📚 Year 11 SQA Politics: Formula & Theorem Quick Reference Handbook | Year 11 SQA 政治：公式定理速查手册
This quick reference handbook brings together the essential formulas, indices, and theorems you will encounter in the Year 11 SQA Politics course. It is designed to help you revise mathematical aspects of electoral systems, power distributions, voter behaviour, and party systems – all of which are fundamental to National 5 and beyond. Use it alongside your class notes to master the quantitative toolkit of political analysis.
这本速查手册汇集了你在 SQA Year 11 政治课程中将会遇到的核心公式、指数和定理。它旨在帮助你复习选举制度、权力分配、选民行为和政党体系中的数学内容——这些都是 National 5 及更高级别政治学的基础。配合你的课堂笔记使用，掌握政治分析的量化工具。
1. The d’Hondt Method | 德洪特法
The d’Hondt method is a highest‑averages formula used to allocate seats in proportion to votes, especially in party‑list proportional representation systems (for example, Scottish Parliament regional lists). Each party’s votes are divided by a sequence of divisors (1, 2, 3, …). For each round of allocation, the party with the highest quotient wins a seat, and its vote total is then divided by the next divisor.
德洪特法是一种最高平均数公式，用于根据选票比例分配席位，尤其常见于政党名单比例代表制（例如苏格兰议会地区名单）。每个政党的票数逐一除以一组除数（1、2、3……）。在每一轮分配中，商数最高的政党获得一个席位，之后该党的票数继续除以下一个除数。
quotient = party votes ÷ divisor
商数 = 政党票数 ÷ 除数
Example: Party A has 40,000 votes, Party B 30,000, Party C 20,000, with 3 seats to fill. Quotients (votes/1, /2, /3): A 40k, 20k, 13.3k; B 30k, 15k, 10k; C 20k, 10k, 6.7k. The three highest quotients are 40k (A), 30k (B), 20k (A or C). Since 20k appears for both A and C, A’s 20k from votes/2 is compared with C’s 20k from votes/1 – the party with more total votes (A) is often prioritised. In practice, A wins 2 seats, B wins 1 seat, C wins 0.
示例：A 党得票 40,000，B 党 30,000，C 党 20,000，争夺 3 个席位。商数（票数/1、/2、/3）：A 40k、20k、13.3k；B 30k、15k、10k；C 20k、10k、6.7k。三个最高商数为 40k（A）、30k（B）、20k（A 或 C）。由于 A 和 C 均出现 20k，通常优先凭借较高总票数的 A（其 20k 来自票数/2）。结果 A 得 2 席，B 得 1 席，C 得 0 席。
2. The Sainte‑Laguë Method | 圣拉格法
The Sainte‑Laguë method (also known as Webster’s method) works in the same way as d’Hondt, but uses the divisor sequence 1, 3, 5, 7, … instead of 1, 2, 3, … This larger gap between divisors makes it more favourable to smaller parties, as their vote totals are divided by smaller initial numbers and can win seats earlier compared with d’Hondt.
圣拉格法（亦称韦伯斯特法）的工作原理与德洪特法相同，但除数列为 1、3、5、7……而非 1、2、3……除数之间的较大间距使其对较小政党更为有利，因为其票数除以较小的初始数字，相比德洪特法更容易尽早赢得席位。
quotient = party votes ÷ (2s – 1), where s = 1, 2, 3, …
商数 = 政党票数 ÷ (2s – 1)，其中 s = 1, 2, 3, …
Using the same vote totals 40k, 30k, 20k for 3 seats, the first divisors give A/1=40k, B/1=30k, C/1=20k; then A/3≈13.3k, B/3=10k, C/3≈6.7k; then A/5=8k, etc. The highest three quotients are 40k (A), 30k (B), 20k (C). Each party receives one seat – a more proportional outcome than d’Hondt in this scenario.
