate with the most votes wins the seat, and all other votes are effectively discarded. This is the operating principle behind First-Past-the-Post (FPTP).
赢者通吃定理描述了单名多数制（SMP），即得票最多的候选人赢得席位，而其他选票实际上被丢弃了。这是简单多数制（FPTP）背后的运作原则。
The theorem states that a party can win a majority of seats with far fewer than 50% of the popular vote, and small parties with geographically dispersed support are systematically disadvantaged. It creates a 'manufactured majority'.
这条定理表明，一个政党能以远低于 50% 的普选票赢得多数席位，而在地理上分散获支持的小党则系统性地处于劣势。它创造了“人造多数”。
For instance, in the 2019 UK General Election, the Conservative Party won 56% of seats with only 43.6% of the vote. This disproportionality is a direct outcome of the winner-takes-all mechanism in each constituency.
例如，在 2019 年英国大选中，保守党仅以 43.6% 的得票率赢得了 56% 的席位。这种不成比例性正是每个选区赢者通吃机制的直接结果。
5. Duverger's Law | 杜瓦杰定律
Duverger's Law is a well-known theorem in political science that links electoral systems to party systems. It states that single-member plurality elections tend to produce a two-party system.
杜瓦杰定律是政治学中的一个著名定理，它将选举制度与政党制度联系起来。定律指出，单名多数选举往往产生两党制。
The psychological and mechanical effects drive this outcome: voters do not want to waste their votes on small parties (psychological), and the electoral formula itself disadvantages small parties (mechanical). Over time, this dual pressure squeezes out third parties.
心理和机械效应驱动了这一结果：选民不想把选票浪费在小党身上（心理效应），而选举公式本身对小党不利（机械效应）。随着时间推移，这一双重压力将第三党排挤出去。
Countries using FPTP, such as the United States and (to a large extent) the United Kingdom, demonstrate this pattern, whereas proportional representation systems tend to support multiparty systems, as noted in Duverger's hypothesis.
采用 FPTP 的国家，如美国和（在很大程度上的）英国，展示了这种模式；而比例代表制则倾向于支持多党制，正如杜瓦杰假说所指出的那样。
6. Effective Number of Parties Formula | 有效政党数公式
The effective number of parties index measures how many parties are genuinely relevant in a party system, weighting them by their size. It refines the raw count by giving larger parties greater weight.
有效政党数指数衡量一个政党体制中真正具有影响力的政党数量，并根据其规模进行加权。它通过对较大政党赋予更大权重来精炼原始的政党计数。
N = 1 / Σ pi²
In this formula, pi represents each party's share of seats (expressed as a decimal fraction), and Σ pi² is the sum of the squares of those shares. A result close to 2.0 indicates a two-party system; a higher number signals a more fragmented multiparty system.
在这个公式中，pi 代表每个政党的席位份额（用小数表示），Σ pi² 是这些份额平方之和。结果接近 2.0 表示两党制；数字越大，说明多党制越碎片化。
For a parliament where three parties hold 45%, 40% and 15% of seats, pi 0.0225, summing to 0.385. N = 1 / 0.385 ≈ 2.60, indicating a two-and-a-half party system.
对于一个三党分别拥有 45%、40% 和 15% 席位的议会，pi² 值分别为 0.2025、0.16 和 0.0225，总和为 0.385。N = 1 / 0.385 ≈ 2.60，表明一个两党半体制。
7. Separation of Powers Theorem | 权力分立定理
The separation of powers theorem, formulated by Montesquieu, establishes that the legislative, executive and judicial branches of government should be distinct institutions with separate personnel and functions. This prevents the concentration of power and safeguards liberty.
由孟德斯鸠提出的权力分立定理确认，政府的立法、行政和司法分支应该是不同的机构，拥有不同的人员和职能。这可以防止权力集中并保障自由。
paration( L, E, J ) ), where the function defines that no organ may exercise the whole power of another. Pure separation hardly exists, but its essence shapes constitutional design.
在规范解读中，此定理像一个公式：自由 = f( 分立(L, E, J) )，其中函数规定没有任何机关能够行使另一机关的全部权力。纯粹的分立几乎不存在，但其要义塑造了宪制设计。
The United States presidential system implements the theorem with a strict separation of institutions: the President (Executive) cannot dissolve Congress (Legislative), and judges are appointed for life, reinforcing independence.
美国的总统制以严格的机构分立贯彻这一原理：总统（行政）不能解散国会（立法），法官终身任职，强化了独立性。
8. Checks and Balances Theorem | 制衡定理
While separation divides power, the checks and balances theorem prescribes that each branch must have limited oversight and control over the others. It creates relationships of mutual accountability, often expressed as a cycle of vetoes and overrides.
