📚 A-Level AQA Politics: Formula & Theorem Quick Reference Handbook | A-Level AQA 政治：公式定理速查手册
In AQA A-Level Politics, mastering key formulas and theorems is essential for analysing electoral systems, power distributions, and political behaviour. This quick reference handbook compiles the most important quantitative and theoretical 'equations' you need to revise, from Droop quota to the median voter theorem, ensuring you can apply them confidently in exams and essays.
在AQA A-Level政治学科中，掌握关键公式与定理对于分析选举制度、权力分配和政治行为至关重要。这本速查手册汇编了复习所需的最重要的量化与理论“方程式”，从德鲁普配额到中间选民定理，确保你能够自信地在考试和论文中运用它们。
1. Droop Quota and STV | 德鲁普配额与单一可转移投票制
The Droop quota determines the minimum number of votes needed for a candidate to be elected under the Single Transferable Vote (STV) system. It minimises wasted votes and ensures proportional outcomes. The formula is:
德鲁普配额决定了在单一可转移投票制下候选人当选所需的最低票数。它能最大限度地减少浪费的选票并确保比例结果。其公式为：
Quota = (Total valid votes ÷ (Seats + 1)) + 1
) + 1 = 10,001. Any candidate reaching or exceeding this threshold in any count is elected, and surplus votes are transferred according to voter preferences.
计算结果总是向下取整至最接近的整数。例如，在争夺5个席位、拥有6万张有效票的选举中，配额为(60000 ÷ (5+1)) + 1 = 10001票。任何候选人在任一轮计票中达到或超过这一门槛即告当选，多余选票则按选民偏好转移给其他候选人。
STV is used for the Northern Ireland Assembly and Scottish local elections. Understanding the quota is crucial for explaining how smaller parties can gain representation under proportional systems.
STV制用于北爱尔兰议会和苏格兰地方选举。理解配额对于解释小党如何在比例代表制下获得议席至关重要。
2. Hare Quota and Largest Remainder Method | 黑尔配额与最大余数法
The Hare quota is the simplest proportional allocation formula, used in some list-PR systems and for the initial allocation of seats. It is calculated as:
黑尔配额是最简单的比例分配公式，用于一些名单比例代表制以及席位的初步分配。其计算方式为：
Hare Quota = Total valid votes ÷ Total seats
If parties receive vote totals above this quota, they are awarded full quotas automatically. Any remaining seats are distributed using the largest remainder method: each party's remaining votes (remainder) are compared, and seats go to the highest remainders in descending order.
如果政党的得票数超过配额，则自动获得相应整数席位。剩余空缺席位采用最大余数法分配：比较每个政党扣除配额后的余票，按余数从高到低分配席位。
For instance, with 100,000 votes and 10 seats, the Hare quota is 10,000. A party with 25,000 votes gets 2 seats initially and a remainder of 5,000. The party with the highest remainder wins the next seat. This method is simpler than the Droop quota but can produce slightly less proportional results.
例如，10万张选票、10个席位时，黑尔配额为1万票。获得2.5万票的政党最初获得2个席位，余数为5000票。余数最高的政党赢得下一个席位。这种方法比德鲁普配额更简单，但比例性略低。
3. Gallagher Index of Disproportionality | 加拉格尔不比例指数
The Gallagher index (also called the least squares index) measures how closely the distribution of seats matches the distribution of votes in an election. A lower score means a more proportional result. The formula is:
加拉格尔指数（也称最小二乘指数）衡量选举中席位分布与选票分布的匹配程度。得分越低，结果的比例性越强。公式为：
LSq = √(½ Σ(vi – si)²) × 100%
Where vi is the vote share (%) for party i, and si is its seat share (%). The differences are squared, summed, halved, and then the square root is taken. The result is expressed as a percentage; values usually range from 0 to 30.
其中 vi 是政党 i 的得票率（%），si 是其席位率（%）。将差值平方后求和、减半，再取平方根。结果以百分比表示，通常介于0到30之间。
In the 2019 UK general election (FPTP), the Gallagher index was around 11.6, showing significant disproportionality. In contrast, the 2021 Scottish Parliament election (AMS) produced a much lower score, demonstrating how PR systems reduce the gap between votes and seats.
在2019年英国大选（FPTP制）中，加拉格尔指数约为11.6，显示出显著的不比例性。相比之下，2021年苏格兰议会选举（AMS制）的得分要低得多，表明比例代表制如何缩小选票与席位之间的差距。
4. Downs' Median Voter Theorem | 唐斯中间选民定理
Downs' median voter theorem states that in a two-party system with a single-dimensional policy spectrum, rational vote-seeking parties will converge towards the ideological position of the median voter. The party that positions itself closer to the centre wins the majority of votes.
