📚 Year 12 CIE Politics: Formulas & Theorems Quick Reference Handbook | CIE 政治公式定理速查手册
This handbook gathers the essential quantitative tools and theoretical models that appear across the Year 12 CIE Politics syllabus. From electoral quotas to party system indices and from rational choice equations to game theory matrices, these formulas and theorems help you analyse political institutions, voting behaviour and power dynamics with precision.
本手册汇整了 Year 12 CIE 政治课程中反复出现的定量工具与理论模型。从选举配额到政党制度指数，从理性选择等式到博弈论矩阵，这些公式与定理能帮助你精准分析政治制度、投票行为与权力运作。
1. Electoral Quotas: Droop and Hare | 选举配额：德鲁普配额与黑尔配额
In proportional representation systems that use the single transferable vote (STV) or other list‑PR methods, a quota determines the number of votes a candidate must secure to win a seat. The two most commonly taught quotas are the Hare quota and the Droop quota.
在采用单记可转让投票（STV）或其他名单比例代表制的方法中，配额决定候选人赢得一个席位所需的票数。最常见的两种配额是黑尔配额和德鲁普配额。
Hare Quota: Q = V ÷ S
Droop Quota: Q = ⌊V ÷ (S + 1)⌋ + 1
Where V is the total number of valid votes cast and S is the number of seats to be filled in the constituency. The Droop quota is usually preferred in STV elections because it is slightly lower than the Hare quota, reducing the number of wasted votes while still ensuring that only S candidates can reach the threshold. In the formula, the floor notation ⌊ ⌋ means we take the integer part of the division before adding 1, effectively rounding down any remainder and then incrementing.
其中 V 代表有效选票总数，S 代表选区内应选席位数。德鲁普配额在 STV 选举中更常用，因为它比黑尔配额略低，可减少废票数，同时依然确保只有 S 名候选人能达到门槛。公式中的取整符号 ⌊ ⌋ 表示先只取除法的整数部分，再加 1，即忽略余数后加 1。
2. Effective Number of Parties and Fragmentation Index | 有效政党数与碎片化指数
When comparing party systems, simply counting the number of parties can be misleading. Political scientists use the effective number of parties (ENP) and the Rae fragmentation index to measure how fragmented a party system really is, weighting parties by their size.
比较政党制度时，简单计数政党个数可能产生误导。政治学者采用有效政党数（ENP）和雷氏碎片化指数来衡量政党制度的真实碎片化程度，并根据政党规模进行加权。
Effective Number of Parties (Laakso–Taagepera): N = 1 ÷ Σ pᵢ²
Rae’s Fragmentation Index: F = 1 − Σ pᵢ²
n of votes (or seats) won by party i. N gives a weighted count of parties; for example, in a two‑ 0. F ranges from 0 (one party takes all the seats) to almost 1 (extreme fragmentation). The term Σ pᵢ² is often called the Herfindahl–Hirschman index of concentration.
在这两个公式中，pᵢ 是政党 i 获得的选票（或席位）比例。N 给出的是加权后的政党个数；例如在两个政党势均力敌的体制中，N ≈ 2.0。F 的取值从 0（一党独占所有席位）到接近 1（极端碎片化）。Σ pᵢ² 通常被称为赫芬达尔–赫希曼集中度指数。
3. The Cube Law: Converting Votes into Seats | 立方定律：选票转化为席位
The cube law describes the relationship between votes and seats in first‑past‑the‑post systems, particularly in two‑party contests. It captures how single‑member plurality elections tend to amplify the lead of the largest party, granting it a seat bonus disproportionate to its vote share.
立方定律描述了简单多数制下选票与席位之间的关系，尤其适用于两党竞争的情形。它揭示了单席选区简单多数选举如何倾向于放大最大党的领先优势，使其获得与其得票份额不成比例的席位红利。
Cube Law: S₁ ÷ S₂ = (V₁ ÷ V₂)³
seats won by the two main parties, and V₁ and V₂ are their total votes. If party A wins 55 % of the two‑party vote while party B wins 45 % law predicts a seat ratio of (0.55 ÷ 0.45)³ ≈ 1.83, meaning party A would hold roughly 65 % n vary, but the cube law remains a useful benchmark for analysing majoritarian distortion.
式中 S₁ 和 S₂ 是两个主要政党所获席位数，V₁ 和 V₂ 是其得票总数。若甲党在两党票数中赢得 55%、乙党赢得 45%，立方定律预测的席位比为 (0.55 ÷ 0.45)³ ≈ 1.83，即甲党约可获得 65% 的席位。现实中指数可能偏离 3，但立方定律始终是分析多数制偏差的实用基准。
4. Gallagher Least Squares Index of Disproportionality | 加拉格尔最小二乘不成比例指数
The Gallagher index is the standard tool for measuring how closely the distribution of legislative seats mirrors the distribution of votes. It is especially useful when comparing the proportionality of different electoral systems.
