📚 Cambridge Year 9 Politics: Quick Reference Handbook of Key Formulas & Theorems | 剑桥九年级政治：公式定理速查手册
This quick reference guide brings together the essential formulas, theorems, and quantitative concepts that underpin the Cambridge Year 9 Politics syllabus. From measuring voting power in a committee to understanding why two-party systems emerge, these tools help you analyse political systems with precision. Each entry presents a clear definition, the core formula or statement where applicable, and a brief explanation of its significance.
本速查手册汇集了支撑剑桥九年级政治课程的核心公式、定理与量化概念。无论是衡量委员会中的投票权力，还是理解两党制为何形成，这些工具都能帮助你精确分析政治体制。每个条目都给出了清晰的定义、适用的核心公式或命题，并简要说明其意义。
1. Voting Power: The Banzhaf Index | 投票权力：班扎夫指数
The Banzhaf index measures the power of a voter in a weighted voting system by counting how often a voter can change a losing coalition into a winning one. For a voter i, the Banzhaf power is the number of times i is critical divided by the total number of critical votes across all voters.
班扎夫指数通过计算一个投票者能将失败联盟变为胜利联盟的次数，来衡量其在加权投票系统中的权力。对投票者i而言，班扎夫权力等于i作为关键投票者的次数除以所有投票者的关键投票总次数。
Banzhaf power of voter i = (number of times i is critical) ÷ (total critical counts for all voters)
投票者i的班扎夫权力 = （i成为关键投票者的次数）÷（所有投票者关键次数的总和）
A coalition is winning if its total weight meets or exceeds the quota. A voter is critical if removing that voter causes the coalition to lose. This index reveals that voting weight does not always equal real power.
一个联盟若其总权重达到或超过法定人数即为胜利联盟。如果移除某投票者会导致联盟由胜转败，则该投票者是关键投票者。该指数揭示出投票权重并不总等于实际权力。
2. Shapley-Shubik Power Index | 沙普利-舒比克权力指数
The Shapley-Shubik index evaluates voting power by considering all possible orders in which voters join a coalition. A voter is pivotal if, when added in a specific order, the coalition first becomes winning. The power index for a voter is the number of times they are pivotal divided by the total number of orderings (n! for n voters).
沙普利-舒比克指数通过考虑投票者加入联盟的所有可能顺序来评估投票权力。若在某种顺序中，加入某投票者后联盟首次成为胜利联盟，则该投票者具有决定性。其权力指数等于该投票者成为决定性投票者的次数除以总排列数（n个投票者的全排列为n!）。
Shapley-Shubik power of voter i = (number of orderings where i is pivotal) ÷ n!
投票者i的沙普利-舒比克权力 = （i成为决定性投票者的排列数）÷ n!
For example, in a three-person committee where members A, B, C have votes of 40, 30, and 30 with a quota of 51, each member is pivotal in exactly one-third of the orderings, so they have equal power despite different vote weights.
例如，在一个三人委员会中，成员A、B、C分别拥有40、30、30票，法定人数为51票，每位成员均在三分之一的排列中成为决定性投票者，因此尽管票数不同，他们拥有平等的权力。
3. Median Voter Theorem | 中间选民定理
The median voter theorem, developed by Anthony Downs, states that in a majority-rule election with voters arranged along a single ideological dimension, the candidate or party whose position is closest to the median voter will win, assuming voters have single-peaked preferences and vote sincerely.
由安东尼·唐斯提出的中间选民定理指出，在多数决选举中，若选民沿单一意识形态维度排列，且偏好呈单峰状并诚实投票，则立场最接近中间选民的候选人或政党将获胜。
This leads to a powerful convergence effect: in a two-candidate race, both candidates are driven to adopt positions near the centre to capture the median voter. The theorem helps explain why mainstream parties often sound similar.
这产生了一个强大的趋同效应：在两人竞选中，双方都会被驱使采纳接近中心的立场以争取中间选民。该定理有助于解释为什么主流政党常常显得相似。
4. Duverger’s Law | 杜瓦杰法则
Duverger’s law states that single-member district plurality (SMDP) electoral systems tend to favour a two-party system. The mechanical effect (only the largest party in each district wins a seat) combines with the psychological effect (voters avoid wasting votes on minor parties) to reduce the number of effective parties.
杜瓦杰法则指出，单一选区相对多数制倾向于催生两党制。机械效应（每个选区内只有最大党赢得席位）与心理效应（选民不愿把票浪费在小党身上）共同作用，减少了有效政党数量。
This is often expressed informally as: “the simple-majority single-ballot system favours the two-party system.” Proportional representation, by contrast, is associated with multiparty systems.
这通常被非正式地表述为：“简单多数单轮投票制有利于两党制。”相比之下，比例代表制则与多党制相关。
5. Effective Number of Parties | 有效政党数
The effective number of parties translates the raw count of parties and their vote or seat shares into a single number that reflects the fragmentation of a party system. The most common measure is the Laakso-Taagepera index, calculated from vote shares (ENPᵥ) or seat shares (ENPₛ).
