📚 Debates on why different electoral systems are used in the UK | 英国使用不同选举制度的数学辩论
The United Kingdom employs a patchwork of electoral systems for its various legislatures, including First Past the Post for Westminster, the Additional Member System for the Scottish Parliament, and the Single Transferable Vote for Northern Ireland Assembly elections. From a mathematical perspective, each system embodies a distinct algorithm that converts votes into seats, balancing conflicting criteria such as proportionality, local representation, and simplicity. This article explores the key mathematical arguments that illuminate why different systems are used in the UK, examining concepts like seat allocation algorithms, disproportionality indices, strategic voting, and the limits imposed by Arrow’s impossibility theorem.
英国在其各级议会选举中采用了多种不同的选举制度，如威斯敏斯特的下议院使用“简单多数制”，苏格兰议会采用“附加成员制”，北爱尔兰议会则使用“单记可让渡投票制”。从数学角度来看，每种制度都是一种独特的将选票转化为席位的算法，需要在比例性、地方代表性和简单性等相互冲突的标准之间进行权衡。本文探讨了有助于理解英国为何使用不同制度的核心数学论证，涉及席位分配算法、不相称性指数、策略投票以及阿罗不可能性定理所施加的限制等概念。
1. Electoral Systems as Mathematical Mappings | 作为数学映射的选举制度
An electoral system can be defined as a function f that takes a set of vote distributions across candidates or parties and maps them to a seat distribution. In the simplest case, First Past the Post (FPTP) applies a “winner-takes-all” rule: in each single-member constituency, the candidate with the largest number of votes wins, regardless of whether this number exceeds a majority. Mathematically, if constituency k has votes vᵢ for candidate i, the seat goes to the candidate maximising vᵢ, and all other votes are discarded in terms of representation. This operation discards a significant amount of information.
选举制度可以定义为一种函数，它将候选人或者政党之间的选票分布映射为席位分布。在最简单的情形下，简单多数制（FPTP）采用“赢者通吃”的规则：在每一个单议席选区中，获得最多选票的候选人当选，无论这一票数是否超过半数。从数学上说，如果选区 k 中候选人 i 获得 vᵢ 票，席位属于使 vᵢ 最大化的候选人，而其他所有选票在代表层面均被丢弃。这一过程丢弃了大量的信息。
In contrast, proportional representation (PR) systems attempt to minimise the discrepancy between the share of votes and the share of seats. A typical seat allocation algorithm uses a divisor method, such as the D’Hondt formula, which calculates successive quotients qᵢ = Vᵢ / (sᵢ + 1), where Vᵢ is the party’s total votes and sᵢ the seats already allocated. The highest quotient wins the next seat. This iterative process approximates proportional representation by reducing large deviations between vote shares and seat shares.
相比之下，比例代表制（PR）则试图最小化得票份额与席位份额之间的差异。一种典型的席位分配算法使用除数法，例如汉狄公式，它计算连续的商数 qᵢ = Vᵢ / (sᵢ + 1)，其中 Vᵢ 是政党的总票数，sᵢ 是该党已获得的席位。最高的商数赢得下一个席位。这一迭代过程通过缩小得票份额与席位份额之间的较大偏差来近似实现比例代表。
2. Plurality Voting and the Cube Law | 多数投票制与三次方定律
A long-observed empirical regularity in FPTP systems is the cube law, which relates the ratio of seats won by two major parties to the cube of the ratio of their votes. If Party A receives a vote share x and Party B receives (1−x) in a two-party contest, the seat ratio S_A / S_B is approximately (x / (1−x))³. This implies that small shifts in vote share are mathematically amplified into large seat gains, creating “manufactured majorities.” The cube law explains why FPTP tends to produce strong single-party governments, a central argument in favour of its use for UK general elections.
