📚 AS Eduqas Politics: Formulas & Theorems Quick Reference Handbook | AS Eduqas 政治：公式定理速查手册
This quick reference handbook brings together the most important formulas, theorems and indices that appear throughout the AS Eduqas Politics specification. Understanding these quantitative and qualitative tools is essential for analysing electoral systems, party competition, coalition behaviour and the logic of collective decision-making. Each entry presents the core definition in English, followed by a Chinese translation, together with worked examples and key implications.
本速查手册汇集了 AS Eduqas 政治课程中最重要的公式、定理与指数。掌握这些定量与定性工具对于分析选举制度、政党竞争、联盟行为以及集体决策的逻辑至关重要。每个条目先用英文给出核心定义，再提供中文翻译，并附有计算示例和关键推论。
1. Duverger’s Law | 杜瓦杰定律
Duverger’s Law states that single‑member district plurality (SMDP) electoral systems tend to produce a two‑party system. The mechanism operates through two effects: the mechanical effect, where smaller parties are denied seats because they cannot secure a plurality in any single district, and the psychological effect, where voters avoid ‘wasting’ their votes on unlikely winners and instead support one of the two front‑runners.
杜瓦杰定律指出，单一选区相对多数制（SMDP）选举制度倾向于产生两党制。其机制通过两种效应发挥作用：机械效应，即小党因无法在任何一个选区获得相对多数而得不到席位；心理效应，即选民为避免“浪费”选票给胜算渺茫的候选人，转而支持两个领先政党中的一个。
The classic example is the United Kingdom House of Commons elections, where the first‑past‑the‑post rule has sustained the dominance of the Labour and Conservative parties. Duverger did note exceptions where regional parties can thrive, but the law remains a powerful predictor of party system shape under plurality rules.
典型例子是英国下议院选举，其“领先者当选”规则维持了工党和保守党的主导地位。杜瓦杰也指出地区性政党可能兴盛的特例，但该定律仍能有力地预测相对多数制下的政党体系形态。
2. Downs’s Median Voter Theorem | 唐斯中位选民定理
Downs’s median voter theorem predicts that in a two‑party election with a single policy dimension, both parties will converge towards the ideological centre. The equilibrium policy position is the ideal point of the median voter – the voter who is exactly in the middle of the distribution, with half of the voters more left‑wing and half more right‑wing. Any deviation from that centre would allow the opponent to capture a majority by moving closer to the median.
唐斯中位选民定理预测，在单一政策维度的两党选举中，两个政党都会向意识形态中心靠拢。均衡的政策位置就是中位选民的理想点——该选民恰好位于分布的正中央，一半选民比其更左，另一半比其更右。任何偏离中心的举动都会使对手有机会通过向中位选民靠拢而赢得多数。
This theorem helps explain why major parties in winner‑takes‑all systems often have broadly similar manifestos. Downs acknowledged that turnout and multi‑party competition complicate the picture, but the centripetal logic remains a cornerstone of spatial voting theory.
该定理有助于解释为何赢者通吃体系中的主要政党往往拥有颇为相似的政纲。唐斯承认投票率和多党竞争会令情况复杂化，但这种向心力逻辑仍是空间投票理论的基石。
3. Effective Number of Parties (Laakso‑Taagepera Formula) | 有效政党数（拉索‑塔格培拉公式）
The effective number of parties (ENP) is given by:
ENP = 1 / Σ(pᵢ²)
where pᵢ is the vote share (or seat share) of party i, expressed as a decimal fraction. ENP tells us how many hypothetical equal‑sized parties would produce the same degree of fragmentation. If one party has 100 % of the vote, ENP = 1. If two parties each have 50 %, ENP = 2. As votes become more evenly divided among many parties, ENP rises.
有效政党数（ENP）公式如下：
ENP = 1 / Σ(pᵢ²)
其中 pᵢ 是政党 i 的得票比例（或席位比例），以小数表示。ENP 告诉我们，需要多少个假想中同等规模的政党才能产生同样的碎片化程度。若一个政党获得 100% 选票，则 ENP = 1；若两个政党各获 50%，则 ENP = 2；票数在多个政党间越分散，ENP 越高。
For example, in an election where Party A has 40 % (0.4), Party B has 30 % (0.3) and Party C has 30 % (0.3), the calculation is: Σ(pᵢ²) = 0.4² + 0.3² + 0.3² = 0.16 + 0.09 + 0.09 = 0.34. Hence ENP = 1 / 0.34 ≈ 2.94, indicating a near‑three‑party system.
例如，在一次选举中，A 党得票 40%（0.4），B 党 30%（0.3），C 党 30%（0.3），则 Σ(pᵢ²) = 0.4² + 0.3² + 0.3² = 0.16 + 0.09 + 0.09 = 0.34。因此 ENP = 1 / 0.34 ≈ 2.94，表明接近三党体系。
4. Gallagher Index of Disproportionality | 加拉格尔比例性偏差指数
The Gallagher index measures the overall distance between vote shares and seat shares. Its formula is:
LSq = √(½ Σ(vᵢ – sᵢ)²)
where vᵢ is the vote share and sᵢ is the seat share of party i. The index uses squared differences to give greater weight to larger discrepancies, and the square root brings the value back to the original percentage scale. A score of 0 indicates perfect proportionality; higher values indicate more disproportional results.
