📚 The Case for Reform of the UK Democratic System | 英国民主制度改革之数学论证
In A-Level Mathematics, we often model real-world systems using ratios, percentages, indices and probability. The UK electoral system provides a rich case study: its ‘first-past-the-post’ rules can be analysed quantitatively to assess whether representation is proportional. This article builds a mathematical case for reform by measuring seat-vote distortion, wasted votes, constituency inequality and voting paradoxes.
在 A-Level 数学中，我们常用比例、百分比、指数和概率为现实系统建模。英国选举制度提供了一个丰富案例：其“简单多数制”可以通过定量分析来评估代表性是否成比例。本文通过衡量席位-选票扭曲、浪费选票、选区不平等和投票悖论，从数学角度构建改革论据。
1. The Mathematical Lens on Democracy | 用数学视角审视民主
A democratic election can be represented as a mapping from a set of vote shares to a set of seat shares. For each party i, we define its vote share vᵢ as the percentage of total votes won, and its seat share sᵢ as the percentage of total seats won. A perfectly proportional system would satisfy vᵢ = sᵢ for all parties.
民主选举可以表示为从得票份额集合到席位份额集合的一种映射。对于每个政党 i，我们定义其得票份额 vᵢ 为赢得的总选票百分比，席位份额 sᵢ 为赢得的总席位百分比。完全比例代表制应满足所有政党的 vᵢ = sᵢ。
Mathematically, proportionality is a linear transformation. If the conversion from votes to seats is linear and identical for all parties, the system preserves proportionality. Any deviation from vᵢ = sᵢ produces what statisticians call a distortion error, which can be measured using standard indices.
从数学上讲，比例性是一种线性变换。如果从选票到席位的转换对所有政党都是同一线性规则，则制度保持比例性。任何偏离 vᵢ = sᵢ 的情况都会产生统计学家所说的扭曲误差，这种误差可以用标准指数来衡量。
2. First-Past-the-Post and Seat-Vote Distortion | 简单多数制与席位-选票扭曲
The UK House of Commons uses the first-past-the-post (FPTP) system: in each constituency, the candidate with the most votes wins the single seat, even if that candidate receives far less than 50% of the votes. This makes the national seat total highly sensitive to the geographical distribution of votes, not just the overall vote count.
英国下议院采用简单多数制（FPTP）：在每个选区，得票最多的候选人赢得唯一席位，即使该候选人得票远低于 50%。这使得全国席位总数对选票的地理分布高度敏感，而不仅仅取决于总票数。
The table below shows actual data from the 2019 UK general election. The vote share and seat share are given as percentages.
下表展示了 2019 年英国大选的实际数据。得票份额和席位份额以百分比表示。
| Party | Vote share vᵢ (%) | Seat share sᵢ (%) | Difference sᵢ − vᵢ |
|---|---|---|---|
| Conservative | 43.6 | 56.2 | +12.6 |
| Labour | 32.1 | 31.2 | −0.9 |
| Liberal Democrats | 11.6 | 1.7 | −9.9 |
| SNP | 3.9 | 7.4 | +3.5 |
| Green | 2.7 | 0.2 | −2.5 |
A party with 43.6% of votes won 56.2% of seats, while a party with 11.6% of votes won only 1.7% of seats. This is a clear quantitative example of seat-vote distortion, where small differences in vote share are amplified into large differences in legislative power.
获得 43.6% 选票的政党赢得了 56.2% 的席位，而获得 11.6% 选票的政党只赢得 1.7% 的席位。这是席位-选票扭曲的清晰定量例子，即选票份额的微小差异被放大为立法权力的巨大差异。
3. Measuring Disproportionality with Indexes | 用指数衡量不成比例程度
To quantify how far an election result is from perfect proportionality, political scientists and mathematicians use disproportionality indices. The Gallagher index, also called the least squares index, is widely used in comparative politics.
为了量化选举结果与完全比例代表制的差距，政治学家和数学家使用不成比例指数。加拉格尔指数（Gallagher index），也称为最小二乘指数，在比较政治学中被广泛使用。
Gallagher index = √( ½ Σ (vᵢ − sᵢ)² )
Here the summation runs over all parties, and the factor ½ accounts for the fact that any over-representation of one party must be matched by under-representation elsewhere. The index is always between 0 and 100, where 0 means perfect proportionality and larger values mean greater distortion.
这里的求和覆盖所有政党，系数 ½ 反映了这样一个事实：一个政党的过度代表必然对应其他政党的代表性不足。该指数始终介于 0 到 100 之间，其中 0 表示完全比例代表制，数值越大表示扭曲越严重。
Using the 2019 data, the Gallagher index is approximately 11.8. For comparison, most proportional representation systems have values below 5. This single number gives a compact mathematical summary of the UK system’s disproportionality.