在简单多数制下，一条长期观察到的经验规律是三次方定律，它将两大政党赢得的席位比与其得票比的立方联系起来。如果政党 A 获得选票份额 x，政党 B 获得 (1−x)，在双党竞争中，席位比 S_A / S_B 近似为 (x / (1−x))³。这意味着得票份额的微小变动在数学上被放大为巨大的席位变化，从而产生“人为制造的多数”。三次方定律解释了为何简单多数制倾向于产生强势的单党政府，这也是支持在英国大选中使用该制度的一个核心论点。
Mathematically, the cube law can be derived from the distribution of constituency-level vote shares. Under certain assumptions about the variance of these shares, the number of constituencies won by a party follows a log-normal distribution, leading to the cubic amplification. Critics argue that this distortion is not a desirable property when the goal is fair representation, because it systematically disadvantages third parties whose support is dispersed rather than concentrated in particular constituencies.
从数学上讲，三次方定律可以从选区层面得票份额的分布推导出来。在对这些份额的方差做出某些假设的情况下，一个政党赢得的选区数量服从对数正态分布，从而导致了立方放大效应。批评者认为，当目标是公平代表时，这种扭曲并不是一种理想的性质，因为它系统性地不利于那些支持者分散而非集中于特定选区的第三党。
3. Seat Allocation Algorithms: D’Hondt versus Sainte-Laguë | 席位分配算法：汉狄法与圣拉格法
Proportional systems rely on clear mathematical rules for translating votes into seats. The D’Hondt method, widely used in the UK for European Parliament elections (when the UK was a member) and for distributing top-up seats in the Additional Member System, uses the divisors 1, 2, 3, … The Sainte-Laguë method (employed in Germany and New Zealand) uses divisors 1, 3, 5, …, which gives a slight advantage to smaller parties. The difference arises because D’Hondt sets the first divisor to 1 and increments by 1, making it harder for a small party to win its first seat when a large party has already allocated several seats.
比例代表制依赖于明确的数学规则将选票转化为席位。汉狄法在英国欧洲议会选举（英国还是成员国时）以及附加成员制的增补席位分配中被广泛使用，它使用的除数为 1, 2, 3, … 圣拉格法（在德国和新西兰使用）则使用除数 1, 3, 5, …，这略微有利于小党派。这种差异的产生是因为汉狄法将第一个除数设为 1 并且增量为 1，这使得当大党已经分配了若干席位后，小党派赢得其第一个席位的难度加大。
Formally, the D’Hondt algorithm selects the next seat by maximising Vᵢ / (sᵢ + 1) over all parties i. The Sainte-Laguë algorithm maximises Vᵢ / (2sᵢ + 1). The choice between these algorithms is not politically neutral; it is a mathematical decision that alters the seat–vote relationship. In the UK, the Additional Member System uses D’Hondt for the regional lists, which tends to favour the larger parties and maintain a balance with the constituency FPTP seats.
形式上，汉狄算法通过在所有政党 i 中最大化 Vᵢ / (sᵢ + 1) 来选择下一个席位。圣拉格算法则最大化 Vᵢ / (2sᵢ + 1)。在这些算法之间进行选择并非政治中立，而是一个会改变席位–选票关系的数学决策。在英国，附加成员制中的地区名单席位采用汉狄法，这往往有利于较大的政党，并与选区简单多数席位保持平衡。
4. The Single Transferable Vote and the Droop Quota | 单记可让渡投票制与德鲁普基数
The Single Transferable Vote (STV), used for Northern Ireland Assembly and Scottish local government elections, is a preference-based proportional system. In STV, voters rank candidates, and a quota determines the threshold for election. The Droop quota is given by:
Quota = ⌊(total valid votes) / (seats + 1)⌋ + 1
This formula ensures that no more candidates can reach the quota than there are seats to fill. Surplus votes from elected candidates are transferred to remaining candidates according to second preferences, and if no candidate reaches the quota, the lowest-polling candidate is eliminated and their votes redistributed.
