📚 A-Level Eduqas Politics: Formula & Theorems Quick Reference Handbook | A-Level Eduqas 政治：公式定理速查手册
This quick reference handbook brings together the key quantitative formulas, electoral rules, and empirical laws that feature in the Eduqas A-Level Politics course. Whether you are studying UK politics, political ideas, or comparative US government, understanding these tools will strengthen your analysis of voting systems, party competition, and the distribution of power. Use this guide to revise essential concepts and their mathematical expressions, all presented side‑by‑side in English and Chinese.
本速查手册汇集了 Edexcel (Eduqas) A-Level 政治课程中关键的量化公式、选举规则与经验定律。无论你正在学习英国政治、政治思想还是比较美国政府，掌握这些工具都能提升你对投票制度、政党竞争与权力分配的分析能力。此指南以中英双语对照呈现核心概念及其数学表达式，供你快速复习。
1. Duverger’s Law | 迪韦尔热定律
Duverger’s Law states that the single‑member plurality (first‑past‑the‑post) electoral system tends to produce a two‑party system, whereas proportional representation encourages multi‑party competition. The psychological and mechanical effects discourage voters from ‘wasting’ their ballots on smaller parties and funnel support towards the two main contenders.
迪韦尔热定律指出，单一选区简单多数制（领先者当选）倾向于产生两党制，而比例代表制则鼓励多党竞争。心理效应与机械效应会阻止选民把选票“浪费”在小党身上，将支持集中到两大主要政党。
FPTP → two‑party system | PR → multi‑party system
2. The Cube Law | 立方律
Originally observed in UK general elections, the cube law describes the relationship between the ratio of seats won by two major parties and the ratio of votes they receive. Because of the winner‑takes‑all geography, a small vote lead is amplified into a much larger seat advantage. The formula is approximated by:
最初在英国大选中观察到的立方律描述了两大政党所获席位比与其得票比之间的关系。由于赢者通吃的地理分布，微小的得票领先会被放大为巨大的席位优势。其近似公式为：
S₁ / S₂ = (V₁ / V₂)³
where S₁, S₂ represent seats for parties 1 and 2, and V₁, V₂ represent their respective vote totals. The cube law has weakened in recent decades due to more volatile voting patterns and regional fragmentation.
其中 S₁, S₂ 代表政党一与政党二的席位数，V₁, V₂ 代表两党各自的总得票数。由于投票模式更加波动且地域碎片化加剧，近几十年来立方律的效应有所减弱。
3. Effective Number of Parties | 有效政党数
Developed by Laakso and Taagepera (1979), the effective number of parties translates the raw count of parties into a weighted index that accounts for their relative size. It is a vital measure for comparing party systems across countries or over time. The formula is:
由拉克索与塔格佩拉（1979）提出的有效政党数，将原始的政党数目转化为考虑政党相对规模的加权指数。这是比较不同国家或不同时期政党体系的关键指标。其计算公式为：
N = 1 / Σ pᵢ²
where pᵢ is the vote share (or seat share) of party i expressed as a decimal. A two‑party system typically records an effective number near 2.0, whereas a fragmented multi‑party system yields a higher value, often above 4.0.
其中 pᵢ 为政党 i 的得票率（或席位率），以小数表示。两党制的有效政党数通常接近2.0，而碎片化的多党制则会产生较高的数值，常高于4.0。
4. Iron Law of Oligarchy | 寡头铁律
Robert Michels’ iron law of oligarchy asserts that all large organisations, including political parties, inevitably become ruled by a small elite, regardless of how democratic they intend to be. The necessity of organisational efficiency, specialised knowledge, and bureaucratic hierarchy concentrates power at the top.
米歇尔斯的寡头铁律断言，一切大型组织（包括政党）最终都不可避免地由少数精英统治，无论其初衷多么民主。组织效率的需求、专业知识以及官僚等级制度将权力集中到上层。
Democracy → Oligarchy (in large organisations)
This theorem is often used to critique the internal dynamics of major UK parties, where central leadership frequently dominates policy‑making despite grassroots participation.
这个定理常被用来批判英国主要政党的内部运作：尽管有草根参与，中央领导层往往主导政策制定。
5. D’Hondt Method | 德洪特法
The D’Hondt method (Jefferson method) is a highest‑averages formula used to allocate seats under party‑list proportional representation, such as in elections to the European Parliament (formerly for the UK) and the Scottish Parliament (regional lists). Votes for each party are divided successively by 1, 2, 3, … after every seat won. Seats are awarded to the largest resulting quotients.
德洪特法（杰斐逊法）是一种最高平均数公式，用于在政党名单比例代表制下分配席位，例如（英国前）欧洲议会选举和苏格兰议会（地区名单）选举。各政党得票数在每赢得一个席位后，依次除以1, 2, 3, …。席位授予所得商数最大的那一方。
Q = V / (s + 1)
where V is the party’s vote total and s is the number of seats already allocated to that party. D’Hondt slightly favours larger parties compared to other PR formulas.