使用 2019 年数据，加拉格尔指数约为 11.8。相比之下，多数比例代表制国家的数值低于 5。这个单一数字为英国制度的不成比例程度提供了一个简洁的数学概括。
4. Wasted Votes and Tactical Voting | 浪费选票与策略性投票
A wasted vote is any vote that does not contribute to electing a candidate. Under FPTP, all votes for losing candidates, and all surplus votes above the winner’s required plurality, are mathematically wasted. In a close three-way constituency, a candidate can win with only 35% support, meaning 65% of the votes are wasted.
浪费选票是指任何对候选人当选没有贡献的选票。在简单多数制下，所有投给落选候选人的选票，以及获胜者超过相对多数所需的所有多余选票，在数学上都是浪费的。在一个三方势均力敌的选区，候选人可能仅以 35% 的支持率获胜，这意味着 65% 的选票被浪费。
Across the UK, the wasted vote proportion is often above 50%. This creates a strong incentive for tactical voting, where a voter chooses not their preferred party but a less disliked party that has a realistic chance of winning. Tactical voting can be modelled using game theory: voters maximise the utility of their vote by considering the probability of each candidate winning.
在整个英国，浪费选票的比例通常超过 50%。这产生了策略性投票的强烈动机，即选民不投给自己最偏好的政党，而是投给一个不太讨厌且有望获胜的政党。策略性投票可以用博弈论建模：选民通过考虑每位候选人获胜的概率来最大化自己选票的效用。
A system with many wasted votes violates the mathematical principle of equal treatment, because each vote does not have the same expected influence on the outcome. This is a key quantitative argument for reform.
浪费选票过多的制度违反了平等待遇的数学原则，因为每张选票对结果的期望影响力并不相同。这是改革的一个关键定量论据。
5. Constituency Size Inequality | 选区规模不平等
For representation to be fair, each constituency should contain roughly the same number of electors. In the UK, constituency electorates vary significantly: some contain under 60,000 voters while others exceed 90,000. The standard deviation of constituency size is large compared with the mean.
为了实现公平代表，每个选区的选民人数应大致相同。在英国，选区选民人数差异显著：有的不到 60,000 人，有的超过 90,000 人。选区规模的标准差相对于平均值较大。
If we treat each constituency as an observation, the coefficient of variation (standard deviation divided by mean) can exceed 10%. This means that a vote in a small constituency carries more weight than a vote in a large constituency, because it takes fewer votes to elect an MP. This is mathematically equivalent to unequal weighting in a weighted average.
如果将每个选区视为一个观测值，变异系数（标准差除以平均值）可能超过 10%。这意味着小选区中的一张选票比大选区中的一张选票权重更大，因为选举一名议员所需的选票更少。这在数学上等价于加权平均中的不等权重。
6. The Duverger Effect and Party Count | 迪韦尔热效应与政党数量
The Duverger effect states that single-member district plurality systems tend to produce two-party dominance, while proportional representation encourages multi-party systems. This can be quantified using the effective number of parties, a formula from the Laakso-Taagepera index.
迪韦尔热效应指出，单一选区相对多数制往往产生两党主导，而比例代表制鼓励多党制。这可以用有效政党数量来量化，该数量来自拉克索-塔格佩拉指数。
N = 1 / Σ pᵢ²
Here pᵢ is the proportion of seats (or votes) won by party i. If one party wins all seats, N = 1. If two parties each win 50% of seats, N = 1 ÷ (0.5² + 0.5²) = 2. In the 2019 UK House of Commons, using seat shares, N is approximately 2.4, confirming the strong two-party squeeze.
这里 pᵢ 是政党 i 赢得的席位（或选票）比例。如果一个政党赢得所有席位，则 N = 1。如果两个政党各赢得 50% 席位，则 N = 1 ÷ (0.5² + 0.5²) = 2。在 2019 年英国下议院中，使用席位份额计算，N 约为 2.4，证实了强烈的两党挤压效应。
A mathematical reform case can be built on the principle that N should reflect the diversity of voter preferences, not be artificially compressed by the electoral formula.
改革的数学论据可以建立在这样一个原则上：有效政党数量 N 应反映选民偏好的多样性，而不应被选举公式人为压缩。
7. Voting Paradoxes and Arrow’s Theorem | 投票悖论与阿罗不可能定理
Voting systems can produce logical paradoxes. The Condorcet paradox shows that with three or more options, majority preferences can be cyclic: a majority prefers A to B, B to C, but also C to A. This means there is no clear ‘will of the majority’ under simple pairwise voting.
投票制度可能产生逻辑悖论。孔多塞悖论表明，当存在三个或更多选项时，多数偏好可能是循环的：多数人偏好 A 优于 B，B 优于 C，但也可能偏好 C 优于 A。这意味着在简单的两两投票下，不存在明确的“多数意志”。
Arrow’s impossibility theorem extends this: no ranked voting system can simultaneously satisfy a small set of seemingly reasonable criteria, including non-dictatorship, Pareto efficiency, independence of irrelevant alternatives and unrestricted domain. This theorem does not prove that reform is impossible; it proves that every system involves trade-offs.