加拉格尔指数衡量选票份额与席位份额之间的总体差距。其公式为：
LSq = √(½ Σ(vᵢ – sᵢ)²)
其中 vᵢ 和 sᵢ 分别为政党 i 的得票比例和席位比例。该指数采用平方差放大较大偏差的权重，再取平方根使数值回到原来的百分比量纲。零分代表完全比例性，分值越高说明越不成比例。
For instance, in the 2019 UK general election the Gallagher index was around 11.7, reflecting a large bonus for the winning party under first‑past‑the‑post. By contrast, many proportional representation systems yield indices below 3.
例如，2019 年英国大选的加拉格尔指数约为 11.7，反映出领先者当选制下胜选政党的巨额红利。相较之下，许多比例代表制选举的指数则低于 3。
5. Rae’s Index of Proportionality | 雷伊比例性指数
Rae’s index is a simple average of the absolute differences between vote and seat shares. Its formula is:
I = (1/n) Σ|vᵢ – sᵢ|
where n is the number of parties included. This index is easy to compute but can be sensitive to the inclusion of very small parties, as they contribute equally to the average. A high Rae index signals significant disproportionality.
雷伊指数是各政党得票与席位份额绝对差值的简单平均。其公式为：
I = (1/n) Σ|vᵢ – sᵢ|
其中 n 为纳入计算的政党数量。该指数计算简便，但会因纳入极小政党而变得敏感，因为它们同样参与平均。雷伊指数高则意味着比例性偏差显著。
If three parties have |vᵢ – sᵢ| values of 0.05, 0.08 and 0.02, then Rae’s I = (0.05 + 0.08 + 0.02)/3 = 0.05, or 5 percentage points average deviation per party.
若三个政党的 |vᵢ – sᵢ| 值分别为 0.05、0.08 和 0.02，则雷伊指数 = (0.05+0.08+0.02)/3 = 0.05，即每个政党平均偏差 5 个百分点。
6. Loosemore‑Hanby Index | 卢斯莫尔‑汉比指数
The Loosemore‑Hanby index sums the absolute vote‑seat differences and divides by two. It represents the proportion of seats that would need to be redistributed to achieve perfect proportionality. The formula is:
D = ½ Σ|vᵢ – sᵢ|
This index is especially intuitive: a result of 0.12 means that 12 % of seats would have to be reallocated to make the outcome perfectly proportional. Like Rae’s index, it does not square the differences, so it does not disproportionately punish large deviations.
卢斯莫尔‑汉比指数将绝对差值之和除以二。它表示要达到完全比例性所需重新分配的席位比例。其公式为：
D = ½ Σ|vᵢ – sᵢ|
该指数特别直观：结果为 0.12 即意味着 12% 的席位需要重新分配才能实现完全比例性。与雷伊指数一样，它不进行平方处理，因此不会过度惩罚大的偏差。
Using the earlier example with |vᵢ – sᵢ| of 0.05, 0.08 and 0.02, the sum is 0.15, so D = 0.075. This suggests 7.5 % of seats would need to change hands.
结合前述 |vᵢ – sᵢ| 为 0.05、0.08 和 0.02 的例子，总和为 0.15，因此 D = 0.075，表明 7.5% 的席位需要易手。
7. Natural Electoral Threshold Formula | 自然选举门槛公式
In proportional representation systems using multi‑member districts, the effective threshold a party must cross to win its first seat can be estimated by:
T ≈ 75% / (M + 1)
where M is the district magnitude (number of seats per constituency). This formula, derived from empirical observation, gives the approximate vote percentage below which a party is unlikely to win any representation. For instance, in a district with M = 5, T ≈ 75/6 = 12.5 %, meaning a party typically needs at least 12.5 % of the vote to secure a seat.
在采用复数选区的比例代表制中，政党赢得其首个席位所需跨越的实际门槛可用以下公式估算：
T ≈ 75% / (M + 1)
其中 M 为选区规模（每个选区的席位数）。这一经验公式给出了一个政党很难赢得任何席位的最低得票百分比。例如，在 M = 5 的选区，T ≈ 75/6 = 12.5%，意味着一个政党通常需要至少 12.5% 的选票才能取得席位。
The natural threshold is important for understanding why countries with small district magnitudes (such as Spain, where most provinces elect 3‑6 MPs) have a less proportional outcome than those with large districts or a single national district.
理解自然门槛有助于明白，为何选区规模较小的国家（如西班牙，多数省份选举 3‑6 名议员）比那些采用大选区或单一全国选区的国家比例性更差。
8. The Cube Law | 立方律
The cube law describes the relationship between votes and seats in two‑party single‑member district systems. It states that the ratio of seats won by the two major parties tends to equal the cube of their vote ratio:
Sₐ / Sᵦ ≈ (Vₐ / Vᵦ)³
where Sₐ and Sᵦ are the number of seats won by parties A and B, and Vₐ and Vᵦ are their total votes. For example, if the leading party wins 55 % of the two‑party vote and the rival wins 45 %, the ratio Vₐ/Vᵦ = 55/45 ≈ 1.22. The cube of 1.22 is about 1.81, so we would expect the seat ratio to be roughly 1.81 : 1, heavily favouring the winner.