有效政党数将政党的原始数量及其选票或席位份额转化为一个反映政党体系碎片化的单一数字。最常用的指标是拉阿科索-塔格佩拉指数，可根据选票份额（ENPᵥ）或席位份额（ENPₛ）计算。
ENP = 1 ÷ Σ(pᵢ²), where pᵢ is the fraction of votes (or seats) won by party i
ENP = 1 ÷ Σ(pᵢ²)，其中pᵢ为政党i赢得的选票（或席位）比例
If all parties are equal in size, ENP equals the actual number of parties. If one party dominates, ENP is close to 1. The index helps compare party systems across countries and over time.
如果所有政党规模相等，ENP等于政党实际数量；若一党独大，ENP则接近1。该指数有助于在国家间和不同时期比较政党体制。
6. Arrow’s Impossibility Theorem | 阿罗不可能定理
Kenneth Arrow proved that no ranked voting system can simultaneously satisfy a set of seemingly reasonable fairness criteria when there are three or more candidates. The conditions are: unrestricted domain, non-dictatorship, Pareto efficiency, and independence of irrelevant alternatives.
肯尼斯·阿罗证明，当存在三名或以上候选人时，任何排序投票制都无法同时满足一组看似合理的公平标准。这些条件是：不受限制的定义域、非独裁性、帕累托效率以及无关选项独立性。
The theorem implies that every democratic voting method involves trade-offs. For instance, ranked-choice voting can violate the independence of irrelevant alternatives because the presence of a third candidate can tip the outcome between the top two.
该定理意味着每种民主投票方法都涉及权衡。例如，排序复选制可能违反无关选项独立性，因为第三名候选人的存在可能改变前两名之间的胜负结果。
7. Condorcet Paradox | 孔多塞悖论
The Condorcet paradox occurs when collective preferences become cyclical even though each individual’s preferences are transitive. In an election with three candidates A, B, and C, it is possible that a majority prefers A over B, B over C, and yet C over A, creating no clear Condorcet winner.
孔多塞悖论指的是，即使每位个体的偏好都具有传递性，集体偏好却可能出现循环。在三名候选人A、B、C的选举中，可能出现多数人偏好A胜于B、B胜于C、但C又胜于A的情况，从而无法产生明确的孔多塞赢家。
This paradox shows that majority rule can be inconsistent. It underpins much of social choice theory and motivates the search for voting rules that minimise such cycles.
这一悖论表明多数决可能产生不一致的结果，它构成了社会选择理论的重要基础，并推动了对能最小化此类循环现象的投票规则的研究。
8. Collective Action Problem and Olson’s Logic | 集体行动问题与奥尔森逻辑
Mancur Olson argued that rational, self-interested individuals will not automatically act to achieve their common group interests, especially in large groups. The benefit a person gains from a collective good is often smaller than the cost of participation, leading to free riding.
曼瑟尔·奥尔森认为，理性的、自利的个人并不会自动为实现共同群体利益而行动，特别是在大群体中。个人从集体物品中获得的收益往往小于参与成本，从而导致搭便车现象。
The logic is captured by the calculus of participation: an individual will contribute only if the personal benefit exceeds the personal cost. Selective incentives—benefits available only to participants—can overcome this problem.
参与的计算逻辑为：只有当个人收益超过个人成本时，个体才会做出贡献。选择性激励——即仅向参与者提供的利益——可以克服这一问题。
Collective action succeeds when: selective incentives + individual benefit ≥ cost
集体行动成功的条件：选择性激励 + 个人收益 ≥ 成本
9. Electoral Thresholds | 选举门槛
Many proportional representation systems use an electoral threshold—a minimum share of the national vote that a party must win to gain any seats. The threshold T is usually expressed as a percentage, commonly 3% or 5%. This reduces parliamentary fragmentation.
许多比例代表制设有选举门槛——即政党为获得席位而必须赢得的最低全国选票比例。门槛T通常以百分比表示，常见为3%或5%。这有助于减少议会的碎片化程度。
Parties with vote share v < T receive no seats. The effective threshold can also be calculated from district magnitude using the formula: T ≈ 75% ÷ (M + 1), where M is the average number of seats per district.
得票率v < T的政党将得不到席位。有效门槛也可由选区规模计算得出，近似公式为：T ≈ 75% ÷ (M + 1)，其中M为每个选区的平均席位数。
v < T ⇒ zero seats; Natural threshold ≈ 75% ÷ (M + 1)
v < T ⇒ 零席次；自然门槛 ≈ 75% ÷ (M + 1)
10. Cube Rule of Seats-Votes | 席次-选票立方定律
The cube rule is an empirical observation about the relationship between the ratio of seats won by two major parties and the ratio of votes they receive in a single-member district system. It states that the seat ratio tends to approximate the cube of the vote ratio.
立方定律是关于单一选区制下两大政党所获席次比与得票比之间关系的经验观察。它指出，席次比往往近似等于得票比的立方。
Seats(A) / Seats(B) ≈ (Votes(A) / Votes(B))³
席次(A) / 席次(B) ≈ (选票(A) / 选票(B))³
For example, if Party A wins 55% of the two-party vote and Party B wins 45%, the expected seat ratio is (55/45)³ ≈ 1.83, meaning A would win roughly 65% of the seats. This magnifies the winner’s advantage and reinforces the two-party system.
例如，若A党赢得两党票数的55%，B党为45%，则预期席次比为(55/45)³ ≈ 1.83，意味着A党将赢得约65%的席位。这放大了胜者优势，并强化了两党制。
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