📚 Year 13 Cambridge Politics: Key Formulas & Theorems Quick Reference Handbook | Year 13 剑桥政治：核心公式定理速查手册
This handbook brings together the most important quantitative and logical "formulas" and theorems encountered in Year 13 Cambridge Politics, covering comparative electoral systems, voting behaviour, power indices, game theory and organisational dynamics. Each entry presents the core statement in a succinct, revision-ready format.
本手册汇集了 Year 13 剑桥政治课程中最重要的定量和逻辑 "公式" 与定理,涵盖比较选举制度、投票行为、权力指数、博弈论和组织动力学等领域。每个条目以简洁的复习格式呈现核心内容。
1. Duverger's Law | 杜瓦杰定律
Duverger's Law states that single-member plurality (SMP) electoral systems tend to favour a two-party system. The mechanical effect reduces small parties' seat shares, while the psychological effect discourages voters from wasting their votes on weak candidates.
杜瓦杰定律指出,单名选区相对多数决制 (SMP) 通常有利于两党制。机械效应会降低小党获得的席位数,而心理效应则使选民不愿将选票 "浪费" 在弱势候选人身上。
Under pure SMP, the winning threshold is simply one more vote than the nearest rival. The effective seat share of a third party with 20% nationwide support often collapses to near zero, reinforcing duopolistic competition.
在纯粹 SMP 下,获胜门槛仅仅是比第二名多一票。一个在全国拥有 20% 支持率的第三党,其实际议席份额往往会骤降至接近零,从而强化双头垄断的竞争格局。
2. Effective Number of Parties (Laakso–Taagepera Index) | 有效政党数 (拉克索–塔格佩拉指数)
The effective number of electoral or parliamentary parties is given by the Laakso–Taagepera formula:
有效选举或议会政党数由拉克索–塔格佩拉公式给出：
N = 1 / Σ pᵢ²
= 1/(0.25+0.25) = 2.0. A fragmented parliament with many small parties yields a higher N.
其中 pᵢ 表示政党 i 的席位份额（或得票份额）,以小数表示。如果两个政党各占 0.5,则 N = 1/(0.25+0.25) = 2.0。一个由许多小党组成的分散议会会产生更大的 N 值。
This index captures not the raw count of parties but their relative weight, making it indispensable for comparing the concentration of party systems across proportional and majoritarian setups.
该指数捕捉的不是政党的简单数量,而是它们的相对权重,因此是跨比例代表制和多数决制比较政党体系集中度的必备工具。
3. Seat Allocation Formulas: D'Hondt & Sainte-Laguë | 席位分配公式：德洪特法与圣拉格法
D'Hondt divisors are 1, 2, 3, …, n. Each party’s votes are divided successively by these integers, and seats are awarded according to the highest resulting quotients. It slightly favours larger parties.
德洪特法除数依次为 1, 2, 3, …, n。将每个政党的得票数依次除以这些整数,席位按所得商数从高到低分配。该规则对大党略微有利。
Sainte-Laguë uses odd divisors 1, 3, 5, 7, … (or modified with a first divisor of 1.4). It produces a more proportional outcome and is used in, for instance, New Zealand and Germany (modified version).
圣拉格法使用奇数除数 1, 3, 5, 7…（或进行修改,将第一个除数设为 1.4）。它能产生更比例化的结果,在新西兰和德国（改良版）等国家被采用。
arty’s total vote, and d is the current divisor for that party’s next seat. The algorithm iterates until all seats are filled.
分配公式可表示为商数 Q = V / d,其中 V 是政党总得票数,d 是该党下一个席位所对应的当前除数。算法迭代进行,直至所有席位分配完毕。
4. Downs' Rational Voter Equation | 唐斯理性选民公式
Anthony Downs modelled the decision to vote as a cost–benefit calculus:
安东尼·唐斯将投票决策建模为成本–收益计算：
R = P × B − C + D
where R is the expected utility of voting, P the probability of one vote being decisive, B the differential benefit of one’s preferred candidate winning, C the cost of voting, and D a civic duty satisfaction term.
其中 R 代表投票的预期效用,P 为一票起决定作用的概率,B 为心仪候选人获胜带来的差异收益,C 为投票成本,D 则表示公民义务的满足感。
Since P is infinitesimally small in large electorates, purely instrumental voting cannot explain turnout. The "paradox of voting" is resolved largely through D, captured by those who feel a moral or expressive benefit from participation.
在规模大的选民群体中,P 极低,纯粹工具性的投票动机无法解释投票率。"投票悖论"主要通过 D 来解释,即那些从参与中感受到道德或表达性收益的选民。
5. Arrow's Impossibility Theorem | 阿罗不可能定理
Kenneth Arrow proved that no rank-order voting system can simultaneously satisfy four minimal fairness conditions when there are three or more alternatives: unrestricted domain, non-dictatorship, Pareto efficiency, and independence of irrelevant alternatives.
肯尼斯·阿罗证明,当存在三个或以上选项时,没有任何排序投票制度能同时满足四个最低限度的公平条件：不受限域的偏好、非独裁性、帕累托效率和无关选项独立性。
Formally, if a social welfare function respects these axioms, it must be dictatorial. This reshaped democratic theory by showing that all aggregation methods involve trade-offs, and that majority cycling is an inherent logical possibility.