仍以 40k、30k、20k 的选票争夺 3 席为例，首批除数结果为 A/1=40k、B/1=30k、C/1=20k；接着 A/3≈13.3k、B/3=10k、C/3≈6.7k；再接着 A/5=8k 等。三个最高商数是 40k（A）、30k（B）、20k（C）。每个政党各得一席——在此情景下比德洪特法更具比例性。
3. Quotas in STV: Hare and Droop | 单一可转让投票中的配额：黑尔配额与德鲁普配额
In the Single Transferable Vote (STV) system, candidates need to reach a quota to be elected. Two common quotas are the Hare quota and the Droop quota. The Hare quota is the simplest: total valid votes divided by the number of seats to be filled. The Droop quota is designed to be the smallest number that guarantees a candidate cannot be beaten by a full coalition of remaining candidates, and it is used in many STV elections (e.g., Republic of Ireland).
在单一可转让投票（STV）制中，候选人需要达到一个配额才能当选。最常见的两种配额是黑尔配额和德鲁普配额。黑尔配额最为简单：总有效票数除以应选席位数。德鲁普配额旨在成为保证候选人不会被剩余候选人完全联盟击败的最小数字，并在许多 STV 选举（如爱尔兰）中使用。
Hare quota QH = total valid votes ÷ number of seats
黑尔配额 QH = 总有效票数 ÷ 席位数
Droop quota QD = ⌊ total valid votes ÷ (seats + 1) ⌋ + 1
德鲁普配额 QD = ⌊ 总有效票数 ÷ (席位数 + 1) ⌋ + 1
For example, with 1000 valid votes and 4 seats, the Hare quota is 250; the Droop quota is floor(1000 / 5) + 1 = 200 + 1 = 201. A candidate reaching 201 votes in the first count is elected, and any surplus is transferred.
例如，有效票数 1000，4 个席位，黑尔配额为 250；德鲁普配额为 floor(1000 / 5) + 1 = 200 + 1 = 201。首轮计票中获得 201 票的候选人即当选，任何超额票数将被转移。
4. Imperiali Quota | 因佩里亚利配额
The Imperiali quota is another threshold used in some STV and limited‑vote systems. It is more favourable to larger parties because the denominator is larger, making the quota lower and thus easier to reach with a concentrated vote. The formula divides total valid votes by (seats + 2) instead of (seats + 1).
因佩里亚利配额是某些 STV 和限制投票制中使用的另一种门槛。它对大党更有利，因为分母更大，配额更低，因而凭借集中选票更容易达到。公式将总有效票数除以（席位数 + 2）而非（席位数 + 1）。
Imperiali quota = total valid votes ÷ (seats + 2)
因佩里亚利配额 = 总有效票数 ÷ (席位数 + 2)
In the earlier example of 1000 votes and 4 seats, the Imperiali quota would be 1000 ÷ 6 ≈ 166.7. Candidates must exceed this lower threshold, which tends to award more seats to the party with the largest vote share. It is used in some local elections but is often considered less proportional.
在前述 1000 票、4 席的例子中，因佩里亚利配额为 1000 ÷ 6 ≈ 166.7。候选人必须超过这一更低的门槛，这往往会把更多席次分配给得票率最高的政党。该配额用于部分地方选举，但常被认为比例性较差。
5. Banzhaf Power Index | 班扎夫权力指数
The Banzhaf power index measures the voting power of a player in a weighted voting game (for example, a committee where members have different vote weights). A player is critical if, by leaving a winning coalition, that coalition would no longer reach the quota. The absolute Banzhaf index for player i is the number of times they are critical. The normalised Banzhaf index gives the proportion of total critical events held by that player.
班扎夫权力指数用于衡量加权投票游戏中参与者的投票权力（例如成员持有不同票数的委员会）。如果某参与者离开一个获胜联盟后，该联盟便不再满足配额门槛，则该参与者被视为关键参与者。参与者 i 的绝对班扎夫指数是其成为关键参与者的次数。标准化班扎夫指数则是该参与者所占关键事件总数的比例。
Normalised Banzhaf index β(i) = Bi ÷ Σ Bj
标准化班扎夫指数 β(i) = Bi ÷ Σ Bj
Where Bi is the number of winning coalitions where i is critical. In a three‑player weighted vote [5; 3, 2, 1] (quota 5, weights A=3, B=2, C=1), winning coalitions and critical players can be enumerated. Often the Banzhaf index reveals that power is not simply proportional to weight.