唐斯的中间选民定理指出，在一个单维度政策光谱的两党制中，理性、追求选票的政党会向中间选民的意识形态立场靠拢。定位更靠近中心的政党将赢得多数选票。
Formally, if voters are distributed along a left-right scale and each votes for the party nearest their ideal point, the equilibrium strategy for both parties is to adopt the median voter's favoured policy. This explains why the Conservative and Labour parties often shift towards the centre during general election campaigns.
在形式上，如果选民沿左右光谱分布且每人都投票给离自己理想点最近的政党，那么双方的均衡策略都是采纳中间选民偏好的政策。这就解释了为什么保守党和工党在大选期间常向中间地带靠拢。
However, the theorem assumes perfect information and no entry of third parties. In reality, internal party factions and safe seats can prevent full convergence, as seen when Labour moved sharply left under Jeremy Corbyn.
然而，该定理假设信息完全且不存在第三方进入。现实中，党内派系和安全选区会阻止完全趋同，正如杰里米·科尔宾领导时期工党大幅左转所见。
5. Duverger's Law | 迪韦尔热定律
Duverger's law asserts that single-member plurality (FPTP) electoral systems tend to produce a two-party system. This occurs through two effects: the mechanical effect, where smaller parties win few seats even with substantial votes, and the psychological effect, where voters avoid 'wasting' votes on third parties and instead vote tactically for the least disliked major party.
迪韦尔热定律断言，单一选区相对多数制（FPTP）倾向于产生两党制。这通过两种效应发生：机械效应，即小党即使得票可观也难获席位；心理效应，即选民不愿将选票“浪费”在第三党上，转而策略性地投给自己最不讨厌的大党。
For the UK, the dominance of the Conservatives and Labour until the 2010s seemed to confirm the law. The rise of the Liberal Democrats and the SNP challenges the 'pure' two-party outcome, but FPTP still heavily penalises third parties, making them regionally concentrated.
就英国而言，直到2010年代保守党与工党的主导地位似乎证实了这一定律。自由民主党和苏格兰民族党的崛起对“纯粹”两党制结果提出了挑战，但FPTP制依然严重惩罚第三党，迫使其形成地区集中。
Duverger also noted that proportional representation tends to create multiparty systems. This can be considered the 'Duverger's hypothesis', highlighting the interwoven relationship between electoral systems and party systems.
迪韦尔热还指出，比例代表制倾向于创造多党体系。这可被视为“迪韦尔热假说”，凸显了选举制度与政党制度之间的交织关系。
6. The Cube Law of Elections | 选举立方定律
The cube law is an empirical relationship found in FPTP systems, stating that the ratio of seats won by two parties approximates the cube of the ratio of their votes. The formula is often written as:
立方定律是在FPTP制中发现的一种经验关系，即两大党所获席位之比约等于其得票之比的立方。公式常写作：
S₁ ÷ S₂ = (V₁ ÷ V₂)³
y B, the cube law predicts that Party A will win roughly 1.5³ = 3.375 times as many seats.
其中 S₁ 和 S₂ 是两党赢得的席位数，V₁ 和 V₂ 是它们得到的票数。如果A党得票是B党的1.5倍，立方定律预测A党赢得的席位约是B党的 1.5³ = 3.375 倍。
This 'winner's bonus' exaggerated Labour majorities in 1997 and 2001. While the law has weakened due to increased voter volatility and multi-party competition, it remains a useful shorthand for explaining why FPTP amplifies leads and often produces landslide victories.
这种“胜者红利”在1997年和2001年放大了工党的席位优势。尽管由于选民流动性增强和多党竞争，这一定律已有所减弱，但它仍然是解释为何FPTP制会放大领先优势并经常产生压倒性胜利的简便工具。
7. Rational Choice Voting Model (R = PB – C + D) | 理性选择投票模型
The rational choice model explains the decision to vote using a cost-benefit analysis. The expected utility of voting is expressed as:
理性选择模型运用成本-收益分析来解释投票决策。投票的期望效用表达为：
R = P × B – C + D
P is the probability that one's vote will be decisive; B is the benefit difference between the preferred candidate winning rather than losing; C is the cost of voting (time, effort); and D is the civic duty or expressive benefit derived from voting.
P 是个人的一票成为决定性票的概率；B 是所偏好候选人当选而非落选带来的利益差；C 是投票成本（时间、精力）；D 是从投票行为中获得的公民责任感或表达性收益。
t that rational individuals would not vote. The fact that millions do vote is known as the 'paradox of voting' and is often explained by strong D (duty) values making R positive despite negligible P×B.
由于除极小型选举外 P 值极小，模型似乎预测理性个体不会投票。然而现实中数百万人投票的现象被称为“投票悖论”，其通常解释为强烈的 D（公民义务）值使得 R 为正，抵销了微不足道的 P×B。
8. Shapley-Shubik Power Index | 夏普利-舒比克权力指数
The Shapley-Shub
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