Comparing Congress and Parliament: A Mathematical Study | 美国国会与英国议会的数学比较

📚 Comparing Congress and Parliament: A Mathematical Study | 美国国会与英国议会的数学比较

Comparing the United States Congress and the United Kingdom Parliament is a classic A-level topic, but it is often taught through qualitative description. This article uses a mathematical lens to compare the two legislatures: we calculate seat-vote distortion, effective party numbers, voting cohesion, constituency ratios, and productivity. These indices turn institutional differences into measurable quantities, helping you write sharper analytical answers.

比较美国国会与英国议会是 A-level 的经典主题，但教学中往往采用定性描述。本文用数学视角比较这两个立法机构：我们计算席位-选票扭曲、有效政党数、投票一致性、选区比例和立法效率。这些指数将制度差异转化为可测量的数量，帮助你写出更清晰的分析答案。

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1. Why Use Maths to Compare Legislatures? | 为什么用数学比较立法机构？

Legislatures can be compared through institutional arithmetic such as seat totals, term lengths, electoral formulas, and voting indices. Mathematical comparison reduces vague terms like “more proportional” or “more disciplined” into exact numbers. For A-level essays, using a few indices and calculations shows analytical depth.

立法机构可以通过制度算术来比较，例如席位总数、任期长度、选举公式和投票指数。数学比较将“更具比例性”或“更有纪律”这类模糊表述转化为确切数字。在 A-level 论文中，使用若干指数和计算能展现分析深度。

• Electoral system indices measure proportionality, such as the Gallagher index.
• Party system indices measure fragmentation, such as the effective number of parties.
• Voting behaviour indices measure internal cohesion, such as the Rice index.

• 选举制度指数测量比例性，例如加拉格尔指数。
• 政党体系指数测量碎片化，例如有效政党数。
• 投票行为指数测量内部凝聚力，例如赖斯指数。

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2. Core Structure: Seats, Chambers, and Terms | 核心结构：席位、议院与任期

The US Congress is bicameral and fully elected, with 435 voting members in the House of Representatives and 100 in the Senate. By contrast, the UK Parliament has 650 elected MPs in the House of Commons, while the House of Lords is unelected and usually excluded from comparative indices.

美国国会是两院制且全部由选举产生，众议院有 435 名投票成员，参议院有 100 名。相比之下，英国议会下议院有 650 名选举产生的议员，而上议院非选举产生，通常在比较指数中被排除。

Feature	US Congress	UK Parliament
Elected voting members	535 (House 435 + Senate 100)	650 (House of Commons)
Term length	House 2 years, Senate 6 years	Commons up to 5 years
Second chamber	Senate, directly elected	House of Lords, appointed
Fixed election cycle	Yes, every 2 years for House	No fixed date; maximum 5 years

This structural difference means the two legislatures produce different mathematical patterns in seat allocation and party competition.

这种结构差异意味着两个立法机构在席位分配和政党竞争中产生不同的数学模式。

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3. Electoral Systems and the Gallagher Index | 选举制度与加拉格尔指数

Both countries use first-past-the-post for most national elections, but the US has two elected chambers with different district sizes, while the UK uses FPTP for the Commons only. The Gallagher index measures the deviation between vote shares and seat shares:

两国大多数全国性选举使用简单多数制，但美国有两个选区规模不同的民选议院，而英国仅下议院使用 FPTP。加拉格尔指数衡量选票份额与席位份额之间的偏差：

LSq = √( ½ × Σ (vᵢ – sᵢ)² )

Here vᵢ is the vote share of party i and sᵢ is its seat share. A low LSq means a highly proportional result; a high LSq means a large distortion.

其中 vᵢ 是政党 i 的选票份额，sᵢ 是其席位份额。LSq 低表示结果高度比例化；LSq 高表示扭曲较大。

Using the main UK parties as an approximation, the 2019 UK general election gives LSq ≈ 11.6%. The 2020 US House election gives LSq ≈ 1.0%. This shows that US House outcomes can be much more proportional than UK outcomes, despite both using FPTP, because the two-party structure dominates US House contests.

以英国主要政党作近似计算，2019 年英国大选的 LSq 约为 11.6%。2020 年美国众议院选举的 LSq 约为 1.0%。这表明尽管都使用 FPTP，美国众议院结果可能比英国结果更具比例性，因为两党结构主导美国众议院选举。

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4. Effective Number of Parties | 有效政党数

The effective number of parties counts how many meaningful parties compete in a party system. The most common formula is the Laakso-Taagepera index:

有效政党数衡量政党体系中有多少个有意义的政党在竞争。最常用的公式是拉克索-塔格佩拉指数：

Nₑ = 1 ÷ (Σ pᵢ²)

Here pᵢ is the decimal vote share or seat share of party i. If one party has 100% of seats, Nₑ = 1. If two parties each have 50%, Nₑ = 2.

其中 pᵢ 是政党 i 的选票或席位份额（小数）。如果一个政党拥有 100% 席位，Nₑ = 1。如果两个政党各占 50%，Nₑ = 2。

For the 2019 UK general election, the effective number of parties based on votes is approximately 3.2, while based on seats it is about 2.4. For the 2020 US House election, both vote-based and seat-based Nₑ are close to 2.0. The UK system contains more meaningful competitors, especially the SNP and Liberal Democrats.