分立是为了划分权力，而制衡定理则规定每个分支必须对其他分支有有限的监督和控制。它创造了相互问责的关系，常表现为否决与推翻的循环。
re checks include presidential veto, legislative override, judicial review, and confirmation of appointments. The balance prevents any single branch from dominating.
一个简化定理指出：良治 = 分立 + 制衡，其中制衡包括总统否决、立法推翻、司法审查以及任命确认。这种平衡可防止任何一个分支独大。
For Cambridge IGCSE, a key example is the American system: Congress passes bills, but the President can veto; the President appoints judges, but the Senate confirms; courts can declare laws unconstitutional, but the Constitution can be amended by supermajorities.
对剑桥 IGCSE 而言，关键例子是美国体制：国会通过法案，总统可以否决；总统任命法官，参议院确认；法院可以宣布法律违宪，但宪法可由超多数修改。
9. Social Contract Theorem | 社会契约论定理
The social contract theorem explains the origin of government legitimacy. It states that individuals agree, explicitly or implicitly, to surrender some freedoms and submit to authority in exchange for security and order. In formulaic terms, consent transforms power into authority.
社会契约论定理解释了政府合法性的起源。它阐明，个人明确或默认同意放弃部分自由并服从权威，以换取安全和秩序。用公式化的语言来说，同意将权力转化为权威。
Hobbes' l rights (life, liberty, property), making government conditional on continued consent. Rousseau's General Will represents the collective sovereign.
霍布斯的版本提出：绝对主权 = 生命保全 / 自然状态。洛克调整了等式：政府权力 = 保护自然权利（生命、自由、财产），使政府以持续同意为条件。卢梭的公意代表集体主权。
These theorems are examined through questions about legitimacy and the right to rebel. Students must be ready to apply the formula: If sovereign breaches protection, then contract is void and citizens may exercise the right to depose tyranny.
这些定理通过关于合法性和反抗权的问题进行考查。学生必须准备好应用这个公式：如果主权者破坏了保护，则契约失效，公民可行使推翻暴政的权利。
10. Left-Right Political Spectrum Model | 左右翼政治光谱模型
The traditional unidimensional political spectrum positions ideologies on a left-right axis according to attitudes toward economic equality and state intervention. While not a precise mathematical formula, the model functions as a mapping function of policy preferences.
传统的单维政治光谱根据对经济平等和国家干预的态度，将意识形态置于左右翼坐标轴上。虽然不是精确的数学公式，但该模型起到了政策偏好映射函数的作用。
vidual responsibility. Contemporary models often add a social dimension (libertarian-authoritarian axis) to form a two-dimensional political compass.
基本版本：左翼 = 高度国家干预 + 公有制；右翼 = 自由市场 + 个人责任。当代模型常加入社会维度（自由意志-权威主义轴）形成一个二维政治指南针。
| Position | Economic stance | Example ideology |
|---|---|---|
| Far left | Full state ownership | Communism |
| Centre left | Welfare state, regulation | Social democracy |
| Centre right | Market economy, low tax | Conservatism |
| Far right | Minimal state, traditional values | Libertarianism / authoritarian nationalism |
Students should use the spectrum not as a rigid formula but as a reference tool to compare how parties, such as Labour and Conservative in the UK, position themselves on economic and social issues.
学生应将光谱用作参考工具，而非死板的公式，用以比较各政党（如英国的工党和保守党）如何在经济与社会议题上自我定位。
11. Balance of Power Theorem | 力量平衡定理
In international relations, the balance of power theorem asserts that national security is best preserved when military power is distributed so that no one state dominates. It can be formulated as a systemic equilibrium: peace is maintained when alliances counteract any rising hegemon.
在国际关系中，力量平衡定理断言，当军事力量得以分散、使得没有任何一国能占主导时，国家安全得以最好地维护。它可以被表述为一种系统性均衡：当联盟能制衡任何崛起的霸权时，和平得以维持。
iance). This is the logic that drove the Concert of Europe and Cold War bipolar balancing.
经典版本通过这样一种机制运作：如果 力量(A) > 力量(B)，那么 B 将构建联盟（B + C + D）直到 力量(A) ≈ 力量(联盟)。这正是驱动维也纳体系和冷战两极平衡的逻辑。
ence fails if either factor is zero. Cambridge exams may ask how the balance of power theory applies to multilateral organisations such as the UN Security Council.
该定理也解释了为什么超级大国会进行军备竞赛。一个简化的战略公式是：威慑 = 能力 × 可信度。若任一因素为零，威慑便失效。剑桥考试可能问道，力量平衡理论如何适用于联合国安理会等多边组织。
12. Federal Distribution of Powers Principle | 联邦分权原则
t competences. The general formula is: Political authority = Federal powers + State powers – Concurrent powers, where concurrent
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