从形式上说,如果某个社会福利函数满足这些公理,它必定是独裁的。该定理重塑了民主理论,表明所有加总方法都涉及取舍,多数决循环是一种内在于逻辑的可能性。
6. Banzhaf Power Index | 班扎夫权力指数
The Banzhaf index measures a voter’s probability of being pivotal in a weighted voting game. A voter is pivotal if a coalition changes from losing to winning when they join. The index is calculated as
班扎夫指数衡量加权投票博弈中选民成为关键者的概率。如果某个参与者加入能使联盟从失败变为获胜,则该参与者即为关键选民。指数计算如下：
βᵢ = bᵢ / Σ bⱼ
where bᵢ is the number of winning coalitions in which voter i is pivotal, and the denominator sums all voters' pivotal positions. It is often normalised so that the sum equals 1.
其中 bᵢ 表示选民 i 成为关键者的获胜联盟数量,分母为所有选民关键次数之和。该指数通常按和为 1 的方式标准化。
In councils with weighted voting (e.g., EU Council, IMF), raw vote weights often hide true power; some states may have zero Banzhaf power despite holding votes, while others can wield disproportionate influence.
在采用加权投票的机构（如欧盟理事会、国际货币基金组织）中,原始票权重常掩盖真实权力；有些国家虽有票数但班扎夫权力可能为零,而另一些国家则拥有不成比例的影响力。
7. Shapley–Shubik Power Index | 沙普利–舒比克权力指数
This index evaluates a voter’s power based on the order in which they join a coalition. For n voters, consider all n! possible sequential orders. Voter i is pivotal if, upon their addition, the cumulative votes reach or exceed the quota for the first time.
该指数根据选民加入联盟的次序来评估其权力。对 n 个选民,考虑所有 n! 种可能的排列顺序。如果选民 i 加入时累计票数首次达到或超过法定门槛,那么该选民即为关键者。
The Shapley–Shubik index for voter i is the proportion of orderings in which i is pivotal. Like the Banzhaf index, it illuminates the gap between formal voting rights and substantive influence in legislatures and international organisations.
选民 i 的沙普利–舒比克指数即是 i 成为关键者的排序数占总排序数的比例。与班扎夫指数一样,它揭示了立法机构和国际组织中形式投票权与实质影响力之间的差距。
8. Prisoner's Dilemma & Payoff Matrix | 囚徒困境与支付矩阵
The Prisoner's Dilemma models strategic interdependence where individually rational choices lead to a collectively suboptimal outcome. The standard payoff matrix (with Temptation > Reward > Punishment > Sucker) ensures that defection is a dominant strategy.
囚徒困境模拟了一种策略性相互依赖,个体理性选择会导致集体次优结果。标准支付矩阵（诱惑 > 奖励 > 惩罚 > 受骗收益）确保背叛成为占优策略。
| B cooperates | B defects | |
| A cooperates | R=3, R=3 | S=0, T=5 |
| A defects | T=5, S=0 | P=1, P=1 |
In global politics, arms races and climate negotiations often resemble repeated Prisoner's Dilemmas. Cooperation can stabilise through iteration, reciprocity (tit-for-tat) and institutional design, linking individual incentives to collective benefits.
在全球政治中,军备竞赛和气候谈判常常类似于重复囚徒困境。通过重复互动、互惠策略（如"以牙还牙"）和制度设计,合作能够趋于稳定,从而将个体激励与集体利益联系起来。
9. Olson's Logic of Collective Action | 奥尔森集体行动的逻辑
Mancur Olson argued that rational, self-interested individuals will not voluntarily contribute to a common good unless the group is small or selective incentives exist. The core inequality is
曼瑟·奥尔森指出,理性的、自利的个体不会自愿为公共品作贡献,除非群体规模小或存在选择性激励。核心不等式如下：
Cᵢ < Vᵢ × F
ndividual. In a large latent group, F → 0, so the inequality rarely holds without coercion or separate private rewards.
其中 Cᵢ 是贡献成本,Vᵢ 是个体对集体物品的估值,F 是该个体能获得的那部分收益的比例。在大型隐性群体中,F → 0,因此除非有强制或额外的私人回报,否则这个不等式极少成立。
This logic explains why trade union membership, environmental activism and international burden-sharing require institutional design to overcome free-riding. It underpins many exam questions on interest groups and global governance.
该逻辑解释了为何工会成员资格、环保行动和国际责任分担需要制度设计来克服搭便车行为。它是许多关于利益团体和全球治理的考试题目的理论基础。
10. Michels' Iron Law of Oligarchy | 米歇尔斯的寡头统治铁律
Robert Michels stated, "Who says organisation, says oligarchy." Even democratic mass parties inevitably concentrate power in a small leadership elite. This "iron law" emerges from the technical indispensability of leadership, communication control and the apathy of the rank-and-file.
罗伯特·米歇尔斯断言："言组织,即言寡头。"即便是民主的大众型政党,权力也必然集中于一个小型领导精英群体。这条 "铁律" 来源于领导层的技术必要性、沟通控制以及普通成员的冷漠。
While not a mathematical formula, the logic can be expressed as a functional relationship: as organisational complexity (O) increases, the concentration of real decision-making power (P) grows disproportionately, so that P ∝ O² in a simplified structural reading.
这虽不是数学公式,但其逻辑可以表达为一个函数关系：随着组织复杂性 (O) 增加,实际决策权力的集中程度 (P) 会不成比例地增长,在简化的结构解读中可记为 P ∝ O²。
It remains a powerful lens for analysing political parties, protest movements and international bureaucracies, reminding students that democratic structures constantly face oligarchic tendencies.
米歇尔斯铁律始终是分析政党、抗议运动和国际官僚机构的有力视窗,提醒学生民主结构时刻面临着寡头化趋势。
Published by TutorHao | Cambridge Politics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图