2019 年英国大选按选票计算的有效政党数约为 3.2，按席位计算约为 2.4。2020 年美国众议院选举中，基于选票和席位的 Nₑ 都接近 2.0。英国体系中包含更多有意义的竞争者，尤其是苏格兰民族党和自由民主党。

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5. Seat-Vote Distortion and Winner Bonus | 席位-选票扭曲与胜者红利

First-past-the-post often gives the largest party a seat bonus: its seat share exceeds its vote share. We can define the winner bonus as s₁ – v₁, where party 1 is the largest party. A positive winner bonus means the system exaggerates the lead of the winner.

简单多数制通常给予第一大党席位红利：其席位份额超过选票份额。我们可以将胜者红利定义为 s₁ – v₁，其中政党 1 是最大党。正值胜者红利意味着制度夸大了获胜者的领先优势。

In the 2019 UK election, the Conservative Party won about 43.6% of votes but 56.2% of seats, giving a winner bonus of approximately +12.6 percentage points. In the 2020 US House election, the Democratic Party won about 50.8% of votes and 51.0% of seats, giving a winner bonus close to +0.2 points. The UK system therefore produces a much stronger winner bonus.

2019 年英国大选中，保守党赢得约 43.6% 的选票但获得 56.2% 的席位，胜者红利约为 +12.6 个百分点。2020 年美国众议院选举中，民主党赢得约 50.8% 的选票和 51.0% 的席位，胜者红利接近 +0.2 个百分点。因此英国制度产生更强的胜者红利。

This difference occurs because the US House has many safe single-party districts and fewer three-party contests, while the UK has regional parties and sharper local swings.

这种差异的原因在于美国众议院有许多单一政党安全选区，三方竞争较少；而英国存在地区性政党和更剧烈的本地波动。

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6. Party Discipline and the Rice Index | 政党纪律与赖斯指数

Party discipline measures how often members vote with their own party. A simple measure is the Rice index for a single vote:

政党纪律衡量议员在多大程度上与自己的政党一起投票。一个简单指标是单次投票的赖斯指数：

R = |yes – no| ÷ (yes + no)

Here yes and no are the numbers of votes on each side. If all members vote one way, R = 1. If the chamber splits exactly 50-50, R = 0. A high average Rice index indicates strong party cohesion.

其中 yes 和 no 是双方票数。如果所有成员投同一方向，R = 1。如果议院正好 50 比 50，R = 0。高平均赖斯指数表明政党凝聚力强。

In the UK House of Commons, average Rice scores for whipped votes often exceed 0.90, meaning party discipline is very strong. In the US Congress, average Rice scores vary by year and chamber, but in recent decades they have usually ranged between 0.80 and 0.95, with the Senate sometimes lower due to more individual independence.

在英国下议院，党鞭投票的平均赖斯指数通常超过 0.90，说明政党纪律非常强。在美国国会，平均赖斯指数因年份和议院而异，但近几十年来通常在 0.80 到 0.95 之间，参议院有时较低，因为议员个人独立性更强。

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7. Constituency Size and Representation Ratio | 选区规模与代表比

Another mathematical comparison is the average number of people or voters per constituency. The US House of Representatives had an average district population of about 761,000 after the 2020 census. The UK House of Commons has about 73,000 registered voters per constituency on average, or roughly 100,000 total residents.

另一个数学比较是每个选区平均人口或选民数。2020 年人口普查后，美国众议院选区平均人口约为 761,000。英国下议院平均每个选区约有 73,000 名登记选民，或约 100,000 名总居民。

The representation ratio = total population ÷ number of elected representatives. For the US House, this is roughly 761,000 people per representative. For the UK Commons, it is roughly 100,000 people per MP. This means each US House member represents a far larger population than a UK MP, although the US also has a Senate that represents states equally.

代表比 = 总人口 ÷ 民选代表人数。美国众议院约为每位代表 761,000 人。英国下议院约为每位议员 100,000 人。这意味着每位美国众议员代表的人口远多于一位英国下议院议员，尽管美国还有平等代表各州的参议院。

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8. Turnout and Citizen Participation Rates | 投票率与公民参与率

Turnout can be compared as a percentage of registered voters or voting-age population. The 2019 UK general election had a turnout of about 67.3% of registered voters. The 2020 US presidential election had a turnout of about 66% of the voting-age population, but US midterm congressional elections are usually much lower, around 50%.

投票率可以按登记选民或投票年龄人口的百分比来比较。2019 年英国大选登记选民投票率约为 67.3%。2020 年美国总统大选投票年龄人口投票率约为 66%，但美国中期国会选举通常低得多，约为 50%。

When comparing Congress and Parliament, it is useful to compare turnout in the most recent House of Commons election with turnout in the most recent US House midterm election. For example, the 2022 US House midterm turnout was about 52% of the voting-age population, notably lower than the 67% UK figure for 2019.

比较国会和议会时，可以比较最近一次下议院选举与最近一次美国众议院中期选举的投票率。例如，2022 年美国众议院中期选举投票率约为投票年龄人口的 52%，明显低于 2019 年英国 67%

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