单记可让渡投票制（STV）用于北爱尔兰议会和苏格兰地方政府选举，它是一种基于偏好排序的比例制。在 STV 中，选民为候选人排序，一个基数决定了当选的门槛。德鲁普基数由以下公式给出：
基数 = ⌊(有效总票数) / (席位 + 1)⌋ + 1
这个公式确保达到基数的候选人数不会超过应选席位数。已当选候选人的剩余选票根据第二偏好转移给其他候选人，如果没有候选人达到基数，则淘汰得票最低的候选人并将其选票重新分配。
Mathematically, STV is a computationally intensive algorithm that requires iterative transfers and eliminations. It maximises the impact of voters’ preferences and minimises wasted votes, defined as votes that do not contribute to the election of a candidate. Proponents argue that STV gives a finer-grained representation; opponents note that the counting process is far less transparent than FPTP, and the results can be sensitive to the quota formula and transfer rules.
从数学上看，STV 是一种计算密集型的算法，需要进行反复的选票转移和淘汰。它最大化了选民偏好的影响力，并最小化了“废票”——即那些对候选人当选未做出贡献的选票。支持者认为 STV 提供了更精细的代表性；反对者则指出，计票过程的透明度远低于简单多数制，并且结果可能对基数公式和转移规则十分敏感。
5. Measuring Disproportionality: The Gallagher Index | 衡量不相称性：加拉格尔指数
One way to evaluate an electoral outcome is to calculate a disproportionality index. The most common is the Gallagher least-squares index:
LSq = √(½ Σ (Vᵢ − Sᵢ)² )
where Vᵢ is the vote share and Sᵢ is the seat share of party i, expressed as percentages. A low value indicates a close match between votes and seats; FPTP elections in the UK typically produce values around 10–15, while PR systems often yield values below 5.
评价选举结果的一种方法是计算不相称性指数。最常用的是加拉格尔最小二乘指数：
LSq = √(½ Σ (Vᵢ − Sᵢ)² )
这里 Vᵢ 是政党 i 的得票份额，Sᵢ 是其席位份额，二者均以百分数表示。较低的值意味着选票与席位匹配度较高；英国简单多数制选举通常产生 10–15 左右的值，而比例代表制下的选举往往得到低于 5 的值。
This index is used in academic debates on electoral reform. Supporters of FPTP might accept larger disproportionality as a price for stable government, whereas reformers argue that minimising LSq should be a primary objective. The choice of index itself is mathematically debatable: other measures, like the Loosemore–Hanby index (LD = ½ Σ |Vᵢ − Sᵢ|), focus on the total deviation, while Gallagher’s squares penalise larger errors more heavily.
该指数被用于关于选举改革的学术辩论中。简单多数制的支持者可能接受较高的不相称性作为稳定政府的代价，而改革派则认为最小化 LSq 应当成为首要目标。指数的选择本身在数学上也是可辩论的：其他度量，如卢斯莫尔–汉比指数（LD = ½ Σ |Vᵢ − Sᵢ|），关注的是总偏差，而加拉格尔指数通过平方更严厉地惩罚较大的误差。
6. Strategic Voting and Game Theory | 策略投票与博弈论
Electoral systems influence how rational voters behave. Under FPTP, supporters of a minor party may decide to vote for a less preferred but viable candidate to avoid “wasting” their vote. This strategic voting can be modelled using game theory. In a simple coordination game, voters receive a payoff u depending on whether their most preferred candidate wins, but they also consider the probability of influencing the outcome. The equilibrium often leads to a two-party system, as predicted by Duverger’s law, which is evident in the UK’s Westminster elections.
选举制度影响理性选民的行为方式。在简单多数制下，小党的支持者可能会决定投票给一个不那么偏好但有机会当选的候选人，以避免“浪费”选票。这种策略投票可以用博弈论来建模。在简单的协调博弈中，选民根据其最偏好候选人是否胜选获得效用 u，但他们也会考虑影响选举结果的概率。博弈均衡常常导致两党制，正如杜瓦杰定律所预测的那样，这在英国威斯敏斯特选举中表现得尤为明显。
Under STV or the Additional Member System, the incentives change. In STV, voters can rank their sincere favourite first without fear of helping their least preferred candidate, because the single transferable vote mechanism allows a vote to flow to a viable candidate if the favourite is eliminated. Game-theoretic analysis shows that these systems reduce the pressure for strategic voting, leading to outcomes that better reflect sincere preferences.
