📚 Debates on why different electoral systems are used in the UK | 英国使用不同选举制度的辩论
The United Kingdom is often described as a laboratory of electoral systems. Different tiers of government use different rules for converting votes into seats, and each choice reflects a mathematical trade-off between proportionality, simplicity, local representation and stable government.
英国常被形容为选举制度的实验室。不同层级的政府采用不同的规则将选票转化为席位，而每一种选择都体现了比例性、简单性、地方代表性与稳定政府之间的数学权衡。
1. The Electoral Menu in the UK | 英国的选举制度菜单
The UK Parliament at Westminster uses First Past the Post (FPTP) for general elections. The Scottish Parliament, Welsh Senedd and London Assembly use the Additional Member System (AMS). The Northern Ireland Assembly uses the Single Transferable Vote (STV), and local elections in Scotland and Northern Ireland also use STV. Before Brexit, UK elections to the European Parliament used regional list proportional representation.
英国西敏寺议会的大选采用简单多数制（FPTP）。苏格兰议会、威尔士议会和伦敦议会使用附加议员制（AMS）。北爱尔兰议会使用单记可转移投票制（STV），苏格兰和北爱尔兰的地方选举也使用STV。在英国脱欧之前，欧洲议会选举使用区域名单比例代表制。
| System | Used for | Key rule |
|---|---|---|
| FPTP | Westminster general elections | Largest vote wins the single seat |
| AMS | Scottish Parliament, Welsh Senedd, London Assembly | Constituency seats plus party-list top-up seats |
| STV | Northern Ireland Assembly, Scottish local elections | Rank candidates; surplus transferred above quota |
| Regional list PR | Former European Parliament elections | Seats allocated by party list in large regions |
2. First Past the Post: Plurality Logic | 简单多数制：相对多数逻辑
In FPTP, the candidate with the largest number of votes wins the single seat. No quota or absolute majority is required. If candidate A receives v₁ votes and candidate B receives v₂ votes, A wins if v₁ > v₂, even if v₁ is far below 50% of the total vote. This is mathematically simple but can produce highly disproportional outcomes.
在简单多数制中，得票最多的候选人赢得单一席位。无需达到配额或绝对多数。如果候选人A获得v₁票，候选人B获得v₂票，只要v₁ > v₂，A就获胜，即使v₁远低于总票数的50%。这在数学上非常简单，但可能产生严重不成比例的结果。
Winning condition: v₁ > v₂ for all other candidates
3. Proportional Representation: Quota and Remainder | 比例代表制：配额与余数
Proportional representation systems allocate seats by quotas. The Hare quota is Q_H = V ÷ S, where V is total valid votes and S is the number of seats. The Droop quota is Q_D = (V ÷ (S + 1)) + 1. Parties receive seats in proportion to their vote shares, which reduces the number of wasted votes.
比例代表制通过配额分配席位。黑尔配额为 Q_H = V ÷ S，其中V为总有效票数，S为席位数。德鲁普配额为 Q_D = (V ÷ (S + 1)) + 1。政党按得票比例获得席位，从而减少了废票数量。
Q_H = V ÷ S and Q_D = (V ÷ (S + 1)) + 1
4. Additional Member System: Mixing Two Principles | 附加议员制：混合两种原则
AMS combines constituency FPTP seats with party-list top-up seats. Voters have two votes: one for a local constituency candidate and one for a regional party list. The list seats are allocated to correct disproportionality. The allocation often uses the D’Hondt divisor method, where each party’s vote total is divided by 1, 2, 3, … and the highest quotients win seats.
附加议员制将选区简单多数席位与政党名单增补席位结合起来。选民有两票：一票投给地方选区候选人，一票投给区域政党名单。名单席位用于纠正不成比例的问题。席位分配通常采用洪德除数法，即各政党得票总数除以1、2、3、…，商数最高者获得席位。
Divisor sequence: 1, 2, 3, …
5. Single Transferable Vote: Rankings and Surplus Transfer | 单记可转移投票：排序与盈余转移
STV requires voters to rank candidates in order of preference. A candidate is elected when they reach the Droop quota. Any surplus votes above the quota are transferred to the next preferences at a fractional value. This maximises voter choice and produces proportional outcomes within multi-member constituencies.
