📚 The Additional Member System, Single Transferable Vote and Supplementary Vote: Advantages and Disadvantages | 附加成员制、单一可转移投票制与补充投票制：优势与劣势
Electoral systems translate votes into seats, and their mathematical properties directly shape representation. The Additional Member System (AMS), Single Transferable Vote (STV) and Supplementary Vote (SV) are three distinct models used in the UK and beyond. This article compares their mechanisms, evaluates their proportionality through numerical indices like the Gallagher index, and analyses wasted votes using simple algebraic formulations. Understanding the mathematical underpinnings of each system helps to assess fairness, voter power and the strategic incentives they create.
选举制度将选票转化为席位，其数学性质直接影响代议制的质量。附加成员制（AMS）、单一可转移投票制（STV）和补充投票制（SV）是英国及其他地区使用的三种不同模式。本文比较它们的运作机制，通过加拉格尔指数等数值指标评估比例性，并运用简单的代数公式分析废票。理解每个制度背后的数学原理，有助于评估公正性、选民权力及其产生的策略激励。
1. Overview of the Three Electoral Systems | 三种选举制度概述
AMS, STV and SV each employ different algorithms for seat allocation. AMS combines single-member constituencies with regional party lists, typically using the d’Hondt divisor method to distribute additional seats. STV asks voters to rank candidates in multi-member constituencies, applying the Droop quota to determine winners. SV restricts voters to marking a first and second preference, effectively conducting an instant runoff if no candidate secures a majority of first-choice votes.
AMS、STV和SV各自采用不同的席位分配算法。AMS将单选区与区域政党名单相结合，通常使用东特除数法分配额外席位。STV要求选民在多席位选区中对候选人排序，利用德鲁普配额确定当选者。SV限制选民只标记第一和第二偏好，若无候选人获得过半数第一选择票，则实质上进行即时决选。
The mathematical core of each system defines its character: AMS uses a two-tier formula, STV relies on iterative surplus transfers, and SV compresses a two-round process into a single ballot. All three attempt to reconcile local representation with overall proportionality, but they differ sharply in how they count and value each vote.
每个系统的数学核心决定了其特性：AMS使用双层公式，STV依赖于迭代的剩余票转移，SV则将两轮选举压缩为一次投票。三者都试图调和地方代表性与整体比例性，但在如何计数和衡量每张选票的价值上存在显著差异。
2. Understanding the Additional Member System (AMS) | 理解附加成员制
In AMS, each voter receives two ballots: one for a constituency representative elected by first-past-the-post (FPTP), and one for a party list in a larger region. The regional seats are allocated to compensate for disproportionalities created by the constituency results. The number of regional seats a party receives is determined by the formula: adjusted votes = regional list votes ÷ (constituency seats already won + 1), iteratively assigning seats to the highest adjusted vote until all regional seats are filled.
在AMS中，每位选民有两张选票：一张选票通过简单多数制（FPTP）选举选区代表，另一张选票投给较大区域的政党名单。区域席位的分配旨在补偿选区结果造成的比例失调。一个政党获得的区域席位数由公式决定：调整后票数 = 区域名单票数 ÷ (已赢得的选区席位数 + 1)，反复将席位分配给调整后票数最高者，直至所有区域席位分配完毕。
Mathematically, the d’Hondt divisor sequence is 1, 2, 3, … which favours larger parties slightly compared to the Sainte-Laguë method (divisors 1, 3, 5, …). This algorithm means that a party winning many constituency seats will find it harder to gain additional regional seats, creating a built-in corrective mechanism.
从数学上看，东特法除数序列为1, 2, 3, …，与圣拉古法（除数1, 3, 5, …）相比略有利于大党。这一算法意味着赢得许多选区席位的政党更难获得额外的区域席位，形成一种内在的纠正机制。
3. Advantages of AMS | 附加成员制的优势
AMS achieves a balance between local representation and proportionality. The Gallagher index, which measures the deviation between vote share and seat share, is typically much lower under AMS than under pure FPTP. For instance, if a party wins 40% of the list votes but only 30% of constituency seats, the top-up seats narrow the gap, reducing the disparity index towards zero.
AMS在地方代表性与比例性之间取得了平衡。衡量得票率与席位率偏差的加拉格尔指数在AMS下通常远低于纯粹FPTP。例如，若一个政党获得40%的名单票却只赢得30%的选区席位，额外议席会缩小差距，使差异指数趋近于零。
From a mathematical perspective, each voter has two independent vote vectors, allowing split-ticket voting. This increases the expressiveness of the ballot: a voter can support a local candidate from one party while endorsing another party regionally, optimising personal representation without wasting the list vote.