在 STV 或附加成员制下，激励结构发生了变化。在 STV 中，选民可以真诚地将他们最喜爱的候选人排在第一位，而不必担心会帮助最不喜欢的候选人，因为单记可让渡投票机制使得如果最喜爱者被淘汰，选票可以流向有当选希望的候选人。博弈论分析表明，这些制度降低了策略投票的压力，从而使结果更好地反映真实的偏好。
7. Arrow’s Impossibility Theorem and the Limits of Aggregation | 阿罗不可能性定理与聚合的极限
No discussion of electoral systems is complete without Arrow’s impossibility theorem. Arrow proved that no ranked-voting electoral system can simultaneously satisfy a set of seemingly reasonable conditions: unrestricted domain, non-dictatorship, Pareto efficiency, and independence of irrelevant alternatives. This theorem demonstrates that all voting methods involve mathematical trade-offs. For instance, FPTP satisfies non-dictatorship and Pareto efficiency but violates independence of irrelevant alternatives, as a third candidate can change the winner between the two main contenders.
任何关于选举制度的讨论都离不开阿罗不可能性定理。阿罗证明，没有任何一种基于排序的投票制度能同时满足一组看似合理的条件：无限制定义域、非独裁、帕累托最优，以及不相关备选方案的独立性。这一定理表明，所有投票方法都涉及数学上的权衡取舍。例如，简单多数制满足非独裁和帕累托最优，却违反了不相关备选方案的独立性，因为第三位候选人可以改变两位主要竞争者之间的胜负。
The theorem justifies the coexistence of different systems in the UK: a single system cannot be universally ideal, so different tiers of government adopt systems that prioritise different mathematical properties. Westminster values decisiveness and local accountability, while devolved bodies may prioritise proportionality and inclusiveness. Thus, the multiplicity of systems reflects a pragmatic response to the theorem’s constraints.
这一定理为英国不同制度并存提供了理由：单一制度无法在普遍意义上是理想的，因此不同层级的政府采用了优先考虑不同数学性质的制度。威斯敏斯特重视决断力和地方问责制，而权力下放后的机构可能更优先考虑比例性和包容性。因此，多种制度的并存反映了一种对这一定理约束的务实回应。
8. Wasted Votes and Efficiency Gaps | 废票与效率差
The concept of wasted votes further illuminates the mathematical debate. In a single-member district, all votes for the losing candidate and all votes for the winning candidate beyond the threshold needed to win are considered wasted. The efficiency gap EG is defined as:
EG = (Wasted votes for Party A − Wasted votes for Party B) / Total votes
A large absolute efficiency gap indicates that one party has translated its votes into seats far more efficiently, often due to the geographical distribution of supporters. In UK general elections, the efficiency gap frequently favours the largest party, contributing to criticisms that FPTP systematically advantages parties with geographically concentrated support.
废票的概念进一步阐明了这场数学辩论。在一个单议席选区中，投给落选候选人的所有选票，以及投给当选者但超过胜选所需门槛的选票，都被视为废票。效率差 EG 的定义为：
EG = (政党 A 的废票 − 政党 B 的废票) / 总票数
效率差的绝对值很大，表明某个政党将自己的选票转化为席位的效率远高于对手，这常常是由于支持者的地理分布所致。在英国大选中，效率差经常有利于最大的政党，这引发了这样的批评：简单多数制系统性地偏袒那些支持者在地理上集中的政党。
Proposals for electoral reform, including the 2011 referendum on the Alternative Vote (AV), were partly motivated by the desire to reduce wasted votes. AV, a version of instant-runoff voting, ensures that the winner must achieve a majority by eliminating candidates and redistributing preferences. While AV does not guarantee proportionality, it reduces some strategic voting and the efficiency gap in individual constituencies. The referendum’s defeat meant FPTP was maintained, reinforcing the dominance of two large parties and preserving higher wasted vote totals.
