Scottish Parliament and Government: The Mathematics of the AMS Electoral System | 苏格兰议会与政府：附带议员制选举中的数学

📚 Scottish Parliament and Government: The Mathematics of the AMS Electoral System | 苏格兰议会与政府：附带议员制选举中的数学

The Scottish Parliament and Scottish Government are central institutions in the United Kingdom’s devolution settlement. This article approaches the topic from a mathematical perspective, focusing on the Additional Member System (AMS) used to elect Members of the Scottish Parliament (MSPs). The AMS involves the D’Hondt allocation algorithm, percentage calculations, rounding and proportionality checks, all of which link directly to the numerical reasoning expected in A-Level Mathematics.

苏格兰议会和苏格兰政府是英国权力下放架构中的核心机构。本文从数学角度切入，重点分析苏格兰议会议员选举所采用的附带议员制（AMS）。AMS 涉及 D’Hondt 席位分配算法、百分比计算、舍入以及比例性检验，这些内容与 A-Level 数学所要求的数值推理能力直接相关。

1. The Scottish Parliament: A Devolved Legislature | 苏格兰议会：权力下放的立法机构

The Scottish Parliament was established in 1999 and consists of 129 MSPs elected for a fixed term, normally every five years. It legislates on devolved matters such as education, health, transport and justice, while reserved matters remain with the UK Parliament at Westminster.

苏格兰议会成立于 1999 年，由 129 名苏格兰议会议员组成，通常每五年选举一次。它负责教育、卫生、交通和司法等权力下放事务的立法，而保留事务仍由英国威斯敏斯特议会处理。

From a mathematical standpoint, the fixed number of 129 seats is the total output of an electoral algorithm. The key variables are the number of constituency seats, the number of regional seats, the votes cast for each party, and the allocation rule that converts votes into seats.

从数学角度看，129 个席位的固定数量是选举算法的总输出。关键变量包括选区席位数、地区席位数、各政党获得的票数，以及将选票转化为席位的分配规则。

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2. How the Scottish Government is Formed | 苏格兰政府如何组成

The Scottish Government is formed by the party or coalition of parties that can command a majority in the Scottish Parliament. The First Minister is nominated by MSPs and formally appointed by the monarch. The government’s stability depends on the arithmetic of seat distribution after an election.

苏格兰政府由能够在苏格兰议会中获得多数席位的政党或政党联盟组成。首席部长由苏格兰议会议员提名，并由君主正式任命。政府的稳定性取决于选举后席位分配的算术结果。

Mathematically, forming a government requires checking whether a party or coalition holds at least 65 of the 129 seats. This is a simple majority condition: 129 ÷ 2 = 64.5, so the smallest integer greater than 64.5 is 65.

从数学上讲，组建政府需要检验一个政党或联盟是否至少拥有 129 个席位中的 65 个。这是一个简单多数条件：129 ÷ 2 = 64.5，因此大于 64.5 的最小整数是 65。

Majority threshold = ⌊129 ÷ 2⌋ + 1 = 65

This threshold calculation is an application of integer arithmetic and the floor function, both of which appear in A-Level problem-solving questions.

这个门槛计算是整数运算和向下取整函数的应用，两者都出现在 A-Level 的问题解决题型中。

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3. Mixed Electoral System: FPTP + Regional List | 混合选举制：简单多数制 + 地区名单制

The Scottish Parliament uses a mixed electoral system. Of the 129 MSPs, 73 are elected in single-member constituencies by first-past-the-post (FPTP), and 56 are elected from eight regional lists, with seven regional MSPs per region.

苏格兰议会采用混合选举制。在 129 名苏格兰议会议员中，73 名通过简单多数制在单席位选区选举产生，56 名从八个地区名单中选举产生，每个地区有 7 名地区议员。

The AMS is therefore a combination of a majoritarian component and a proportional component. The regional list seats are allocated to correct, at least partially, the disproportionality created by FPTP results in the constituencies.

因此，AMS 是多数制成分与比例制成分的结合。地区名单席位用于至少部分地纠正选区简单多数制所造成的非比例性。

This dual structure generates interesting mathematical tasks: comparing constituency vote shares with seat shares, calculating regional vote percentages, and applying a divisor rule to allocate the additional members.

这种双重结构产生了有趣的数学任务：比较选区得票率与席位率、计算地区得票百分比，以及应用除数规则来分配附加议员。

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4. The D’Hondt Method: An Allocation Algorithm | 东特法：一种席位分配算法

The 56 regional seats in the Scottish Parliament are allocated using the D’Hondt method, also known as the Jefferson method or the highest averages method. It is an iterative algorithm that repeatedly awards one seat at a time to the party with the largest quotient.