加拉格尔指数是衡量立法机构席位分布与选票分布接近程度的标准工具，在比较不同选举制度的比例性时尤为有用。
Gallagher Index: LSq = √[½ Σ(vᵢ − sᵢ)²]
Here vᵢ and sᵢ are the vote percentage and seat percentage for party i respectively. The index equals zero when every party receives exactly the same share of seats as votes. Large values indicate a highly disproportional outcome. For example, recent UK general elections have produced LSq values between 5 and 15, whereas most PR systems yield values below 5.
其中 vᵢ 和 sᵢ 分别是政党 i 的得票百分比和席位百分比。当每个政党的席位份额与其得票份额完全一致时，指数为零；数值越大表明结果越不成比例。例如，近年英国大选的 LSq 值在 5 到 15 之间，而多数比例代表制国家的数值通常低于 5。
5. Duverger’s Law and Its Hypotheses | 迪韦尔热法则及其假设
Duverger’s law is one of the most famous theorems in comparative politics. It states that first‑past‑the‑post, single‑member district electoral systems tend to produce a two‑party system, whereas proportional representation encourages multipartyism. The mechanism combines a ‘mechanical effect’ (small parties are under‑rewarded in seats) and a ‘psychological effect’ (voters and elites avoid wasting support on likely losers).
迪韦尔热法则是比较政治学中最知名的定理之一。它指出：单席选区的简单多数选举制倾向于产生两党制，而比例代表制则会促进多党制。其机制结合了“机械效应”（小党在席位分配上吃亏）和“心理效应”（选民和精英会避免把支持浪费在可能落败的候选人身上）。
Although often presented as a law, political scientists treat it as a probabilistic tendency subject to intervening variables such as social cleavages and presidentialism. In examination answers, you should distinguish Duverger’s law from Duverger’s hypothesis, which suggests that ranked‑ballot majoritarian systems (e.g. instant‑runoff voting) also favour a two‑party system.
尽管常以“法则”的面目出现，政治学者实际上把它视为一种概率性趋势，会受到社会分歧、总统制等干预变量的影响。在答题时，你应当区分迪韦尔热法则与迪韦尔热假设——后者认为，排位投票的多数决制（如即时决选投票）同样有利于两党制。
6. Electoral Thresholds: Legal and Natural | 选举门槛：法定门槛与自然门槛
Electoral thresholds limit the access of very small parties to the legislature. A legal threshold is a rule written into the electoral law (e.g. 5 % of the national vote required for list‑PR seats), whereas a natural threshold arises from the arithmetic of district magnitude.
选举门槛限制了极小政党进入立法机构的机会。法定门槛是写入选举法中的规则（例如名单比例代表制要求获得全国 5% 的选票），而自然门槛则产生于选区规模的算术特性。
Effective threshold (Taagepera): T ≈ 75% ÷ (M + 1)
M denotes the district magnitude — the number of seats elected in a constituency. In a single‑member district (M = 1), the effective threshold is about 37.5 % sier for small parties to gain representation. This formula helps explain why systems with large multi‑member districts are more permissive of multipartism.
M 代表选区规模——即一个选区应选的议席数。在单席选区（M = 1）中，有效门槛约为 37.5%，意味着政党需要获得大约这一比例的选票才有现实的机会赢得席位。随着 M 增大，门槛降低，使小党更容易获得代表权。这一公式有助于解释为何采用大型多名选区制的体系更容易催生多党格局。
7. Downs’ Rational Voting Equation | 唐斯理性投票等式
Anthony Downs applied economic reasoning to voter turnout in An Economic Theory of Democracy. His calculus of voting remains a core theorem for explaining why individuals choose to vote — or abstain — in large‑scale elections.
安东尼·唐斯在《民主的经济理论》中将经济学推理应用于投票行为。他的投票计算依然是解释个人为何在大规模选举中选择投票或弃权的核心定理。
R = pB − C
heir preferred candidate wins instead of the opponent, and C represents the costs of voting (time, information, travel). Since p is infinitesimally small in large electorates, the equation leads to the ‘ ally negative, yet many people still vote. This paradox has prompted extensions that add a ‘duty’ term: D, so R = pB − C + D.
R 是投票的预期效用，p 是单张选票起决定性作用的概率，B 是选民倾向的候选人当选而非对手当选所带来的效用差，C 代表投票成本（时间、信息、交通）。由于在大规模选民中 p 极其微小，该等式导致了“投票悖论”：R 通常为负，但许多人仍然参与投票。这一悖论促使了扩展模型，增加一个“义务”项 D，即 R = pB − C + D。
8. The Prisoner’s Dilemma Payoff Matrix | 囚徒困境收益矩阵
The prisoner’s dilemma is a foundational game‑theoretic model used in politics to explain cooperation and conflict in areas such as arms races, climate negotiations and coalition bargaining. The payoffs are structured so that defection is the dominant strategy for each player, yet mutual cooperation yields a better collective outcome.