其中 Bi 是参与者 i 成为关键人物的获胜联盟数量。在一个三人加权投票 [5; 3, 2, 1]（配额为 5，权重 A=3、B=2、C=1）中，可以枚举获胜联盟和关键参与者。班扎夫指数通常显示，权力并不简单地与权重成正比。
6. Shapley‑Shubik Power Index | 沙普利‑舒比克权力指数
The Shapley‑Shubik power index considers all possible orders in which voters join a coalition one by one. The player who, by joining, causes the coalition to reach or exceed the quota is called the pivot. The index for player i is the proportion of all n! permutations of voters in which i is the pivot. This index is based on cooperative game theory and is especially useful for understanding sequential decision‑making processes.
沙普利‑舒比克权力指数考虑选民逐一加入联盟的所有可能顺序。玩家加入后使联盟达到或超过配额的玩家称为枢纽玩家。玩家 i 的指数是在所有 n! 种选民排列中，i 成为枢纽玩家的比例。该指数基于合作博弈论，尤其有助于理解序贯决策过程。
φ(i) = (number of permutations where i is pivotal) ÷ n!
φ(i) =（i 成为枢纽玩家的排列数）÷ n!
For the same example [5; 3,2,1], we list all 6 permutations of A,B,C. A is pivot in orders where A joins and the cumulative weight first reaches 5; analysis shows A is pivotal in 4 permutations, B in 1, C in 1. So Shapley‑Shubik indices are 4/6, 1/6, 1/6 respectively.
对于同样的例子 [5; 3,2,1]，列出 A、B、C 的 6 种排列。在累计权重首次达到 5 时，A 在 4 种排列中成为枢纽，B 在 1 种，C 在 1 种。因此沙普利‑舒比克指数分别为 4/6、1/6、1/6。
7. Median Voter Theorem | 中位数选民定理
The Median Voter Theorem, formalised by Anthony Downs, states that in a two‑party system with a single‑dimensional policy space (e.g., left–right continuum), both parties will converge towards the position of the median voter. This happens because the median voter occupies the central point where half the electorate is more left‑wing and half more right‑wing; any deviation from the median loses votes.
由安东尼·唐斯形式化的中位数选民定理指出，在一个一维政策空间（如左右连续谱系）的两党制下，两个政党将向中位数选民的立场靠拢。之所以如此，是因为中位数选民占据中心点，其一半的选民较其更左，一半更右；任何偏离中位数立场的行为都会失去选票。
Although the theorem does not involve a single arithmetic formula, the median is defined as the value separating the higher half of a data sample from the lower half. In politics, it explains why competing parties often produce similar manifestos and why radical platforms tend to weaken electoral performance under first‑past‑the‑post.
尽管该定理不涉及单一的算术公式，但中位数被定义为将数据样本的较大一半与较小一半分隔的值。在政治学中，它解释了为何竞争政党常常推出相似的政纲，以及为何在简单多数制下激进纲领往往削弱选举行情。
8. Duverger’s Law | 杜瓦杰定律
Duverger’s Law is a key theorem in political science rather than a mathematical formula. It posits that single‑member district plurality (SMDP/FPTP) electoral systems tend to favour a two‑party system, because smaller parties struggle to win seats unless their support is geographically concentrated, and voters gravitate toward the two most viable parties to avoid wasting their vote.
杜瓦杰定律是政治学中的关键定理，而非数学公式。它假定单席位选区简单多数制（SMDP/FPTP）选举制度倾向于形成两党制，因为小政党除非其支持者高度集中，否则难以赢得席位，而选民为了避免浪费选票，也会向最有可能获胜的两大党聚拢。
Duverger also stated the hypothesis (sometimes called Duverger’s second law) that proportional representation tends to favour multiparty systems. Understanding these structural effects helps explain the shape of party competition in Scotland and the UK.
杜瓦杰还提出了一个假设（有时被称为杜瓦杰第二定律），即比例代表制倾向于促进多党制。理解这些结构效应有助于解释苏格兰和英国政党竞争的形态。
9. Effective Number of Parties (Laakso‑Taagepera Index) | 有效政党数（拉科索‑塔格佩拉指数）
The effective number of parties measures how many parties are relevant in a party system, weighted by their relative size. The Laakso‑Taagepera index uses the squared vote shares (or seat shares) of each party. It is a widely used formula in comparative politics to describe party system fragmentation
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