苏格兰议会的 56 个地区席位采用 D’Hondt 法分配，该方法也称为杰斐逊法或最高平均数法。它是一种迭代算法，每次将一个席位分配给商数最大的政党。

For each party, the quotient is calculated by dividing its regional vote total V by one more than the number of seats s that the party has already won in that region, including constituency seats and any regional seats already awarded.

对于每个政党，商数通过将其地区票数 V 除以该政党在该地区已经赢得的席位数 s 加 1 来计算，这里 s 包括选区席位和已经分配的地区席位。

q = V ÷ (s + 1)

The algorithm repeats this calculation until all seven regional seats for the region have been allocated. This divisor sequence 1, 2, 3, 4, … is what gives the D’Hondt method its name.

该算法重复此计算，直到该地区的全部七个地区席位分配完毕。除数序列 1、2、3、4……正是 D’Hondt 法名称的由来。

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5. Worked Example: Allocating Regional Seats | 实例演算：地区席位分配

Consider a region with four parties, A, B, C and D. The regional vote totals are 42 000 for A, 31 000 for B, 20 000 for C and 7 000 for D. In the constituency contests, Party A has already won 3 seats, Party B has won 1 seat, and Parties C and D have won 0 seats.

考虑一个地区，共有四个政党 A、B、C 和 D。地区票数分别为：A 党 42 000 票、B 党 31 000 票、C 党 20 000 票、D 党 7 000 票。在选区竞争中，A 党已赢得 3 席，B 党已赢得 1 席，C 党和 D 党赢得 0 席。

We allocate four regional seats in sequence. The table below shows the quotient q = V ÷ (s + 1) after each round. The highest quotient in each round is shown in bold.

我们依次分配四个地区席位。下表显示了每轮后的商数 q = V ÷ (s + 1)。每轮中的最高商数用粗体表示。

Round	Party A	Party B	Party C	Party D	Seat awarded
1	42 000 ÷ 4 = 10 500	31 000 ÷ 2 = 15 500	20 000 ÷ 1 = 20 000	7 000 ÷ 1 = 7 000	C
2	42 000 ÷ 4 = 10 500	31 000 ÷ 2 = 15 500	20 000 ÷ 2 = 10 000	7 000 ÷ 1 = 7 000	B
3	42 000 ÷ 4 = 10 500	31 000 ÷ 3 ≈ 10 333.33	20 000 ÷ 2 = 10 000	7 000 ÷ 1 = 7 000	A
4	42 000 ÷ 5 = 8 400	31 000 ÷ 3 ≈ 10 333.33	20 000 ÷ 2 = 10 000	7 000 ÷ 1 = 7 000	B

The final regional allocation is Party B with 2 seats, Party A with 1 seat, Party C with 1 seat and Party D with 0 seats. Adding constituency seats gives totals of A 4, B 3, C 1 and D 0 in the region.

最终地区席位分配结果为：B 党 2 席，A 党 1 席，C 党 1 席，D 党 0 席。加上选区席位后，该地区总席位为 A 党 4 席、B 党 3 席、C 党 1 席、D 党 0 席。

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6. Mathematical Conditions: Quota and Threshold | 数学条件：配额与门槛

A key condition in proportional allocation is the quota. The exact quota for a party is its vote share multiplied by the total number of seats to be allocated. In a region with 7 regional seats and total regional votes T, the quota for party i is given by:

比例分配中的一个关键条件是配额。一个政党的精确配额等于其得票率乘以待分配的总席位数。在一个有 7 个地区席位、总地区票数为 T 的地区，政党 i 的配额为：

Qᵢ = (Vᵢ ÷ T) × 7

In the worked example, the total regional vote T is 100 000. The quotas are A = 2.94, B = 2.17, C = 1.40 and D = 0.49. The D’Hondt method tends to round some quotas down and others up, but it guarantees that each party receives either the floor or the ceiling of its quota in many practical cases.

在上述实例中，地区总票数 T 为 100 000。配额分别为：A 党 2.94，B 党 2.17，C 党 1.40，D 党 0.49。D’Hondt 法倾向于将某些配额向下取整、另一些向上取整，但在许多实际情形中它保证各党获得其配额的向下取整或向上取整值。

There is no separate legal threshold in the Scottish AMS, but the divisor itself acts as an implicit threshold. Small parties with very low vote shares rarely receive a regional seat because their quotients remain small throughout the allocation rounds.

苏格兰 AMS 中没有单独的法律门槛，但除数本身起到了隐性门槛的作用。得票率极低的小党很少能获得地区席位，因为在整个分配过程中它们的商数始终很小。

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7. Proportionality and Disproportionality | 比例性与非比例性

Proportionality measures how closely the share of seats matches the share of votes. The Gallagher index is a common measure of disproportionality. It is calculated using the formula:

比例性衡量席位份额与选票份额的接近程度。加拉格尔指数是衡量非比例性的常用指标。其计算公式为：

LSq = √( ½ × Σ (vᵢ − sᵢ)² )

Here vᵢ is the vote share of party i and sᵢ is its seat share, both expressed as decimals. A lower index indicates a more proportional result. FPTP alone tends to produce a higher Gallagher index than AMS because the regional top-up seats reduce the gap between votes and seats.