单记可转移投票要求选民按偏好顺序对候选人进行排序。候选人达到德鲁普配额即当选。超过配额的多余选票按分数价值转移给下一偏好。这在多席位选区内最大化选民选择并产生比例结果。
Surplus = total votes for candidate − Q_D
6. Disproportionality Measures: The Gallagher Index | 偏差度量：加拉格尔指数
To compare systems mathematically, political scientists use the Gallagher index. It measures the square root of half the sum of squared differences between each party’s vote share and seat share. A lower value means a more proportional system. FPTP typically scores much higher than PR systems.
为了在数学上比较各种制度，政治学家使用加拉格尔指数。它计算各政党得票比例与席位比例之差的平方和的一半的平方根。数值越低表示制度越比例。简单多数制的得分通常远高于比例代表制。
LSq = √(½ Σ (vₙ − sₙ)²)
7. Effective Number of Parties and Duverger’s Law | 有效政党数与杜瓦杰定律
The effective number of parties is calculated as ENP = 1 ÷ Σ pₙ², where pₙ is the vote share of party n. FPTP tends to produce an ENP close to 2, supporting Duverger’s law that plurality rules favour two-party systems. PR systems typically yield a higher ENP and multiparty legislatures.
有效政党数计算公式为 ENP = 1 ÷ Σ pₙ²，其中pₙ是第n党的得票比例。简单多数制往往产生接近2的有效政党数，支持杜瓦杰定律，即相对多数规则有利于两党制。比例代表制通常产生更高的有效政党数和多党议会。
ENP = 1 ÷ Σ pₙ²
8. Constituency Link vs Proportional Fairness | 选区联系与比例公平的权衡
A core debate is whether the mathematical fairness of PR justifies losing the direct constituency link. FPTP gives every voter a named local MP, but many votes do not count towards the result. PR makes every vote count more equally, but often weakens the individual constituency-representative bond.
核心辩论在于比例代表制的数学公平是否足以证明失去直接选区联系是合理的。简单多数制为每位选民提供一名具名的地方议员，但许多选票对结果没有影响。比例代表制使每张选票更平等地发挥作用，但往往削弱个别选区与代表之间的联系。
9. Wasted Votes and Tactical Voting | 废票与策略性投票
In FPTP, wasted votes are those cast for losing candidates or surplus votes for safe winners. In a seat with three candidates, if votes are 38%, 32%, 30%, the winning candidate needs only 38%, leaving 62% of voters unrepresented. This creates incentives for tactical voting, where voters abandon their preferred candidate.
在简单多数制中，废票是投给落选候选人的选票或投给稳赢候选人的多余选票。在一个有三名候选人的选区，如果得票率为38%、32%、30%，获胜候选人只需38%，导致62%的选民没有代表。这产生了策略性投票的动机，即选民放弃首选候选人。
10. Stability and Coalition Formation | 稳定性与联合政府形成
PR systems often produce no single-party majority, requiring coalitions. Mathematically, coalition formation is a bargaining problem: parties must reach a combined seat total above 50%. This can lead to more consensual policy but also slower decision-making. FPTP often manufactures a single-party majority from a minority of votes.
比例代表制往往产生无一党多数的结果，需要组建联合政府。从数学上看，联合政府形成是一个讨价还价问题：各党必须达成超过50%的席位总和。这可能导致更具共识性的政策，但决策速度较慢。简单多数制经常从少数选票中制造出一党多数。
11. The Reform Debate: Thresholds and Trade-offs | 改革辩论：门槛与权衡
Reformers argue that the UK should adopt a more proportional system for Westminster. However, there is no perfect system: increasing proportionality may require larger multi-member constituencies, weaker local links, or electoral thresholds. Arrow’s impossibility theorem shows that no voting system can satisfy all reasonable fairness criteria simultaneously.
改革派认为英国下议院应采用更比例的制度。然而，没有完美的制度：提高比例性可能需要更大的多议员选区、更弱的地方联系或选举门槛。阿罗不可能定理表明，没有任何投票制度能同时满足所有合理的公平标准。
12. Conclusion: An Equilibrium of Competing Values | 结论：竞争价值的均衡
The UK uses different electoral systems because different institutions prioritise different values. Westminster prioritises strong local representation and decisive government; devolved bodies prioritise proportionality and inclusion. The debates are ultimately mathematical arguments about trade-offs that cannot be eliminated.
英国使用不同的选举制度，是因为不同机构优先考虑不同的价值。西敏寺优先考虑强有力的地方代表性和果断的政府；权力下放机构优先考虑比例性和包容性。这些辩论本质上是关于无法消除的权衡的数学论证。
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