从数学角度看，每位选民拥有两个独立的选举向量，允许分裂投票。这增加了选票的表达力：选民可以支持一个政党的本地候选人，同时区域性地支持另一个政党，从而在不浪费名单票的情况下优化个人代表权。
4. Disadvantages of AMS | 附加成员制的劣势
AMS creates two classes of representatives – constituency and list – which can lead to accountability ambiguities. Mathematically, the mixed nature means that the overall proportionality depends on the ratio of constituency to list seats. If the list portion is too small, the corrective effect weakens; the system tends towards the disproportionality of FPTP.
AMS制造了两类代表——选区代表和名单代表——可能引发问责模糊。从数学上看，混合性质意味着整体比例性取决于选区席位与名单席位的比例。如果名单部分太小，纠正作用就会减弱，制度会趋向于FPTP的比例失调。
Furthermore, the allocation formula can produce overhang seats when a party wins more constituency seats than its list vote entitlement. The mathematical handling of overhangs (adding extra seats or reducing list allocations) introduces instability and makes the exact seat distribution unpredictable from vote totals alone.
此外，当一个政党赢得的选区席位超过其名单票应得席位数时，分配公式可能产生超额席位。超额席位在数学上的处理方式（增加额外议席或削减名单分配）会带来不稳定性，使得仅凭得票总数难以预测确切的席位分布。
5. Understanding the Single Transferable Vote (STV) | 理解单一可转移投票制
STV uses multi-member constituencies and ranked ballots. The threshold for election is the Droop quota, defined as:
Quota = ⌊Total valid votes ÷ (Seats + 1)⌋ + 1
任何候选人的第一偏好票数达到或超过配额即当选。超出配额的剩余票按照下一有效偏好的权重转移。该过程持续进行，直至所有席位填满，每次转移都重新计算。
Candidates exceeding the quota have their surplus transferred using a fractional value: transfer value = (votes received − quota) ÷ total votes received. This mathematical step preserves the exact influence of each ballot paper throughout the count, minimising wasted votes to a degree impossible under single-member systems.
超出配额的候选人剩余票以分数值转移：转移值 = (得票数 − 配额) ÷ 总得票数。这一数学步骤在计票全程保留了每张选票的影响力，将选票浪费降至单选区制度无法达到的程度。
6. Advantages of STV | 单一可转移投票制的优势
STV maximises proportional representation at the constituency level without the need for party lists. The Gallagher index under well-designed STV with sufficient district magnitude typically falls below 3, indicating near-perfect proportionality. Voter choice is extremely rich: preferences form a complete ranking over all candidates.
STV在选区层面最大化了比例代表制，无需政党名单。在选区规模足够大且设计良好的STV下，加拉格尔指数通常低于3，表明几乎完美的比例性。选民选择极其丰富：偏好构成所有候选人的完整排序。
Mathematically, the iterative elimination and surplus transfer procedure ensures that almost every vote contributes to electing a candidate or expresses an active preference. The concept of ‘wasted votes’ approaches zero asymptotically as district magnitude increases. Furthermore, STV discourages tactical voting because ranking one’s sincere favourite first never penalises more preferred candidates.
从数学上看，迭代淘汰和剩余票转移程序确保了几乎所有选票都有助于选出候选人，或表达了积极偏好。随着选区规模增大，“废票”概念渐近于零。此外，STV不鼓励策略性投票，因为将真心最喜欢的候选人排在第一绝不会损害更偏好的候选人。
7. Disadvantages of STV | 单一可转移投票制的劣势
The counting algorithm is complex and can produce non-monotonic outcomes, where increasing support for a candidate might cause their defeat. The mathematical logic of surplus fractions and random subset selection for transfer sometimes introduces an element of chance that can alter close results.
计票算法复杂，且可能产生非单调的结果，即增加对某候选人的支持反而会导致其落选。剩余票分数转移和用于转移的随机子集抽取的数学逻辑，有时会引入随机性，可能改变势均力敌的结果。
STV requires large constituencies, which weakens the geographical link between voters and representatives. Computationally, the manual count is time-consuming, and even with electronic assistance, recounting or verifying results involves tracing hundreds of thousands of ballot orderings. The Condorcet paradox can emerge where collective preferences are cyclical, meaning no mediative winner exists without eliminating some candidates arbitrarily.
STV要求大选区，这会削弱选民与代表之间的地理联系。在计算上，人工点票耗时费力，即便有电子辅助，重新计票或验证结果也需要追踪数十万张选票的排序。孔多塞悖论可能出现，即集体偏好存在循环，这意味着若不任意淘汰某些候选人，就不存在居中的胜者。
8. Understanding the Supplementary Vote (SV) | 理解补充投票制
SV allows voters to mark a first and a second preference. If no candidate receives more than 50% of first preferences, all but the top two candidates are eliminated. The second preferences of eliminated candidates’ ballots are then examined: any votes for the remaining two candidates are added to their totals, and the candidate with the higher combined score wins.