其中 vᵢ 是政党 i 的得票率，sᵢ 是其席位率，两者均以小数表示。指数越低，结果越具比例性。仅使用简单多数制往往比 AMS 产生更高的加拉格尔指数，因为地区增补席位缩小了选票与席位之间的差距。

In the worked example, the regional vote shares are A 0.42, B 0.31, C 0.20 and D 0.07, while the total seat shares are A 0.50, B 0.375, C 0.125 and D 0. The Gallagher index can be computed to quantify the deviation.

在实例中，地区得票率为 A 党 0.42、B 党 0.31、C 党 0.20、D 党 0.07，而总席位率为 A 党 0.50、B 党 0.375、C 党 0.125、D 党 0。可通过加拉格尔指数量化这一偏差。

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8. Role of Percentages and Rounding | 百分比与舍入的作用

Percentage calculations are essential throughout the AMS process. Before any allocation, analysts convert raw vote totals into percentage shares to compare party strength. For example, Party A’s regional vote share is 42 000 ÷ 100 000 × 100 = 42%.

百分比计算贯穿整个 AMS 过程。在分配之前，分析人员将原始票数转换为得票率，以比较各党实力。例如，A 党的地区得票率为 42 000 ÷ 100 000 × 100 = 42%。

Rounding also appears when quotas are not integers. The D’Hondt method uses integer division in the sense that seats are whole numbers, so the algorithm must round seat counts to the nearest whole number in a systematic way. However, the quotients themselves are not rounded during the comparison; they are compared at their full precision.

当配额不是整数时，舍入也会出现。D’Hondt 法在席位数必须是整数的意义上使用整数分配，因此算法必须系统地将席位计数舍入为整数。然而，商数在比较时本身不进行舍入；它们以完整精度进行比较。

Understanding how rounding and truncation affect the final seat allocation is a valuable skill in A-Level Mathematics, especially in the context of discrete and statistical reasoning.

理解舍入和截断如何影响最终的席位分配是 A-Level 数学中的一项重要技能，尤其是在离散数学和统计推理的背景下。

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9. Algorithmic Steps and Efficiency | 算法步骤与效率

The D’Hondt allocation can be expressed as a repeat-until loop. The steps are as follows:

D’Hondt 分配可以表示为一个重复直到循环。步骤如下：

• Set the number of regional seats already awarded to zero for each party.
• 将每个政党已获得的地区席位数设为零。
• For each party, compute the quotient V ÷ (s + 1), where s includes constituency seats.
• 对于每个政党，计算商数 V ÷ (s + 1)，其中 s 包括选区席位。
• Award one regional seat to the party with the largest quotient.
• 将一个地区席位授予商数最大的政党。
• Increase that party’s seat count s by 1.
• 将该政党的席位数 s 增加 1。
• Repeat until all regional seats have been allocated.
• 重复直到所有地区席位分配完毕。

This is a finite algorithm because each iteration awards exactly one seat, so the number of iterations equals the number of regional seats available. For the Scottish Parliament, this means 56 iterations across eight regions, or seven iterations per region.

这是一个有限算法，因为每次迭代恰好分配一个席位，所以迭代次数等于可分配的地区席位数。对于苏格兰议会，这意味着在八个地区共进行 56 次迭代，或每个地区进行 7 次迭代。

Algorithm efficiency is a topic encountered in decision mathematics, and the D’Hondt method is a practical example of an iterative allocation procedure.

算法效率是决策数学中遇到的课题，D’Hondt 法是迭代分配程序的一个实际例子。

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10. Criticisms and Mathematical Fairness | 批评与数学公平性

Although AMS improves proportionality, it is not perfectly fair in a mathematical sense. The D’Hondt method tends to favour larger parties because the divisor s + 1 increases more slowly for parties that already have many seats. This property is called the large-party bias.

尽管 AMS 提高了比例性，但从数学意义上并非完全公平。D’Hondt 法倾向于有利于大党，因为对于已经拥有较多席位的政党，除数 s + 1 增长较慢。这一特性称为大党偏向。

Alternative divisor methods, such as the Sainte-Laguë method, use divisor sequences like 1, 3, 5, 7, … to reduce this bias. The choice of divisor sequence can materially change the allocation outcome, which shows that electoral mathematics involves normative judgements about fairness.

其他除数法，如圣拉格法，使用 1、3、5、7……这样的除数序列来减小这种偏向。除数序列的选择可能实质性地改变分配结果，这表明选举数学涉及关于公平性的规范性判断。

In the Scottish context, the inclusion of constituency seats in the divisor s is specifically designed to compensate parties that did well in constituencies but poorly in the regional vote, or vice versa.

Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com

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