囚徒困境是博弈论的基础模型，在政治学中用于解释军备竞赛、气候谈判和联合政府谈判等领域的合作与冲突。其收益结构使背叛成为每个参与者的优势策略，然而相互合作却能带来更好的集体结果。
| Player B Cooperates | Player B Defects | |
| Player A Cooperates | (−1, −1) | (−10, 0) |
| Player A Defects | (0, −10) | (−5, −5) |
In the typical payoff matrix above (with negative numbers representing years in prison), the Nash equilibrium is (Defect, Defect), giving (−5, −5), even though (Cooperate, Cooperate) would be Pareto‑superior at (−1, −1). This tension is central to understanding why rational self‑interested actors may fail to achieve mutual gains without enforcement mechanisms or repeated interaction.
在上述典型的收益矩阵中（负数代表监禁年数），纳什均衡为（背叛，背叛），收益为 (−5, −5)，尽管（合作，合作）在帕累托意义上更优，收益为 (−1, −1)。这一张力是理解为何理性的自利行为体在没有执行机制或重复互动的情况下无法实现共同利益的核心所在。
9. Power Indices: Shapley–Shubik and Banzhaf | 权力指数：沙普利–舒比克与班扎夫
When legislatures or committees make decisions by weighted voting, raw seat shares do not always reflect true power. Power indices measure the probability that a player can change a losing coalition into a winning one. The two most widely used indices are the Shapley–Shubik index and the Banzhaf index.
当立法机构或委员会通过加权投票做决策时，原始席位比重未必反映真实权力。权力指数衡量的是某参与者能将失败联盟转变为获胜联盟的概率。最常用的两种指数是沙普利–舒比克指数和班扎夫指数。
Consider a three‑party council with votes [50, 49, 1] and a quota of 51. A coalition is winning if its combined votes reach 51. The Banzhaf index counts how many times a party is critical — i.e. its withdrawal would cause the coalition to lose. Party C’s single vote is critical in coalitions {50,1} and {49,1}, making it just as powerful as the larger parties in certain configurations. The Shapley–Shubik index examines all possible orders in which players join a coalition and identifies the pivotal member.
设想一个三党议会，票权分别为 [50, 49, 1]，法定多数额为 51。若某联盟总票数达到 51 即为获胜。班扎夫指数计算每个政党成为关键者的次数——即一旦其退出联盟就会丧失多数地位。政党 C 的那一票在 {50,1} 和 {49,1} 这两个联盟中均起关键作用，在此情形下其权力与大党相当。沙普利–舒比克指数则检视所有加入次序，找出其中起转折作用的关键成员。
Banzhaf index: βᵢ = cᵢ ÷ Σ cⱼ
Here cᵢ is the number of winning coalitions in which player i is critical, and the denominator is the sum of critical counts across all players. These indices reveal that small parties in coalition governments can wield influence disproportionate to their seat share, an insight vital for analysing minority governments and bargaining power.
式中 cᵢ 是参与者 i 在其中起关键作用的获胜联盟数目，分母为所有参与者关键次数的总和。这些指数表明，联合政府中的小党往往拥有与席位数不成比例的影响力，这对分析少数派政府与议价权力至关重要。
10. Easton’s Political System Model | 伊斯顿政治系统模型
David Easton’s systems theory is a foundational theorem for understanding how political systems process demands and maintain stability through feedback loops. Although it is a conceptual model rather than a mathematical formula, it is a core ‘theorem’ in the CIE Politics course, framing the relationship between the environment, inputs, the ‘black box’ of decision‑making, and outputs.
戴维·伊斯顿的系统理论是理解政治系统如何处理要求并通过反馈环维持稳定的一项基础定理。尽管它是概念模型而非数学公式，却构成了 CIE 政治课程中一项核心“定理”，搭建起环境、输入、决策“黑箱”和输出之间的关系框架。
ports] → Political System (Conversion) → Outputs (Decisions, Policies) → Feedback → Inputs
According to Easton, citizens generate demands (e.g. for better healthcare) and supports (e.g. paying taxes, obeying laws) that feed into the political system. The system converts these into authoritative outputs such as legislation. Over time, the loop of feedback—public reactions that generate new demands or supports—allows the system to self‑correct and persist. This theorem is particularly helpful when analysing policy cycles, legitimacy crises, and the capacity of democratic institutions to respond to societal change.
按照伊斯顿的理论，公民产生要求（例如要求更好的医疗）和支持（例如纳税、守法），这些输入进入政治系统。系统将其转化为权威性输出，如立法。随着时间推移，反馈环——公众反应产生新的要求或支持——使得系统能够自我修正并持续运转。这一定理在分析政策周期、合法性危机以及民主机构回应社会变迁的能力时尤为有力。
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