立方律描述了两党单一选区制中选票与席位的关系。它指出，两大政党赢得的席位比大致等于其选票比的立方：
Sₐ / Sᵦ ≈ (Vₐ / Vᵦ)³
其中 Sₐ 和 Sᵦ 分别为政党 A 和 B 赢得的席位数，Vₐ 和 Vᵦ 为各自的总票数。例如，若领先党赢得两党票数的 55%，对手赢得 45%，则 Vₐ/Vᵦ = 55/45 ≈ 1.22。1.22 的立方约为 1.81，因此我们预期席位比约为 1.81 : 1，显著有利于胜者。
Although the cube law fits many historical UK elections, it can be distorted by third‑party votes and regional concentrations. Nonetheless, it remains a classic illustration of how mechanical and geographical effects amplify the winner’s advantage in majoritarian systems.
虽然立方律与英国许多历史选举吻合，但可能受到第三党票数和地域集中的扭曲。尽管如此，它仍是说明多数制中机械与地理效应如何放大胜者优势的经典例证。
9. Riker’s Minimum Winning Coalition Theorem | 赖克最小获胜联盟定理
Riker’s theorem is a game‑theoretic prediction about coalition formation among rational, office‑seeking politicians. It states that in a legislature where no party has a majority, the coalition that forms will be the smallest possible winning coalition – i.e. the one that commands just over 50 % of the seats while including the minimum number of members. The logic is that by minimising partners, each member of the coalition can command a larger share of cabinet portfolios and policy influence.
赖克定理是博弈论对理性、追求官职的政治家组建联盟行为的预测。它指出，在没有任何政党占多数的立法机构中，所形成的联盟将是最小的可能获胜联盟——即恰好掌控超过 50% 席位且包含最少成员的联盟。其逻辑在于，通过最小化合作伙伴，联盟中的每个成员都能分得更大份额的内阁职位和政策影响力。
An example would be a parliament where seats are distributed as 45 %, 30 %, 15 % and 10 %. A minimal winning coalition might be the 45 % + 15 % parties (total 60 %), but Riker would predict a still smaller combination if one exists: perhaps 30 % + 15 % + 10 % = 55 %. The latter includes more parties but a smaller total seat surplus over 50 %; however, theorists often search for the coalition with the fewest members first, then minimal surplus. Riker’s theorem helps explain why surplus-majority coalitions are rarely observed when politicians are purely office‑seeking.
例如，一个议会席位分布为 45%、30%、15% 和 10%。一个最小获胜联盟可能是 45%+15% 的政党（共 60%），但赖克会预测如果存在更小的组合：如 30%+15%+10% = 55%。后者包含更多政党但席位超出 50% 的幅度更小；不过理论家通常先寻找成员最少的联盟，再考虑最小超出。赖克定理有助于解释，为何当政治人物纯粹追求官职时，罕见出现超出过半席位较多的超大联盟。
10. Arrow’s Impossibility Theorem | 阿罗不可能定理
Arrow’s impossibility theorem is a fundamental result in social choice theory. It demonstrates that no ranked voting system can convert individual preference orderings into a collective ranking while simultaneously satisfying four apparently reasonable conditions: Non‑dictatorship (no single voter determines the outcome); Unanimity (if every voter prefers X over Y, society prefers X over Y); Independence of Irrelevant Alternatives (the societal ranking of X and Y depends only on individual rankings of X and Y); and Transitivity (if society prefers X over Y and Y over Z, it must prefer X over Z).
阿罗不可能定理是社会选择理论的一个基础性结论。它证明，没有任何排序投票制度能够将个人偏好排序转化为集体排序，同时满足四个看似合理的条件：非独裁（没有任何单个选民能决定结果）；一致性（如果所有选民都偏好 X 甚于 Y，则社会也应偏好 X 甚于 Y）；无关选项的独立性（社会对 X 和 Y 的排序仅取决于个体对 X 和 Y 的排序）；以及可传递性（若社会偏好 X 甚于 Y、Y 甚于 Z，则必须偏好 X 甚于 Z）。
Arrows theorem has profound implications for all electoral systems that rely on ranked ballots, such as the Alternative Vote. It tells us that no system can be perfectly fair across all criteria; every rule involves trade‑offs. When studying electoral reform for the AS Eduqas specification, students can use this theorem to evaluate claims that one voting system is unambiguously ‘better’ than another.
阿罗定理对所有依赖排序选票的选举制度（如替代投票制）具有深远影响。它告诉我们，没有任何制度能在所有标准上都绝对公平；每条规则都涉及权衡。在学习 AS Eduqas 课程中选举改革的内容时，学生可用该定理来评价“某种投票制度明显‘更优’”的主张。
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