阿罗不可能定理进一步指出：任何排序投票制度都无法同时满足一组看似合理的标准，包括非独裁、帕累托效率、无关选项独立性和无限制定义域。这一定理并不证明改革不可能；它证明每一种制度都涉及权衡。
From a mathematical perspective, the case for reform is not that a perfect system exists, but that FPTP performs poorly on key measurable criteria such as proportionality and wasted votes. Alternative systems may offer a better trade-off, even if no system is flawless.
从数学角度来看，改革的理由不是存在完美制度，而是简单多数制在比例性和浪费选票等关键可量化标准上表现不佳。其他制度可能提供更好的权衡，即使没有制度是完美无缺的。
8. Mathematical Comparison of Reform Options | 改革方案的数学比较
Several alternative electoral systems have been proposed for the UK. We can compare them using the same mathematical measures: disproportionality index, effective number of parties, and wasted vote share.
针对英国提出了几种替代选举制度。我们可以使用相同的数学指标进行比较：不成比例指数、有效政党数量和浪费选票份额。
| System | Typical Gallagher index | Effective parties N | Wasted vote share |
|---|---|---|---|
| FPTP (current) | 10–15 | 2.0–2.5 | 50–70% |
| Alternative Vote (AV) | 7–12 | 2.5–3.0 | 40–55% |
| Single Transferable Vote (STV) | 4–8 | 3.5–5.0 | 20–35% |
| Mixed Member Proportional (MMP) | 3–7 | 3.0–4.5 | 15–30% |
The mathematical comparison shows that proportional and mixed systems consistently reduce disproportionality and wasted votes, while increasing the effective number of parties. This provides a quantitative basis for evaluating reform options, rather than relying on purely qualitative arguments.
数学比较表明，比例制和混合制持续降低不成比例程度和浪费选票，同时增加有效政党数量。这为评估改革方案提供了定量基础，而不是仅仅依赖定性论点。
9. Simulation and Statistical Forecasts | 模拟与统计预测
Mathematical modelling can simulate what UK election results would look like under different systems. Using Monte Carlo methods, we can take actual constituency-level voting data and apply different seat allocation formulas to generate seat predictions. This allows us to estimate the probability of different outcomes, such as a hung parliament or a single-party majority.
数学模型可以模拟不同制度下英国大选的结果。使用蒙特卡洛方法，我们可以采用实际的选区级投票数据，并应用不同的席位分配公式来生成席位预测。这使我们能够估计不同结果的概率，例如悬浮议会或一党多数。
Suppose we run 10,000 simulations under FPTP and under a proportional list system. We can compute the mean and standard deviation of the effective number of parties and the Gallagher index for each system. The statistical distributions show that FPTP produces a wider range of possible outcomes with higher disproportionality, while proportional systems are more stable and representative.
假设我们在简单多数制和比例名单制下分别运行 10,000 次模拟。我们可以计算每种制度下有效政党数量和加拉格尔指数的均值和标准差。统计分布表明，简单多数制产生更广泛的可能结果且不成比例程度更高，而比例制更稳定、更具代表性。
This use of probability and statistics is directly relevant to A-Level Mathematics: it applies sampling, distributions, mean and variance to a real-world political problem, strengthening the case for reform with numerical evidence.
这种概率和统计的运用与 A-Level 数学直接相关：它将抽样、分布、均值和方差应用于现实政治问题，用数值证据加强了改革的论据。
10. Conclusion: A Quantitative Case for Reform | 结论：基于定量分析的改革理由
The UK democratic system’s first-past-the-post electoral rules create measurable distortions between votes and seats. Using basic mathematical tools such as percentage difference, disproportionality indices, wasted vote calculations, effective party counts and simulation, we can demonstrate that the system systematically over-represents large parties and under-represents smaller ones.
英国民主制度的简单多数制选举规则在选票与席位之间造成了可测量的扭曲。利用百分比差异、不成比例指数、浪费选票计算、有效政党数量和模拟等基本数学工具，我们可以证明该制度系统性地过度代表大党，而对小党代表性不足。
Reform is not simply a matter of political preference; it can be framed as a statistical improvement in fairness and accuracy. A mathematically informed debate can compare FPTP with alternative systems using clear numerical criteria, showing that reform would reduce disproportionality, lower wasted votes and make the system more representative of voter preferences.
改革不仅仅是政治偏好问题；它可以被表述为公平性和准确性方面的统计改进。基于数学的辩论可以使用清晰的数值标准将简单多数制与其他制度进行比较，表明改革将降低不成比例程度、减少浪费选票，并使制度更能代表选民偏好。
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