SV允许选民标记第一和第二偏好。若无候选人获得超过50%的第一偏好票，则只保留前两名候选人，其余淘汰。然后查看被淘汰候选人选票上的第二偏好：所有投向剩余两位候选人的第二偏好票分别加入他们的得票总数，综合得分较高者获胜。
Mathematically, SV applies a majority threshold condition and, if necessary, a supplementary redistribution from a restricted subset of ballots. The final tally is a single-round count of two preference vectors. This is far simpler than STV but also less precise in capturing the order of elimination.
从数学上讲，SV应用多数门槛条件，并在必要时从有限选票子集中进行补充再分配。最终计票是两个偏好向量的单次计数。这远比STV简单，但在捕捉淘汰顺序方面精度较低。
9. Advantages of SV | 补充投票制的优势
SV is straightforward to understand and administer. The ballot asks only for two preferences, reducing voter confusion compared with full ranking systems. The mathematical rule – check majority, then redistribute – makes the outcome determined by a simple two-stage process that can be completed quickly even in manual counts.
SV简单易懂，便于管理。选票只要求两个偏好，与全排序系统相比减少了选民的混淆。其数学规则——检查多数，然后重新分配——使得结果由一个简单的两阶段过程决定，即使人工点票也能迅速完成。
By ensuring that the winner must appeal beyond their core supporters to attract second preferences, SV encourages candidates to build broader coalitions. This can be modelled as a minimisation of negative votes: a polarising figure may lead in first preferences but lose when second preferences are counted, promoting candidates with wider acceptability.
通过确保获胜者必须吸引核心支持者之外的选民的第二偏好票，SV鼓励候选人建立更广泛的联盟。这可以建模为最小化负面选票：一个两极分化的人物可能在第一偏好中领先，但当第二偏好计入时落败，从而促使更具广泛接受度的候选人当选。
10. Disadvantages of SV | 补充投票制的劣势
SV does not guarantee a majority of the total votes cast if many second preferences are exhausted or cast for eliminated candidates. Mathematically, a candidate can win with well under 50% of the original valid vote, creating a legitimacy gap. The system also discards information: all preferences below the second are ignored, meaning it may fail to identify a Condorcet winner.
如果许多第二偏好成为废票或投给了被淘汰候选人，SV不能保证获胜者获得总票数的多数。从数学上看，一个候选人可能以远低于原始有效票数50%的得票获胜，造成合法性差距。该系统还会丢弃信息：所有第二偏好以下的选择都被忽略，这意味着它可能无法选出孔多塞赢家。
The restriction to only two preferences can encourage tactical voting similar to the problems in FPTP. Voters may feel pressured to use their second preference for a front-runner rather than a genuine favourite, for fear of wasting the vote. In close three-way races, the elimination order can produce counter-intuitive results that are sensitive to small shifts in first-preference support.
仅仅限制两个偏好可能鼓励策略投票，类似FPTP的问题。选民可能觉得不得不将第二偏好投给领先者而非真心喜欢的候选人，以免浪费选票。在三人竞争激烈的情况下，淘汰顺序可能产生违背直觉的结果，且对第一偏好的微小变化十分敏感。
11. Comparative Mathematical Analysis | 比较数学分析
Three key metrics can be used to evaluate these systems: the proportionality score (Gallagher index), the effective number of parties (Laakso-Taagepera index), and the voter efficiency (fraction of votes that actually elect someone). The table below summarises typical values derived from UK and international elections.
三个关键指标可用于评估这些制度：比例性得分（加拉格尔指数）、有效政党数（拉克索-塔格佩拉指数）和投票者效率（实际选出某人的选票比例）。下表总结了基于英国和国际选举得出的典型值。
| Metric | AMS | STV | SV |
|---|---|---|---|
| Gallagher Index (≈) | 5-8 | 2-5 | 10-16 |
| Effective parties | 3-4 | 4-6 | 2-3 |
| Wasted vote share | 15-25% | 5-10% | 25-40% |
AMS occupies a middle ground, STV prioritises individual representation and low waste, while SV simplifies the ballot at the cost of reduced proportionality. The choice of system ultimately involves trade-offs between simplicity, local ties, and the mathematical fairness of translating the will of the electorate into seats.
AMS处于中间地带，STV优先考虑个人代表性和低选票浪费，而SV以降低比例性为代价简化了选票设计。制度的选择最终涉及简单性、地方联系与将选民意愿转化为席位的数学公平性之间的权衡。
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