其中 V 为该政党的总得票数，s 为该政党已分配到的席位数。与其他比例代表制公式相比，德洪特法略微偏向大党。
6. Sainte‑Laguë Method | 圣拉格法
The Sainte‑Laguë method (Webster method) is another highest‑averages system, but its divisors are odd numbers: 1, 3, 5, 7, … This makes it more proportional to votes and less favourable to large parties than D’Hondt. Some systems use a modified version with a first divisor of 1.4 to create a higher threshold for very small parties.
圣拉格法（韦伯斯特法）是另一种最高平均数制度，但它的除数是奇数：1, 3, 5, 7, …。这使得席位分配与得票比例更成比例，且不如德洪特法偏向大党。有些国家采用修正版，将第一个除数设为1.4，为极小的政党设置更高门槛。
Q = V / (2s + 1)
Used, for example, in New Zealand’s MMP system and some German regional elections, Sainte‑Laguë is favoured when the goal is high representational accuracy.
圣拉格法被应用于新西兰的混合成员比例制及德国部分州选中。当追求高代表性准确度时，该法更受青睐。
7. Largest Remainder Method | 最大余数法
The largest remainder method relies on a quota rather than a divisor series. The most common is the Hare quota: total valid votes divided by total seats. Parties receive one seat for each full quota they achieve. Remaining seats are distributed to the parties with the largest vote remainders.
最大余数法依赖于配额而非除数序列。最常见的是黑尔配额：总有效票数除以总席位数。各政党每获得一个完整配额即分得一席。剩余席位则按照余数大小依次分配给各党。
Hare quota = Total votes / Total seats
This method is used in some forms of the Single Transferable Vote (STV) and in countries such as Denmark. It tends to help smaller parties, but can occasionally produce frustratingly counter‑intuitive results due to the ‘remainder paradox’.
该方法用于单记可让渡投票制的某些形式及丹麦等国。它往往有利于小党，但偶尔会因“余数悖论”产生违反直觉的结果。
8. Electoral Threshold | 选举门槛
Many proportional systems have a legal vote threshold (Sperrklausel) that a party must cross to win any seats. Typically set at 5% of the national vote, this rule limits parliamentary fragmentation and discourages extremist micro‑parties. In some mixed systems, constituency seat winners bypass the threshold.
许多比例代表制设有政党必须跨越的法定得票门槛，通常设定为全国得票率的5%，以限制议会碎片化并阻止极端微型政党。在某些混合选举制度中，赢得选区席位的政党可绕过该门槛。
Party vote% ≥ Threshold ⇒ eligible for seats
For Eduqas, it is essential to recognise that Germany’s 5% threshold (or alternatively three constituency seats) and Turkey’s 7% (formerly 10%) threshold shape party competition and strategic voting.
对Eduqas政治考试而言，重要的是认识到德国5%的门槛（或替代条件：赢得三个选区席位）与土耳其7%（此前为10%）的门槛如何塑造政党竞争与策略投票。
9. Arrow’s Impossibility Theorem | 阿罗不可能定理
Kenneth Arrow’s theorem proves that no voting system can simultaneously satisfy a set of reasonable, fair conditions when there are three or more candidates. The conditions include unrestricted domain (voters may rank candidates in any order), non‑dictatorship, Pareto efficiency, and independence of irrelevant alternatives.
阿罗定理证明，当有三名或更多候选人时，任何投票制度都无法同时满足一套合理、公平的条件。这些条件包括无限制偏好（选民可任意排序候选人）、非独裁、帕累托有效与无关选择独立性。
No ranked voting rule satisfies all fairness axioms simultaneously.
This informs debates about electoral reform in the UK, highlighting that there is no mathematically perfect system, only trade‑offs between different democratic values.
这为英国的选举改革辩论提供了依据，体现了不存在数学上完美的制度，只有不同民主价值之间的权衡。
10. Condorcet Paradox | 孔多塞悖论
The Condorcet paradox shows that collective preferences can be cyclical even when individual preferences are transitive. In a three‑way contest, it is possible for a majority to prefer A over B, B over C, and C over A, meaning that no single alternative can beat all others in a head‑to‑head vote – the ‘Condorcet winner’ disappears.
孔多塞悖论表明，即使个体偏好具有传递性，集体偏好也可能出现循环。在一场三方角逐中，可能出现多数选民认为A优于B、B优于C，却又认为C优于A的情况，使得没有单一选项能在两两对决中击败所有其他选项——“孔多塞胜者”并不存在。
A > B, B > C, C > A ⇒ no Condorcet winner
This paradox is crucial for understanding why agenda‑setting and the choice of voting method can dramatically influence electoral outcomes, and it supports the rationale for preferential voting designs.
此悖论对理解议程设定与投票方法的选择如何戏剧性地影响选举结果至关重要，也为优先顺序投票设计提供了理论依据。
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