选举改革提案，包括 2011 年关于排序复选制（AV）的公投，其部分动机就是要减少废票。AV 是即时决选制的一种形式，通过淘汰候选人和重新分配偏好来确保胜选者必须获得多数票。AV 虽然不能保证比例性，但可以减少一些策略投票和单个选区内的效率差。公投的失败意味着简单多数制得以维持，从而巩固了两大政党的主导地位，并维持着较高的废票总量。
9. The Additional Member System as a Mathematical Hybrid | 作为数学混合体的附加成员制
The Additional Member System (AMS), used for the Scottish Parliament, the Welsh Senedd, and the London Assembly, is a hybrid combining FPTP constituency seats with regional list seats allocated proportionally. Mathematically, AMS can be seen as a two-stage process. First, constituency seats are filled using FPTP. Then, the list seats are distributed so that the overall seat allocation more closely matches the vote shares. The D’Hondt formula is applied, but each party’s constituency seats are taken into account, so the corrected quotient becomes Vᵢ / (sᵢ_const + sᵢ_list + 1).
用于苏格兰议会、威尔士议会和伦敦议会的附加成员制（AMS）是一种混合制度，它将 FPTP 的选区席位与按比例分配的地区名单席位结合在一起。从数学上讲，AMS 可以被视为一个两阶段过程。首先，选区席位通过 FPTP 产生。然后，分配名单席位使得总体的席位分配更贴近得票份额。汉狄公式被应用，但每个政党的选区席位都要纳入计算，因此修正后的商数变为 Vᵢ / (sᵢ_const + sᵢ_list + 1)。
This design reflects a compromise: the constituency link is preserved, while the overall disproportionality is reduced compared to pure FPTP. The degree of proportionality can be adjusted by changing the ratio of constituency to list seats. In Scotland, 73 constituency seats are complemented by 56 list seats, leading to a fairly proportional outcome. Mathematical simulation shows that if the list seats were reduced, disproportionality would increase, approaching a pure FPTP outcome.
这一设计反映了一种折衷：保留了选区纽带，同时与纯粹的 FPTP 相比降低了总体的不相称性。比例性的程度可以通过改变选区席位与名单席位的比例来调节。在苏格兰，73 个选区席位加上 56 个名单席位，导致结果具有相当高的比例性。数学模拟表明，如果减少名单席位，不相称性就会增加，接近纯粹的 FPTP 结果。
10. Real-World Constraints and Algorithmic Choice | 现实世界的约束与算法选择
The mathematical elegance of a system must be balanced against practical constraints. FPTP is computationally trivial: results can be declared soon after polls close. STV, however, requires multiple rounds of counting and data management, which is feasible only with computerised systems or prolonged manual counts. The complexity of an electoral algorithm affects public trust. In the UK, the speed and simplicity of FPTP are often cited as virtues, while the more mathematically sophisticated systems are accepted only where proportional representation is deemed essential for legitimacy, as in Northern Ireland’s consociational arrangements.
一种制度在数学上的精妙性，必须与现实约束相平衡。FPTP 的计算极为简单：投票结束后很快就可以公布结果。而 STV 则需要多轮计票和数据管理，只有当计算机系统到位或进行长时间的人工计票时才可行。选举算法的复杂性会影响公众信任。在英国，FPTP 的快速和简单常常被引为优点，而那些在数学上更为复杂的制度，只有在比例代表被视为合法性的必要条件时才被接受，例如在北爱尔兰的权力共享安排中。
The decision on which algorithm to adopt is ultimately a normative one, informed by mathematical analysis. Edexcel A-level Mathematics, particularly in the Decision Mathematics strand, equips students with the tools to analyse voting algorithms, compare efficiencies, and recognise the computational trade-offs. Understanding these mathematical underpinnings enriches the debate on why the UK uses different electoral systems for different levels of governance.
采用哪种算法的决策最终还是一个规范性的选择，并由数学分析提供依据。爱德思 A-level 数学，尤其是决策数学部分，让学生掌握分析投票算法、比较效率以及识别计算权衡的方法。理解这些数学基础，可以使关于英国为何在不同治理层级使用不同选举制度的辩论更为丰富。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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